The Octonion Math That Could Underpin Physics

New findings are fueling an old suspicion that fundamental particles and forces spring from strange eight-part numbers called “octonions.”

In the May 2011 issue of Scientific American mathematician John Baez co-authors "The Strangest Numbers in String Theory," an article about the octonions, an eight-dimensional number system that was discovered in the mid–19th century but that has been largely ignored until quite recently. As the name of the article implies, interest in the octonions has been rekindled by their surprising relationship to recent developments in theoretical physics, including supersymmetry, string theory and M-theory. Baez and his co-author John Huerta wrote, "If string theory is right, the octonions are not a useless curiosity; on the contrary, they provide the deep reason why the universe must have 10 dimensions: in 10 dimensions, matter and force particles are embodied in the same type of numbers—the octonions."

The Octonions Math That Could Underlie the Laws of Nature

In 2014, a graduate student at the University of Waterloo, Canada, named Cohl Furey rented a car and drove six hours south to Pennsylvania State University, eager to talk to a physics professor there named Murat Günaydin. Furey had figured out how to build on a finding of Günaydin’s from 40 years earlier — a largely forgotten result that supported a powerful suspicion about fundamental physics and its relationship to pure math.

The suspicion, harbored by many physicists and mathematicians over the decades but rarely actively pursued, is that the peculiar panoply of forces and particles that comprise reality spring logically from the properties of eight-dimensional numbers called “octonions.”

As numbers go, the familiar real numbers — those found on the number line, like 1, π and -83.777 — just get things started. Real numbers can be paired up in a particular way to form “complex numbers,” first studied in 16th-century Italy, that behave like coordinates on a 2-D plane. Adding, subtracting, multiplying and dividing is like translating and rotating positions around the plane. Complex numbers, suitably paired, form 4-D “quaternions,” discovered in 1843 by the Irish mathematician William Rowan Hamilton, who on the spot ecstatically chiseled the formula into Dublin’s Broome Bridge. John Graves, a lawyer friend of Hamilton’s, subsequently showed that pairs of quaternions make octonions: numbers that define coordinates in an abstract 8-D space.

There the game stops. Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided. The first three of these “division algebras” would soon lay the mathematical foundation for 20th-century physics, with real numbers appearing ubiquitously, complex numbers providing the math of quantum mechanics, and quaternions underlying Albert Einstein’s special theory of relativity. This has led many researchers to wonder about the last and least-understood division algebra. Might the octonions hold secrets of the universe?

“Octonions are to physics what the Sirens were to Ulysses,” Pierre Ramond, a particle physicist and string theorist at the University of Florida, said in an email.

Günaydin, the Penn State professor, was a graduate student at Yale in 1973 when he and his advisor Feza Gürsey found a surprising link between the octonions and the strong force, which binds quarks together inside atomic nuclei. An initial flurry of interest in the finding didn’t last. Everyone at the time was puzzling over the Standard Model of particle physics — the set of equations describing the known elementary particles and their interactions via the strong, weak and electromagnetic forces (all the fundamental forces except gravity). But rather than seek mathematical answers to the Standard Model’s mysteries, most physicists placed their hopes in high-energy particle colliders and other experiments, expecting additional particles to show up and lead the way beyond the Standard Model to a deeper description of reality. They “imagined that the next bit of progress will come from some new pieces being dropped onto the table, [rather than] from thinking harder about the pieces we already have,” said Latham Boyle, a theoretical physicist at the Perimeter Institute of Theoretical Physics in Waterloo, Canada.

Decades on, no particles beyond those of the Standard Model have been found. Meanwhile, the strange beauty of the octonions has continued to attract the occasional independent-minded researcher, including Furey, the Canadian grad student who visited Günaydin four years ago. Looking like an interplanetary traveler, with choppy silver bangs that taper to a point between piercing blue eyes, Furey scrawled esoteric symbols on a blackboard, trying to explain to Günaydin that she had extended his and Gürsey’s work by constructing an octonionic model of both the strong and electromagnetic forces.

“Communicating the details to him turned out to be a bit more of a challenge than I had anticipated, as I struggled to get a word in edgewise,” Furey recalled. Günaydin had continued to study the octonions since the ’70s by way of their deep connections to string theory, M-theory and supergravity — related theories that attempt to unify gravity with the other fundamental forces. But his octonionic pursuits had always been outside the mainstream. He advised Furey to find another research project for her Ph.D., since the octonions might close doors for her, as he felt they had for him.

But Furey didn’t — couldn’t — give up. Driven by a profound intuition that the octonions and other division algebras underlie nature’s laws, she told a colleague that if she didn’t find work in academia she planned to take her accordion to New Orleans and busk on the streets to support her physics habit. Instead, Furey landed a postdoc at the University of Cambridge in the United Kingdom. She has since produced a number of results connecting the octonions to the Standard Model that experts are calling intriguing, curious, elegant and novel. “She has taken significant steps toward solving some really deep physical puzzles,” said Shadi Tahvildar-Zadeh, a mathematical physicist at Rutgers University who recently visited Furey in Cambridge after watching an online series of lecture videos she made about her work.

Furey has yet to construct a simple octonionic model of all Standard Model particles and forces in one go, and she hasn’t touched on gravity. She stresses that the mathematical possibilities are many, and experts say it’s too soon to tell which way of amalgamating the octonions and other division algebras (if any) will lead to success.

“She has found some intriguing links,” said Michael Duff, a pioneering string theorist and professor at Imperial College London who has studied octonions’ role in string theory. “It’s certainly worth pursuing, in my view. Whether it will ultimately be the way the Standard Model is described, it’s hard to say. If it were, it would qualify for all the superlatives — revolutionary, and so on.”

Peculiar Numbers

Furey, who is 39, said she was first drawn to physics at a specific moment in high school, in British Columbia. Her teacher told the class that only four fundamental forces underlie all the world’s complexity — and, furthermore, that physicists since the 1970s had been trying to unify all of them within a single theoretical structure. “That was just the most beautiful thing I ever heard,” she told me, steely-eyed. She had a similar feeling a few years later, as an undergraduate at Simon Fraser University in Vancouver, upon learning about the four division algebras. One such number system, or infinitely many, would seem reasonable. “But four?” she recalls thinking. “How peculiar.”

After breaks from school spent ski-bumming, bartending abroad and intensely training as a mixed martial artist, Furey later met the division algebras again in an advanced geometry course and learned just how peculiar they become in four strokes. When you double the dimensions with each step as you go from real numbers to complex numbers to quaternions to octonions, she explained, “in every step you lose a property.” Real numbers can be ordered from smallest to largest, for instance, “whereas in the complex plane there’s no such concept.” Next, quaternions lose commutativity; for them, a × b doesn’t equal b × a. This makes sense, since multiplying higher-dimensional numbers involves rotation, and when you switch the order of rotations in more than two dimensions you end up in a different place. Much more bizarrely, the octonions are nonassociative, meaning (a × b) × c doesn’t equal a × (b × c). “Nonassociative things are strongly disliked by mathematicians,” said John Baez, a mathematical physicist at the University of California, Riverside, and a leading expert on the octonions. “Because while it’s very easy to imagine noncommutative situations — putting on shoes then socks is different from socks then shoes — it’s very difficult to think of a nonassociative situation.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parentheses feel artificial.”

The octonions’ seemingly unphysical nonassociativity has crippled many physicists’ efforts to exploit them, but Baez explained that their peculiar math has also always been their chief allure. Nature, with its four forces batting around a few dozen particles and anti-particles, is itself peculiar. The Standard Model is “quirky and idiosyncratic,” he said.

In the Standard Model, elementary particles are manifestations of three “symmetry groups” — essentially, ways of interchanging subsets of the particles that leave the equations unchanged. These three symmetry groups, SU(3), SU(2) and U(1), correspond to the strong, weak and electromagnetic forces, respectively, and they “act” on six types of quarks, two types of leptons, plus their anti-particles, with each type of particle coming in three copies, or “generations,” that are identical except for their masses. (The fourth fundamental force, gravity, is described separately, and incompatibly, by Einstein’s general theory of relativity, which casts it as curves in the geometry of space-time.)

Sets of particles manifest the symmetries of the Standard Model in the same way that four corners of a square must exist in order to realize a symmetry of 90-degree rotations. The question is, why this symmetry group — SU(3) × SU(2) × U(1)? And why this particular particle representation, with the observed particles’ funny assortment of charges, curious handedness and three-generation redundancy? The conventional attitude toward such questions has been to treat the Standard Model as a broken piece of some more complete theoretical structure. But a competing tendency is to try to use the octonions and “get the weirdness from the laws of logic somehow,” Baez said.

Furey began seriously pursuing this possibility in grad school, when she learned that quaternions capture the way particles translate and rotate in 4-D space-time. She wondered about particles’ internal properties, like their charge. “I realized that the eight degrees of freedom of the octonions could correspond to one generation of particles: one neutrino, one electron, three up quarks and three down quarks,” she said — a bit of numerology that had raised eyebrows before. The coincidences have since proliferated. “If this research project were a murder mystery,” she said, “I would say that we are still in the process of collecting clues.”

The Dixon Algebra

To reconstruct particle physics, Furey uses the product of the four division algebras, R⊗C⊗H⊗O (R for reals, C for complex numbers, H for quaternions and O for octonions) — sometimes called the Dixon algebra, after Geoffrey Dixon, a physicist who first took this tack in the 1970s and ’80s before failing to get a faculty job and leaving the field. (Dixon forwarded me a passage from his memoirs: “What I had was an out-of-control intuition that these algebras were key to understanding particle physics, and I was willing to follow this intuition off a cliff if need be. Some might say I did.”)

Whereas Dixon and others proceeded by mixing the division algebras with extra mathematical machinery, Furey restricts herself; in her scheme, the algebras “act on themselves.” Combined as R⊗C⊗H⊗O, the four number systems form a 64-dimensional abstract space. Within this space, in Furey’s model, particles are mathematical “ideals”: elements of a subspace that, when multiplied by other elements, stay in that subspace, allowing particles to stay particles even as they move, rotate, interact and transform. The idea is that these mathematical ideals are the particles of nature, and they manifest the symmetries of R⊗C⊗H⊗O.

As Dixon knew, the algebra splits cleanly into two parts: C⊗H and C⊗O, the products of complex numbers with quaternions and octonions, respectively (real numbers are trivial). In Furey’s model, the symmetries associated with how particles move and rotate in space-time, together known as the Lorentz group, arise from the quaternionic C⊗H part of the algebra.  The symmetry group SU(3) × SU(2) × U(1), associated with particles’ internal properties and mutual interactions via the strong, weak and electromagnetic forces, come from the octonionic part, C⊗O.

Lucy Reading-Ikkanda/Quanta Magazine

Günaydin and Gürsey, in their early work, already found SU(3) inside the octonions. Consider the base set of octonions, 1, e₁, e₂, e₃, e₄, e₅, e₆ and e₇, which are unit distances in eight different orthogonal directions: They respect a group of symmetries called G2, which happens to be one of the rare “exceptional groups” that can’t be mathematically classified into other existing symmetry-group families. The octonions’ intimate connection to all the exceptional groups and other special mathematical objects has compounded the belief in their importance, convincing the eminent Fields medalist and Abel Prize–winning mathematician Michael Atiyah, for example, that the final theory of nature must be octonionic. “The real theory which we would like to get to,” he said in 2010, “should include gravity with all these theories in such a way that gravity is seen to be a consequence of the octonions and the exceptional groups.” He added, “It will be hard because we know the octonions are hard, but when you’ve found it, it should be a beautiful theory, and it should be unique.”

Holding e₇ constant while transforming the other unit octonions reduces their symmetries to the group SU(3). Günaydin and Gürsey used this fact to build an octonionic model of the strong force acting on a single generation of quarks.

Furey has gone further. In her most recent published paper, which appeared in May in The European Physical Journal C, she consolidated several findings to construct the full Standard Model symmetry group, SU(3) × SU(2) × U(1), for a single generation of particles, with the math producing the correct array of electric charges and other attributes for an electron, neutrino, three up quarks, three down quarks and their anti-particles. The math also suggests a reason why electric charge is quantized in discrete units — essentially, because whole numbers are.

However, in that model’s way of arranging particles, it’s unclear how to naturally extend the model to cover the full three particle generations that exist in nature. But in another new paper that’s now circulating among experts and under review by Physical Letters B, Furey uses C⊗O to construct the Standard Model’s two unbroken symmetries, SU(3) and U(1). (In nature, SU(2) × U(1) is broken down into U(1) by the Higgs mechanism, a process that imbues particles with mass.) In this case, the symmetries act on all three particle generations and also allow for the existence of particles called sterile neutrinos — candidates for dark matter that physicists are actively searching for now. “The three-generation model only has SU(3) × U(1), so it’s more rudimentary,” Furey told me, pen poised at a whiteboard. “The question is, is there an obvious way to go from the one-generation picture to the three-generation picture? I think there is.”

This is the main question she’s after now. The mathematical physicists Michel Dubois-Violette, Ivan Todorov and Svetla Drenska are also trying to model the three particle generations using a structure that incorporates octonions called the exceptional Jordan algebra. After years of working solo, Furey is beginning to collaborate with researchers who take different approaches, but she prefers to stick with the product of the four division algebras, R⊗C⊗H⊗O, acting on itself. It’s complicated enough and provides flexibility in the many ways it can be chopped up. Furey’s goal is to find the model that, in hindsight, feels inevitable and that includes mass, the Higgs mechanism, gravity and space-time.

Already, there’s a sense of space-time in the math. She finds that all multiplicative chains of elements of R⊗C⊗H⊗O can be generated by 10 matrices called “generators.” Nine of the generators act like spatial dimensions, and the 10th, which has the opposite sign, behaves like time. String theory also predicts 10 space-time dimensions — and the octonions are involved there as well. Whether or how Furey’s work connects to string theory remains to be puzzled out.

The Final Theory

Furey mostly demurred on my more philosophical questions about the relationship between physics and math, such as whether, deep down, they are one and the same. But she is taken with the mystery of why the property of division is so key. She also has a hunch, reflecting a common allergy to infinity, that R⊗C⊗H⊗O is actually an approximation that will be replaced, in the final theory, with another, related mathematical system that does not involve the infinite continuum of real numbers.

That’s just intuition talking. But with the Standard Model passing tests to staggering perfection, and no enlightening new particles materializing at the Large Hadron Collider in Europe, a new feeling is in the air, both unsettling and exciting, ushering a return to whiteboards and blackboards. There’s the burgeoning sense that “maybe we have not yet finished the process of fitting the current pieces together,” said Boyle, of the Perimeter Institute. He rates this possibility “more promising than many people realize,” and said it “deserves more attention than it is currently getting, so I am very glad that some people like Cohl are seriously pursuing it.”

Boyle hasn’t himself written about the Standard Model’s possible relationship to the octonions. But like so many others, he admits to hearing their siren song. “I share the hope,” he said, “and even the suspicion, that the octonions may end up playing a role, somehow, in fundamental physics, since they are very beautiful.”

Octonions

In 1878 and 1880, Frobenius and Peirce proved that the only associative real division algebras are real numbers, complex numbers, and quaternions. Adams proved that n-dimensional vectors form an algebra in which division (except by 0) is always possible only for n = 1, 2, 4, and 8. Bott and Milnor proved that the only finite-dimensional real division algebras occur for dimensions n = 1, 2, 4, and 8. Each gives rise to an algebra with particularly useful physical applications (which, however, is not itself necessarily nonassociative), and these four cases correspond to real numbers, complex numbers, quaternions, and Cayley numbers, respectively. The Cayley algebra is the only nonassociative algebra. Hurwitz proved in 1898 that the algebras of real numbers, complex numbers, quaternions, and Cayley numbers are the only ones where multiplication by unit "vectors" is distance-preserving. These most general numbers are also called “octonions.” The mathematical fact that n-dimensional, division algebras are allowed only for n = 2⁰ = 1, 2¹ = 2, 2² = 4 and 2³ = 8 gives meaning to these powers of 2 on one slope of the Platonic Lambda.

It is a powerful example of the ‘Tetrad Principle’ formulated by the author wherein the fourth member of a class of mathematical object (in this case, even numbers) has fundamental significance to physics (in this case, the relevance of octonions to superstring theory). In the musical context of Plato’s cosmological treatise, Timaeus, the numbers of his Lambda generate the musical proportions of the Pythagorean musical scale, successive octaves of which have pitches 2⁰, 2¹, 2², 2³, etc. This demonstrates the archetypal role played by these powers of 2, for they define not only successive musical octaves but also the dimensions of the four possible division algebras. It intimates a connection between the Pythagorean basis of music and octonions and therefore with superstring theory, as was discussed in Article 13. This will be explored in Section 7.

An octonion has the form:

N = a₀+ a₁e₁ + a₂e₂ + a₃e₃ + a₄e₄ + a₅e₅ + a₆e₆ + a₇e₇,

where the a_i (i = 0, 1, 2, 3, 4, 5, 6, 7) are real numbers, the unit octonions e_i (i = 1-7) are imaginary numbers:

e_i² = -1

and are anticommutative(i ≠ j):

e_i e_j = -e_je_i 

Shown on the right is the multiplication table for the 8-tuple of unit octonions: (1, e₁, e₂, e₃, e₄, e₅, e₆, e₇)

It comprises (8×8=64=4³) entries, of which eight are diagonal and real (8 = fourth even integer) and (64–8=56) are off-diagonal and imaginary (28 on one side of the diagonal and 28 on the other side with opposite signs due to their anticommutativity). 28 is the seventh triangular number, where 7 is the fourth odd integer and the fourth prime number. There are 36 entries with a positive sign (one real and 35 imaginary) and 28 entries with a negative sign (7 real and 21 imaginary). 36 is the sum of the first four even integers and the first four odd integers:

36 = (2+4+6+8) + (1+3+5+7).

This illustrates how the Pythagorean Tetrad (4) defines the properties of the octonion multiplication table. The 35 positive imaginary entries consist of five copies of each of the seven imaginary unit octonions and the 21 negative entries comprise three copies of these, that is, the multiplication table contains (ignoring their signs) eight copies of each imaginary octonion. Multiplication yields seven new copies, that is, (7×7=49) imaginary octonions made up of 28 positive in four sets and 21 negative in three sets. Multiplication also generates seven, new, identical, negative numbers -1, so the 56 new entries comprise equal numbers with positive and negative signs (four positive sets of seven and (3+1) negative sets).

Sacred geometry of octonions

As the realisation of the most general type of numbers showing neither associativity nor commutativity, the eight unit octonions are the mathematical counterpart in the Tree of Life of Daath and the seven Sephiroth of Construction.

The differentiation between the non-Sephirah Daath and the Sephiroth of Construction corresponds to the distinction between the real unit octonion 1 and the seven imaginary octonions. The 1:3:4 pattern of new numbers created by their multiplication in pairs (see previous paragraph) corresponds, respectively, to Daath, the triangular array of Chesed, Geburah and Tiphareth and the quartet of Netzach, Hod, Yesod and Malkuth situated at the corners of the tetrahedron at the base of the Tree of Life.

The multiplication table for the seven, imaginary, unit octonions (e₁-e₇) has (7×7=49) entries, of which 42 are imaginary (21 in three positive copies and 21 in three negative copies of the set) and seven are real (-1). As (-ei )×(-ej ) = (ei )×(ej ), the multiplication table for the negative, imaginary unit octonions is the same as Table 2. This 7:42 pattern conforms to the tetractys, the Pythagorean archetypal pattern of wholeness, for the following reason: the set (e₁-e₇) can be assigned to what the author calls the seven ‘hexagonal yods’ of the tetractys, the sets (e₁, e₂, e₃ ) and (e₄, e₅, e₆) So-called because they are located in the tetractys at the corners and centre of a hexagon.

being located at the corners of the two intersecting equilateral triangles forming a Star of David, whilst e₇ is at the centre of the tetractys because it corresponds to Malkuth, which the rules of correspondence between the Tree of Life and the tetractys require to be the central yod of the tetractys. The tetractys of the next higher order contains yods, of which 15 (the fourth triangular number after 

85 = 4⁰ + 4¹ + 4² + 4³ 

1 and the sum of the first four powers of 2 in the Lambda) are corners of tetractyses and (85–15=70) are hexagonal yods. The seven hexagonal yods of the tetractys now become (7×7=49) such yods belonging to their corresponding seven tetractyses. The 49 entries of the multiplication table for the seven imaginary unit octonions can be assigned to them in a way consistent with their 7:42 pattern. The set (-1,-1,-1,-1,-1,-1,-1) corresponds to the central set of seven hexagonal yods, the three sets (-e₁,-e₂,-e₃,-e₄,-e₅,-e₆,-e ₇) correspond to one triangular array of seven yods and the three sets (-e₁,-e₂,-e₃,-e₄,-e₅,-e₆,-e ₇) correspond to the other triangular array of seven yods. The original set of seven octonions, represented by the Pythagorean symbol of wholeness, yields on multiplication another pattern that is really just a more differentiated form of the latter: unity remains unity as new levels of complexity emerge.

M-theory, Octonions and Tricategories

You’ve probably heard rumors that superstring theory lives in 10 dimensions and something more mysterious called M-theory lives in 11. You may have wondered why.

In fact, there’s a nice way to write down theories of superstrings in dimensions 3, 4, 6, and 10 — at least before you take quantum mechanics into account. Of these theories, it seems you can only consistently quantize the 10-dimensional version. But never mind that. What’s so great about the numbers 3, 4, 6 and 10?

What’s so great is that they’re 2 more than 1, 2, 4, and 8.

There are only normed division algebras in dimensions 1, 2, 4, and 8. The real numbers are 1-dimensional. The complex numbers are 2-dimensional. There are also more esoteric options: the quaternions are 4-dimensional, and the octonions are 8-dimensional. When you try to go beyond these, you lose the law that |x y| = |x| |y| and things aren’t so nice.

I’ve spent decades studying the quaternions and octonions, just because they’re weird and interesting. Why do the dimensions double each time in this game? There’s a nice answer. What happens if you go further, to dimension 16? I’ve learned a bit about that too, though I bet there are big mysteries still lurking here.

Most important, what — if anything — do normed division algebras have to do with physics? The jury is still out on this one, but there are some huge clues. Most fundamentally, a normed division algebra of dimension n gives a nice unified way to describe both spin-1 and spin-1/2 particles in (n+2)-dimensional spacetime! The gauge bosons in nature are spin-1 particles, while the fermions are spin-1/2 particles. We’d definitely like a good theory of physics to fit these together somehow.

One cool thing is this. A string is a curve, so it’s 1-dimensional, but as time passes it traces out a 2-dimensional surface. So, if we have a string floating around in some spacetime, we’ve got a 2d surface together with some extra dimensions of spacetime. It turns out to be very good to put complex coordinates on that 2d surface. Then you can describe how the string wiggles in the extra dimensions using equations that have symmetry under conformal transformations.

But for the string to be ‘super’ — for it to have supersymmetry, a symmetry between bosons and fermions — we need a certain special identity to hold, called the 3-\psi’s rule. And this holds precisely when we can take the extra dimensions and think of them as forming a normed division algebra!

So, we need 1, 2, 4 or 8 extra dimensions. So the total dimension of spacetime needs to be 3, 4, 6, or 10. Not at all coincidentally, these are also the dimensions where spin-1 and spin-1/2 particles can be described using a normed division algebra.

(This is a very rough sketch of a complicated argument, of course. I’m leaving out the details, but later I’ll show you where to find them.)

We can also look at theories of ‘branes’, which are like strings but higher-dimensional. Instead of a curve, a 2-brane is a 2-dimensional surface. As time passes, it traces out a 3-dimensional surface. So, if we have a 2-brane floating around in some spacetime, we’ve got a 3-dimensional surface together with some extra dimensions of spacetime. And it turns out that 2-branes can also have supersymmetry when the extra dimensions can be seen as a normed division algebra!

So now the total dimension of spacetime needs to be 3 more than 1, 2, 4, and 8. It needs to be 4, 5, 7 or 11.

When we take quantum mechanics into account it seems that the 11-dimensional theory works best… but the quantum aspects are still mysterious, murky and messy compared to superstring theory, so it’s called M-theory.

In his new paper, John Huerta has shown that using the octonions we can build a ‘super-3-group’, an algebraic structure that seems just right for understanding the symmetries of supersymmetric 2-branes in 11 dimensions.

(More information about the next part linked below)

The detailed story has four parts.

• John Baez and John Huerta, Division algebras and supersymmetry I, in Superstrings, Geometry, Topology, and C*-Algebras, eds. Robert Doran, Greg Friedman and Jonathan Rosenberg, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65–80.

Here we explain how to use normed division algebras to describe vectors (and thus spin-1 particles) and spinors (and thus spin-1/2 particles) in spacetimes of dimensions 3, 4, 6 and 10. We use this description to derive the 3-\psi’s rule, an identity obeyed by three spinors only in these special dimensions. We also explain how the 3-\psi’s rule is important in supersymmetric Yang–Mills theory. This stuff was known before, but not explained all in one place.

• John Baez and John Huerta, Division algebras and supersymmetry II, Adv. Math. Theor. Phys. 15 (2011), 1373–1410.

Here go up a dimension and use normed division algebras to derive a special identity that is obeyed by 4 spinors in dimensions 4, 5, 7 and 11. This is called the 4-\Psi’s rule, and it’s important for supersymmetric 2-branes. More importantly, we start studying how the symmetries of superstrings and super-2-branes arise from the normed division algebras. Mathematicians and physicists use Lie algebras to study symmetry, as well as generalizations called ‘Lie superalgebras’, which describe symmetries that mix bosons and fermions. Here we study categorified versions called ‘Lie 2-superalgebras’ and ‘Lie 3-superalgebras’. It turns out that the 3-\psi’s rule is a ‘3-cocycle condition’ — just the thing you need to build a Lie 2-superalgebra extending the Poincaré Lie superalgebra! Similarly, the 4-\Psi’s rule is a ‘4-cocycle condition’ which lets you build a Lie 2-superalgebra extending the Poincaré Lie superalgebra. Next, try this:

• John Huerta, Division algebras and supersymmetry III.

At this point John Huerta sailed off on his own!

In this paper John cooked up the ‘Lie 2-supergroups’ that govern classical superstrings in dimensions 3, 4, 6 and 10. Just as a group is a category with one object and with all its morphisms being invertible, a 2-group is a bicategory with one object and with all its morphisms and 2-morphisms being weakly invertible. A Lie 2-supergroup is a bicategory internal to the category of supermanifolds. John shows how to derive the pentagon identity for this bicategory from the 3-\psi’s rule!

And here’s his new paper, the last of the series:

• John Huerta, Division algebras and supersymmetry IV.

Here John built the ‘Lie 3-supergroups’ that govern classical super-2-branes in dimensions 3, 4, 6 and 10. A 3-group is a tricategory with one object and with all its morphisms, 2-morphisms and 3-morphisms being weakly invertible. John shows how to derive the ‘pentagonator identity’ — that is, a commutative diagram shaped like the 3d Stasheff polytope — from the 4-\Psi’s rule.

In case you’re wondering: I believe this game stops here. I’m pretty sure there isn’t a nontrivial 5-cocycle (valued in the trivial representation) which gives a Lie 4-superalgebra extending the Poincaré superalgebra in 12 dimensions. But I hope someone proves this, or has proved it already!

Of course, Urs Schreiber and collaborators have done vastly more general things using a more intensely modern point of view. For example:

• Hisham Sati, Urs Schreiber and Jim Stasheff, L∞ algebra connections and applications to String- and Chern-Simons n-transport.
• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models, and super p-branes with tensor multiplet fields.
• Urs Schreiber, Differential cohomology in a cohesive infinity-topos.

But one thing the ‘Division Algebras and Supersymmetry’ series has to offer is a focus on the way normed division algebras help create the exceptional higher algebraic structures that underlie superstring and super-2-brane theories. And with the completion of this series, I can now relax and forget all about these ideas, confident that at this point, the minds of a younger generation will do much better things with them than I could.

E8 tetrahedron grid

As I have already said the E8 lie group is the tetrahedron grid fractal because one of its lower-dimensional forms which is a quasicrystal is made up of tetrahedrons so it is the (64) tetrahedron grid fractal. Also the tetractys forms tetrahedron.

The Rhombic Tria- Contahedron, Zn-Mg-Ho QuasiCrystal Electron Diffractions (Shown on the right)

Quasicrystal and spacetime structure

(83(-7/84=-0.083)+115(231 gates symmetry))×2=396=3311(11:33=693)=19:91=structural formation of 64 tetrahedra, so quasicrystals with the geometry of 396 can manipulate spacetime.

universal math solutions in dimensions 8 and 24

Mathematicians used “magic functions” to prove that two highly symmetric lattices solve a myriad of problems in eight- and 24-dimensional space.

Three years ago, Maryna Viazovska, of the Swiss Federal Institute of Technology in Lausanne, dazzled mathematicians by identifying the densest way to pack equal-sized spheres in eight- and 24-dimensional space (the second of these in collaboration with four co-authors). Now, she and her co-authors have proved something even more remarkable: The configurations that solve the sphere-packing problem in those two dimensions also solve an infinite number of other problems about the best arrangement for points that are trying to avoid each other.

The points could be an infinite collection of electrons, for example, repelling each other and trying to settle into the lowest-energy configuration. Or the points could represent the centers of long, twisty polymers in a solution, trying to position themselves so they won’t bump into their neighbors. There’s a host of different such problems, and it’s not obvious why they should all have the same solution. In most dimensions, mathematicians don’t believe this is remotely likely to be true.

But dimensions eight and 24 each contain a special, highly symmetric point configuration that, we now know, simultaneously solves all these different problems. In the language of mathematics, these two configurations are “universally optimal.”

The sweeping new finding vastly generalizes Viazovska and her collaborators’ previous work. “The fireworks have not stopped,” said Thomas Hales, a mathematician at the University of Pittsburgh who in 1998 proved that the familiar pyramidal stacking of oranges is the densest way to pack spheres in three-dimensional space.

Eight and 24 now join dimension one as the only dimensions known to have universally optimal configurations. In the two-dimensional plane, there’s a candidate for universal optimality — the equilateral triangle lattice — but no proof. Dimension three, meanwhile, is a zoo: Different point configurations are better in different circumstances, and for some problems, mathematicians don’t even have a good guess for what the best configuration is.

“You change the dimension or you change the problem a little bit and then things may be completely unknown,” said Richard Schwartz, a mathematician at Brown University in Providence. “I don’t know why the mathematical universe is built this way.”

Proving universal optimality is much harder than solving the sphere-packing problem. That’s partly because universal optimality encompasses infinitely many different problems at once, but also because the problems themselves are harder. In sphere packing, each sphere cares only about the location of its nearest neighbors, but for something like electrons scattered through space, every electron interacts with every other electron, no matter how far apart they are. “Even in light of the earlier work, I would not have expected this [universal optimality proof] to be possible to do,” Hales said.

“I’m very, very impressed,” said Sylvia Serfaty, a mathematician at New York University. “It’s at the level of the big 19th-century mathematics breakthroughs.”

A Magic Certificate

It might seem strange that dimensions eight and 24 should behave differently from, say, dimension seven or 18 or 25. But mathematicians have long known that packing objects into space works differently in different dimensions. For instance, consider a higher-dimensional sphere, defined simply as the collection of points some fixed distance from a center point. If you compare the sphere’s volume to that of the smallest cube that fits around it, the sphere fills up less and less of the cube as you go up in dimension. If you wanted to ship an eight-dimensional soccer ball in the smallest possible box, the ball would fill less than 2 percent of the box’s volume — the rest would be wasted space.

In each dimension higher than three, it’s possible to construct a configuration analogous to the pyramidal orange arrangement, and as the dimension increases, the gaps between the spheres grow. When you hit dimension eight, there’s suddenly enough room to fit new spheres into the gaps. Doing so produces a highly symmetric configuration called the E8 lattice. Likewise, in dimension 24, the Leech lattice arises from fitting extra spheres into the gaps in another well-understood sphere packing.

For reasons mathematicians don’t fully understand, these two lattices crop up in one area of mathematics after another, from number theory to analysis to mathematical physics. “I don’t know of a single root cause for everything,” said Henry Cohn, of Microsoft Research New England in Cambridge, Mass., one of the new paper’s five authors.

For more than a decade, mathematicians have had strong numerical evidence suggesting that E8 and the Leech lattice are universally optimal in their respective dimensions — but until recently they had no idea how to prove it. Then in 2016, Viazovska took the first step by proving that these two lattices are the best possible sphere packings (she was joined, for the Leech lattice proof, by Cohn and the other three authors of the new paper: Abhinav Kumar, Stephen Miller of Rutgers University and Danylo Radchenko of the Max Planck Institute for Mathematics in Bonn, Germany).

While Hales’ proof for the three-dimensional case filled hundreds of pages and required extensive computer calculations, Viazovska’s E8 proof came in at just 23 pages. The core of her argument involved identifying a “magic” function (as mathematicians have come to call it) that provided what Hales called a “certificate” that E8 is the best sphere packing — a proof that might be hard to come up with, but that once found carries instant conviction. For example, if someone asked you whether any real number x makes the polynomial x² – 6x + 9 negative, you might be hard-pressed to reply. But if you realized that the polynomial equals (x – 3)², you would immediately know that the answer is no, since squared numbers are never negative.

Viazovska’s magic function method was powerful — almost too powerful, in fact. The sphere-packing problem only cares about interactions between nearby points, but Viazovska’s approach seemed as if it might work for long-range interactions as well, like those between distant electrons.

High-Dimensional Uncertainty

To show that a configuration of points in space is universally optimal, one must first specify the universe in question. No point configuration is optimal with respect to every single goal: For instance, when an attractive force acts on the points, the lowest-energy configuration will not be some lattice but just a massive pileup, with all the points at the same spot.

Viazovska, Cohn and their collaborators restricted their attention to the universe of repulsive forces. More specifically, they considered ones that are completely monotonic, meaning (among other things) that the repulsion is stronger when points are closer to each other. This broad family includes many of the forces most common in the physical world. It includes inverse power laws — such as Coulomb’s inverse square law for electrically charged particles — and Gaussians, the bell curves that capture the behavior of entities with many essentially independent repelling parts, such as long polymers. The sphere-packing problem sits at the outer edge of this universe: The requirement that the spheres not overlap translates into an infinitely strong repulsion when their center points are closer together than the diameter of the spheres.

For any one of these completely monotonic forces, the question becomes, what is the lowest-energy configuration — the “ground state” — for an infinite collection of particles? In 2006, Cohn and Kumar developed a method for finding lower bounds on the energy of the ground state by comparing the energy function to smaller “auxiliary” functions with especially nice properties. They found an infinite supply of auxiliary functions for each dimension, but they didn’t know how to find the best auxiliary function.

In most dimensions, the numerical bounds Cohn and Kumar found bore little resemblance to the energy of the best-known configurations. But in dimensions eight and 24, the bounds came astonishingly close to the energy of E8 and the Leech lattice, for every repulsive force Cohn and Kumar tried their method on. It was natural to wonder whether, for any given repulsive force, there might be some perfect auxiliary function that would give a bound exactly matching the energy of E8 or the Leech lattice. For the sphere-packing problem, that’s exactly what Viazovska did three years ago: She found the perfect, “magic” auxiliary function by looking among a class of functions called modular forms whose special symmetries have made them objects of study for centuries.

When it came to other repelling-point problems, such as the electron problem, the researchers knew what properties each magic function would need to satisfy: It would have to take on special values at certain points, and its Fourier transform — which measures the function’s natural frequencies — would need to take on special values at other points. What they didn’t know, in general, was whether such a function actually exists.

It’s usually easy to construct a function that does what you want at your favorite points, but it’s surprisingly tricky to control a function and its Fourier transform at the same time. “When you impose your will on one of them, the other one does something that’s totally different from what you wanted,” Cohn said.

In fact, this persnicketiness is none other than the famous uncertainty principle from physics in disguise. Heisenberg’s uncertainty principle — which says that the more you know about a particle’s position, the less you can know about its momentum, and vice versa — is a special case of this general principle, since a particle’s momentum wave is the Fourier transform of its position wave.

In the case of a repulsive force in dimension eight or 24, Viazovska made a daring conjecture: that the limitations the team wanted to place on their magic function and its Fourier transform lay precisely on the border between the possible and the impossible. Any more limitations, she suspected, and no such function could exist; fewer limitations, and many functions could exist. In the situation the team cared about, she proposed, there should be exactly one function that fit.

“This is, I think, one of the great things about Maryna,” Cohn said. “She’s very insightful and also very bold.”

At the time, Cohn was skeptical — Viazovska’s guess seemed too good to be true — but the team eventually proved her right. Not only did they show that there exists exactly one magic function for each repulsive force, but they gave a recipe for how to make it. As with sphere packing, this construction provided an immediate certificate of the optimality of E8 and the Leech lattice. “It’s kind of a monumental result,” Schwartz said.

The Triangle Lattice

Beyond settling the universal optimality problem, the new proof answers a burning question many mathematicians have had since Viazovska solved the sphere-packing problem three years ago: Just where did her magic function come from? “I think many people were puzzled,” Viazovska said. “They asked, ‘What is the meaning here?’”

In the new paper, Viazovska and her collaborators showed that the sphere-packing magic function is the first in a sequence of modular-form building blocks that can be used to construct magic functions for every repulsive force. “Now it has many brothers and sisters,” Viazovska said.

It still feels somewhat miraculous to Cohn that everything worked out so neatly. “There are some things in mathematics that you do by persistence and brute force,” he said. “And then there are times like this where it’s like mathematics wants something to happen.”

The natural next question is whether these methods can be adapted to prove universal optimality for the only other clear candidate out there: the equilateral triangle lattice in the two-dimensional plane. For mathematicians, the fact that no one has come up with a proof in this simple setting has been a “big embarrassment for the whole community,” said Edward Saff, a mathematician at Vanderbilt University in Nashville.

Unlike E8 and the Leech lattice, the two-dimensional triangular lattice shows up all over the place in nature, from the structure of honeycombs to the arrangement of vortices in superconductors. Physicists already assume this lattice is optimal in a wide range of contexts, based on a mountain of experiments and simulations. But, Cohn said, no one has a conceptual explanation for why the triangular lattice should be universally optimal — something a mathematical proof would hopefully provide.

Dimension two is the only dimension other than eight and 24 in which Cohn and Kumar’s numerical lower bounds work well. This strongly suggests that there should be magic functions in dimension two as well. But the team’s method for constructing magic functions is unlikely to carry over to this new domain: It relies heavily on the fact that the distances between points in E8 and the Leech lattice are especially well-behaved numbers, which is not the case in dimension two. That dimension “seems beyond humanity’s abilities right now,” Cohn said.

For the time being, mathematicians are celebrating their new insight into the strange worlds of eight- and 24-dimensional space. It is, Schwartz said, “one of the best things I’ll probably see in my lifetime.”

Poincare Dodecahedral Space and S3

The following information is from www.valdostamuseum.com/hamsmith/PDS3.html

Spheres are Symmetric Spaces of

(The Symmetric Spaces discussed here can be represented as coset spaces G / K where G is a connected Lie Group and K is a subgroup of G - for a more technically accurate definition and more details, see, for example: Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic 1978; Helgason, Groups and Geometric Analysis, Academic 1984; Edward Dunne's web site; Brocker and tom Dieck, Representations of Compact Lie Groups, Springer-Verlag 1985; Besse, Einstein Manifolds, Springer-Verlag 1987; Rosenfeld, Geometry of Lie Groups, Kluwer 1997; Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, John Wiley 1974. Hermitian Symmetric Spaces are related to Bounded Complex Domains.)

the form Spin(n+1) / Spin(n) for the n-sphere Sn

The rank of a Symmetric Space is the maximal dimension of a flat totally geodesic submanifold of the symmetric space. For example, the rank of an n-sphere Sn is 1, because the maximal flat totally geodesic submanifold is a Great Circle, such as its Equator, that is a 1-dimensional S1.

(Note: Sometimes (thanks to Aaron Bergman for pointing this out to me) I have used the term "Symmetric Space" incorrectly. Some of the spaces that I refer to as "Symmetric Spaces" should really be called Isotropy Irreducible Homogeneous Spaces. Such erroneous terminology still exists on my web site, with respect to, for example, Spin(7) / G2.)

The book Einstein Manifolds by Besse (Springer 1987) lists (on page 203) SO(7) / G2 as a compact non-symmetric strongly isotropy irreducible space, and on pages 179-180 it states that Spin(7) / G2 and S7 are related by Spin(7) acting transitively on S7 with isotropy subgroup K and a 0-dimensional "space of G-invariant Riemannian metrics up to homotheties (i.e., isometries and multiplication by a positive constant)."and that "... G2 [subgroup of] SO(7) ... G2 ...[is]... NOT the full group of isometries of any Riemannian metric on the corresponding sphere [S7]. We recall that any SO(n)-invariant metric on S(n-1) is (proportional to) the canonical one, so the full group of isometries is O(n). Thus even SO(n) is not the full group of isometries of any Riemannian metric on S(n-1). ...".

Joseph A. Wolf, in his book Spaces of Constant Curvature (5th ed) (Publish or Perish 1984), says (at page 302): "... Added in proof. A completes structure theory and classification [Joseph A. Wolf, The Geometry and Structure of Isotropy Irreducible Homogeneous Spaces, Acta Math. 120 (1968) 59-148 and Erratum Acta Math. 152 (1984) 141-142] has now been worked out for the isotropy irreducible riemannian manifolds. Unfortunately it is too long and technical to summarize here. ...".

The book Clifford Algebras and the Classical Groups by Ian Porteous (Cambridge 1995) describes the structure G2 -> Spin(7) -> S7 in terms of exact sequences and coset spaces. 

Note also that I have sometimes used the notation SO(n) when I should have use Spin(n), and vice versa.

Examples of Symmetric Spaces of rank 1 are

• Spin(n+1) / Spin(n) = SO(n+1) / SO(n) = O(n+1) / O(n) - n-spheres Sn
• SU(n+1) / SU(n) = U(n+1) / U(n) - (2n+1)-spheres S(2n+1)
• Sp(n+1) / Sp(n) - (4n+3)-spheres S(4n+3)
• SO(n+1) / O(n) - Real Projective Spaces RPn
• SU(n+1) / S(U(n)xU(1)) - Complex Projective Spaces CPn
• Sp(n+1) / Sp(n)xSp(1) - Quaternionic Projective Spaces QPn
• F4 / Spin(9) - Octonionic Projective Plane OP2
• Spin(4) / U(2)
• Spin(8) / Spin(7) - 7-sphere (round: no torsion, twisting, or squashing)
  • Spin(7) / G2 - 7-sphere with torsion
  • Spin(6) / SU(3) = SU(4) / SU(3) - 7-sphere twisted
  • Spin(5) / Spin(3) = Sp(2) / SU(2) = Sp(2) / Sp(1) - squashed 7-sphere (this leads to an H-space, the Hilton-Roitberg Criminal)

Some examples of Symmetric Spaces of rank 2 are

• Spin(8) / U(4) - compare dimensional reduction of Spin(8) gauge group
• Spin(n+2) / Spin(n)xU(1) - Lie Spheres
• E6 / Spin(10)xU(1) - Rosenfeld's (CxO)P2
• E6 / F4 - set of OP2 in (CxO)P2
• G2 / Spin(4) = G2 / Spin(3)xSpin(3) = G2 / SU(2)xSU(2) = G2 / Sp(1)xSp(1) - Quaternionic subalgebras of Octonions

Some examples of Symmetric Spaces of rank 3 are

• Spin(12) / U(6)
• Spin(n+3) / Spin(n)xSpin(3) - for n at least 6
• E7 / E6xU(1) - set of (CxO)P2 in (QxO)P2

Some examples of Symmetric Spaces of rank 4 are

• Spin(16) / U(8)
• Spin(n+4) / Spin(n)xSpin(4) - for n at least 8
• F4 / Sp(3)xSp(1) - set of QP2 in OP2
• E6 / SU(6)xSp(1) - set of (CxQ)P2 in (CxO)P2
• E7 / Spin(12)xSp(1) - Rosenfeld's (QxO)P2
• E8 / E7xSp(1) - set of (QxO)P2 in (OxO)P2

An example of a Symmetric Space of rank 6 is

• E6 / Sp(4)

An example of a Symmetric Space of rank 7 is

• E7 / SU(8)

Some examples of Symmetric Spaces of rank 8 are

• Spin(n+8) / Spin(n)xSpin(8) - for n at least 16 - compare Cl(n+8) = Cl(n)xCl(8)
• E8 / Spin(16) - Rosenfeld's (OxO)P2

Spheres are related to Lie groups, particularly to the B and D series of Lie groups Spin(n).

Volume of spheres

Conway and Sloane, in Sphere Packings, Lattices, and Groups (3rd ed Springer 1999), say, at pages 9-10:

"... the volume of an n-dimensional sphere [that is, a sphere in n-dimensional Euclidean space] of radius r ... is Vn r^n where Vn, the volume of a sphere or radius 1, is given by Vn = pi^(n/2) / (n/2)! [for even n]= 2^n pi^((n-1)/2) ((n-1)/2)! / n! [for odd n] The surface area of a sphere of radius r is n Vn r^(n-1)...".

For a unit sphere S(2n+1) in (2n+2)-dimensional space, Conway and Sloane's formula would be (2n+2) pi^(n+1) / (n+1)! For a unit sphere S(2n) in (2n+1)-dimensional space, Conway and Sloane's formula would be (2n+1) 2^(2n+1) pi^n n! / (2n+1)!

Clifford Pickover, in his book Surfing Through Hyperspace (Oxford 1999, notes that the dimension in which a sphere of radius r=1 encloses maximum volume is 5, and the maximal volume dimension increases with radius. For r=1.1 it is 7 and for r=1.2 it is 8.

The dimension in which a sphere of radius r=2 encloses maximum volume is 24.

According to a MathWorld web page:

"... the hyper-Surface Area and Content reach Maxima and then decrease towards 0 as [sphere dimension] n increases. ... The point of Maximal hyper-Surface Area ...[and]... The point of Maximal Content ...[cannot]... be solved analytically for n, but the numerical solutions are n = 7.256295... for hyper-Surface Area and n = 5.25695... for Content. As a result, the 7-D [ S6 in R7 ] and 5-D [ S4 in R5 ] hyperspheres have Maximal hyper-Surface Area and Content, respectively. ..".

Sphere Packings

Conway and Sloane, in Sphere Packings, Lattices, and Groups (3rd ed Springer 1999), say, at pages 14-16, 128: "... We see from Fig. 1.5 ... that the laminated latttices /\n are the densest packings known in dimensions < 29, except for dimensions 10-13. In dimension 12 ... the Coxeter-Todd lattice K12 ... the real form of ..[a]... 6-dimensional complex lattice over the Eisenstein integers ... is the densest known, and in dimensions 10, 11, and 13 there are non-lattice packings ... that are denser than any known lattices. ... Minkowski gave a nonconstructive proof in 1905 that there exist lattices with density ... > zeta(n) / 2^(n-1) where zeta(n) ... is the Riemann zeta-function. ... Many generalizations and extensions ... have been found, although no essential improvement is known for large n. In its general form this bound is known as the Minkowski-Hlawka theorem. We still do not know how to construct packings that are as good as ...[ the bound of the Minkowski-Hlawka theorem]... ".

According to an article by George Szpiro in Nature 424 (03 July 2003) 12-13: "... Just under five years ago, Thomas Hales ... declared that he had used a series of computers to prove an idea that has evaded certain confirmation for 400 years. The subject of his message was Kepler's conjecture, proposed by the German astronomer Johannes Kepler, which states that the densest arrangement of spheres is one in which they are stacked in a pyramid - much the same way as grocers arrange oranges. ... But today, Hales's proof remains in limbo. It has been submitted to the prestigious Annals of Mathematics, but is yet to appear in print. Those charged with checking it say that they believe the proof is correct, but are so exhausted with the verification process that they cannot definitively rule out any errors. So when Hales's manuscript finally does appear in the Annals, probably during the next year, it will carry an unusual editorial note - a statement that parts of the paper have proved impossible to check. At the heart of this bizarre tale is the use of computers in mathematics ... In 1977, for example, a computer-aided proof was published for the four-colour theorem, which states that no more than four colours are needed to fill in a map so that any two adjacent regions have different colours1, 2. No errors have been found in the proof, but some mathematicians continue to seek a solution using conventional methods. ...

... Hales, who started his proof at the University of Michigan in Ann Arbor before moving to the University of Pittsburgh, Pennsylvania, began by reducing the infinite number of possible stacking arrangements to 5,000 contenders. He then used computers to calculate the density of each arrangement. Doing so was more difficult than it sounds. The proof involved checking a series of mathematical inequalities using specially written computer code. In all, more than 100,000 inequalities were verified over a ten-year period. ... It was not enough for the referees to rerun Hales's code - they had to check whether the programs did the job that they were supposed to do. Inspecting all of the code and its inputs and outputs, which together take up three gigabytes of memory space, would have been impossible. So the referees limited themselves to consistency checks, a reconstruction of the thought processes behind each step of the proof, and then a study of all of the assumptions and logic used to design the code. A series of seminars, which ran for full academic years, was organized to aid the effort. But success remained elusive. Last July [2002], Fejes Tóth reported that he and the other referees were 99% certain that the proof is sound. They found no errors or omissions, but felt that without checking every line of the code, they could not be absolutely certain that the proof is correct. For a mathematical proof, this was not enough. After all, most mathematicians believe in the conjecture already - the proof is supposed to turn that belief into certainty. The history of Kepler's conjecture also gives reason for caution. In 1993, Wu-Yi Hsiang, then at the University of California, Berkeley, published a 100-page proof of the conjecture in the International Journal of Mathematics ... But shortly after publication, errors were found in parts of the proof. Although Hsiang stands by his paper, most mathematicians do not believe it is valid. ...

...[In January 2003, Hales]... launched the Flyspeck project, also known as the Formal Proof of Kepler. Rather than rely on human referees, Hales intends to use computers to verify every step of his proof. The effort will require the collaboration of a core group of about ten volunteers, who will need to be qualified mathematicians and willing to donate the computer time on their machines. The team will write programs to deconstruct each step of the proof, line by line, into a set of axioms that are known to be correct. If every part of the code can be broken down into these axioms, the proof will finally be verified. ... Pierre Deligne, an algebraic geometer at the Institute for Advanced Study, is one of the many mathematicians who do not approve of computer-aided proofs. "I believe in a proof if I understand it," he says. For those who side with Deligne, using computers to remove human reviewers from the refereeing process is another step in the wrong direction. ...

... Whether or not computer-checking takes off, it is likely to be several years before Flyspeck produces a result. ... Hales estimates that the whole process, from crafting the code to running it, is likely to take 20 person-years of work. Only then will Kepler's conjecture become Kepler's theorem, and we will know for sure whether we have been stacking oranges correctly all these years. ...".

Exotic Spheres

Exotic Spheres are related to Division Algebras

The alternative real division algebras are R, C, Q, and O, the real and complex numbers, quaternions, and octonions, of dimensions 1 = 2^0,  2 = 2^1,  4 = 2^2, and  8 = 2^3

They correspond to the Hopf fibrations of spheres(Image on the right)

Z3 = {1,2,3=0} is the only field such that all of its elements are real prime integers. All of its subgroups Z0, Z1, and Z2 are prime fields. The sedenions of dimension 16 = 2^4 are not a division algebra, and Z4 decomposes into Z2 x Z2.

Kervaire and Milnor have calculated the numbers of different (including exotic) differentiable structures for some spheres: 

Poincare conjecture

Topological = TOP Homeomorphisms, Combinatorial = Piecewise Linear = PL Isomorphisms, and Smooth = Differentiable = DIFF Diffeomorphisms are not equivalent in all dimensions.

According to Freedman and Quinn (Topology of 4-Manifolds, Princeton University Press, 1990, pages 3-4 and 118-119): "... Manifolds of dimension 2 ... have been largely understood for half a century. ... in dimension 3 ... Moise in 1952 reduced topological [TOP] theory to the piecewise linear [PL] (or equivalently smooth [DIFF]) category ... higher dimensions ( > 5 ) were developed next. The two key events were the development of the methods of smooth [DIFF] and PL handlebody theory by Smale in the late 1950s, and the extension to topological [TOP] manifolds by Kirby and Siebenmann in 1969. Dimension 4 was last, with the corresponding results published by the authors in 1982. ... it is the behavior of 2-dimensional disks which separates the dimensions in this way: the generic map of a 2-disk in a 3-manifold has 1-dimensional self-intersections; in a 4-manifold the intersections are isolated points; and in dimensions 5 and above 2-disks are generically embedded. ... Papakyriakopoulos' breakthrough in dimension 3 was a proof of the Dehn lemma for embedding 2-disks in 3-manifolds. In dimension 5 and higher 2-disks are used in the Whitney trick ... New phenomena are ... encountered in dimension 4 ... In other dimensions the principal methods (handlebody theory or embeddedsurface theory) proceed uniformly in the smooth [DIFF], PL, or topological [TOP] categories ... Differences between the categories can for the most part be understood in terms of the associated bundle theories. In dimension 4 qualitatively different ... phenomena occur in the smooth [DIFF] category ... the topological [TOP] theory ... results parallel those of higher dimensions. ... the smooth [DIFF] analogs are known to be false by Donaldson's work ... In dimensions less than 4 the categories DIFF, PL, TOP are equivalent. ... [PL and DIFF] are the same up to isotopy, for manifolds of dimension less than 6. ... ".

John Baez says: "... there are various distinct questions floating around, including:

• A) how many topological manifolds are homotopy-equivalent to the sphere?
• B) how many PL ( = piecewise-linear = combinatorial) manifolds are homeomorphic to the sphere?
• C) how many smooth manifolds are PL equivalent to the sphere?

... In the case of dimension 3, question A is open [ However, as of April 2003, after this was written by John Baez, it may have been proven by Grisha (Grigori) Perelman. ] (the Poincare conjecture), while questions B and C are solved and the answer is 1 ... In the case of dimension 4, question A is solved (in the 1980s, by Freedman) and the answer is 1; question C is solved and the answer is 1, but question B is open (the smooth Poincare conjecture in dimension 4) ...".

Why is question B hard to answer in dimension 4 ? As John Baez says: "... in 4 dimensions we have PL = DIFF, so question B) is equivalent to: how many smooth manifolds are homeomorphic to the sphere ...", and question C) is equivalent to how many smooth manifolds are diffeomorphic to the sphere. As Donaldson and Kronheimer say in their book The Geometry of Four-Manifolds (Oxford 1990): "... Smale's h-cobordism theorem: if X and Y are h-cobordant then they are diffeomorphic ... breaks down in four dimensions ... ".

Why can question A be answered in dimension 4 ? Ian Stewart (Nature 320 (20 March 1986) said: "... Steven Smale developed a method of breaking manifolds into nice pieces, called handles ... [and] ... moving handles around ... Another important tool ... was .. the technique of surgery. This is in essence a systematic study of the effect of removing cerain nice pieces of a manifold and putting them back with a specific twist. ... [In] the three-dimensional case [Poincare's original conjecture] and the four-dimensional case ... both handle theory and surgery fail to work. The number of dimensions is high enough to allow complicated behaviour, but too small to leave adequate room for manoeuvre in eliminating those complications. ... But ... Andrew Casson found a way to make handle theory work in four dimensions. In 1982 Michael Freedman used Casson-handles to prove the four dimensional case of the Poincare conjecture (but not the more specialized differentiable case, which remains open). ...".

Why is question A hard to answer in dimension 3 ? Ian Stewart (Nature 325 (12 February 1987) 579-580) said: "... If evey loop in a given [simply-connected, unlike PDS3] three-dimensional manifold can be shrunk to a point, must that manifold be topologically a hypersphere? ... start with a description of a three-dimensional manifold in terms of a system of links. ... assume that every loop in this manifold can be shrunk, and as what constraint this requirement imposes on the system of links. Then ... find ways to modify the links, using those constraints, until at the end of the process ... [there is] ... a system of links corresponding to the hypersphere. Many complicated moves are needed to effect the simplification. ... The sources of technical difficulty are twofold. The first is the combinatorial complexity of the moves, and the rules stating which one should be applied in circumstances to achieve which tiny step in the simplification. The second is that, in order for the proof to proceed rigorously, a certain amount of technical baggage must be carried along during each move. At the end of hte proof this additional superstructure can be discarded, but until the end is reached it is essential. More than one topologist has attempted to prove the Poincare conjecture by this route, but nobody ... [has] ... succeeded fitting all the pieces together and making them all work at once. ...". One way to describe a 3-sphere in terms of loops is the Hopf Fibration.

According to a 15 April article by Sara Robinson in The New York Times:

"... Grigori Perelman of the Steklov Institute of Mathematics of the Russian Academy of Sciences in St. Petersburg ... is reporting that he has proved the Poincare Conjecture ... Perelman's work ... relies on ideas pioneered by another mathematician, Richard Hamilton. ... Perelman's personal story has parallels to that of ... Andrew Wiles ... who, without confiding in his colleagues, worked alone in his attic on Fermat's Last Theorem. Though his early work has earned him a reputation as a brilliant mathematician, ... Perelman spent the last eight years sequestered in Russia, not publishing. ... Perelman's ... papers say that he has proved what is known as the Geometrization Conjecture, a complete characterization of the geometry of three-dimensional spaces.

Since the 19th century, mathematicians have known that a type of two-dimensional space called a manifold can be given a rigid geometric structure that looks the same everywhere. Mathematicians could list all the possible shapes for two-dimensional manifolds and explain how a creature living on the surface of one can tell what kind of space he is on.

In the 1950's, however, a Russian mathematician proved that the problem was impossible to resolve in four dimensions and that even for three dimensions, the question looked hopelessly complex.

In the early 1970's, ... William P. Thurston, a professor at the University of California at Davis, conjectured that three-dimensional manifolds are composed of many homogeneous pieces that can be put together only in prescribed ways and proved that in many cases his conjecture was correct. ... Thurston won a Fields Medal, the highest honor in mathematics, for his work.

... Perelman's work, if correct, would provide the final piece of a complete description of the structure of three-dimensional manifolds and, almost as an afterthought, would resolve Poincare's famous question.

... Perelman's approach uses a technique known as the Ricci flow, devised by ... Hamilton, who is now at Columbia University. The Ricci flow is an averaging process used to smooth out the bumps of a manifold and make it look more uniform. ... Hamilton uses the Ricci flow to prove the Geometrization Conjecture in some cases and outlined a general program of how it could be used to prove the Geometrization Conjecture in all cases. He ran into problems, however, coping with certain types of large lumps that tended to grow uncontrollably under the averaging process.

... Perelman has ... figure[d] out some new and interesting ways to tame these singularities ...".

As of 15 April 2003, Grisha Perelman has posted two papers: math.DG/0211159 and math.DG/0303109. In the latter paper, he says:

"... This is a technical paper, which is a continuation of [math.DG/0211159]. Here we verify most of the assertions, made in [math.DG/0211159, §13]; the exceptions are (1) the statement that a 3-manifold which collapses with local lower bound for sectional curvature is a graph manifold - this is deferred to a separate paper, as the proof has nothing to do with the Ricci flow, and (2) the claim about the lower bound for the volumes of the maximal horns and the smoothness of the solution from some time on, which turned out to be unjustified, and, on the other hand, irrelevant for the other conclusions.

The Ricci flow with surgery was considered by Hamilton ... unfortunately, his argument, as written, contains an unjustified statement (R_MAX = GAMMA, on page 62, lines 7-10 from the bottom), which I was unable to fix. Our approach is somewhat different, and is aimed at eventually constructing a canonical Ricci flow, defined on a largest possible subset of space-time, - a goal, that has not been achieved yet in the present work. For this reason, we consider two scale bounds: the cutoff radius h, which is the radius of the necks, where the surgeries are performed, and the much larger radius r, such that the solution on the scales less than r has standard geometry. The point is to make h arbitrarily small while keeping r bounded away from zero. ...".

The truth of the smooth Poincare Conjecture in 3 dimensions and the truth of the topological Poincare Conjecture in 4 dimensions may be useful in studying the MacroSpace of Sum-Over-Histories Many-Worlds Quantum Theory.

The following table is of Smooth=Differentiable manifolds that are Combinatorial=Piecewise Linear equivalent to the n-sphere Sn. For Topological manifolds that are homotopy-equivalent to the n-sphere, the answer is unknown for dimension n = 3. For Combinatiorial=Piecewise-Linear manifolds that are homeomorphic to the n-sphere, the answer is unknown for dimension n = 4. (Table on right)

Homotopy

A way to look at spheres is by their Homotopy Groups PI(k)(Sn), which is roughly the number of ways you can wrap a k-sphere around an n-sphere. For example, PI(n)(Sn) is the infinite cyclic group Z, and each element of Z corresponds to a winding number of a wrapping of Sn around Sn.

As noted by John Baez, Homotopy Groups can be related to the 4-dimensional 24-cell, as well as the 24-dimensional Leech lattice, since the (n+3) Homotopy Group of the n-sphere Sn is PI(n+3)(Sn) = Z24  for n greater than 4.

They are also related to the 8-dimensional Witting polytope, since PI(n+7)(Sn) = Z240  for n greater than 8.

Homotopy Groups have Periodicity relations related to Clifford Periodicity.

John Baez, in his week 104 and week 105, looks at octonions, PI(k(O(infinity)), and Bott periodicity  PI(n+8)(O(infinity)) = PI(n(O(infinity)) where O(infinity) is the orthogonal group for infinite-dimensional real space which contains as subgroups all orthogonal groups O(n) for all finite n.

The Clifford algebra reflection group Pin(n) double covers O(n). The spin group Spin(n) double covers SO(n).

John Baez notes that the fact that PI(7)(O(infinity)) = Z corresponds to dimension can also be seen by: forming the classifying space BO(infinity) of O(infinity); noting that PI(n)(G) = PI(n+1)(BG) for any Lie group G so that PI(7)(O(infinity)) = PI(-1)(O(infinity)) = PI(0)(BO(infinity)); and noting that PI(0)(BO(infinity)) = KS0 where KS0 is the real K-theory of S0 which counts the difference in dimension between the two fibres of a vector bundle over the 0-sphere S0.

As John Baez notes, PI(0), PI(1), PI(3), and PI(7) correspond to the alternative real division algebras: real R, complex C, quaternion Q, and octonion O, as their imaginaries are the spheres S0, S1, S3, and S7.

Klein Bottle

PI(0)(O(infinity)) = Z/2   Pin(n)/Spin(n)  connected components O(n)/SO(n)  and reflections.

As described in The Topology of Fibre Bundles, by Norman Steenrod (Princeton 1951), the facts that O(k+1) is the group of a k-sphere bundle whose fibre is Sk, that sphere bundles over Sn correspond 1-1 with elements of PI(n-1)(O(k+1)) up to equivalence under the operations of PI(0)(O(k+1)), and that PI(0)(O(k+1)) has 2 elements, imply that there are two Sk bundles over S1, the untwisted S1 x Sk  and the twisted generalized Klein Bottle.

The twisted Klein(1,k) Bottle is constructed by forming the product of Sk with an interval and matching the ends with an orientation reversing transformation, so that the Klein(1,k) bottle is not orientable.

Since RP1 is homeomorphic to S1, the untwisted S1 x Sk corresponds to the Shilov boundary manifolds RP1 x S3  and  RP1 x S7 used in the D4-D5-E6-E7-E8 physics model.

The corresponding twisted manifolds are Klein(1,3) Bottle  and  Klein(1,7) Bottle WHAT IF PHYSICAL SPACETIME WERE TWISTED KLEIN(1,3) BOTTLE RATHER THAN UNTWISTED RP1 x S3? 

Igor Kulikov has shown that:

for untwisted fields in S1 x R3, decreasing the topological parameter, equivalent to increasing finite temperature, causes the coupling constant to decrease; while for twisted fields in S1 x R3, decreasing the topological parameter,  equivalent to increasing finite temperature, causes the coupling constant to increase; and massless untwisted fields get dynamical mass with decreasing topological parameter, equivalent to increasing finite temperature; while effective mass for twisted fields is eliminated at a critical topological parameter, equivalent to a critical finite temperature.

I Ching Periodicity

Eightfold periodicity is related to the 8 trigrams of the I Ching.

The relationship between homotopy periodicity and Clifford algebra periodicity is shown by this table, in which O represents O(infinity) real Lie group rotations, Sp represents Sp(infinity) quaternionic Lie group rotations, U represents U(infinity) complex Lie group rotations, Cl(n) is the real Clifford algebra Cl(0,n), and Clc(n) is the complex Clifford algebra over complex n-space.

What patterns emerge from this table?

1. For each field K, if the Clifford algebra n is of the form K+K, the origin of the first K of the K+K can be shifted relative to the origin of the second K of the K+K by any integral distance, so that the homotopy PI(n) is Z.
2. For each field K, if the Clifford algebra n is of the form K and preceding Clifford algebra n-1 is of the form K+K, then the homotopy PI(n) has the same order as the number of different multiplicative products of K, because there is a K+K from n-1 inside the K of the n, and the product of the first K of the K+K can be reconciled with the product of the second K of the K+K in that many different ways. For K = C, the complex numbers, there is only one product because i fixes +1 and -1, and PI(n) = 0. For K = R, the real numbers, and for K = Q, the quaternions, there are two different multiplicative products because for R the unit can be +1 or -1 and for Q the product ij can be +k or -k so that the two products are mirror images of each other, and PI(n) = Z/2.  Such a "first Z/2" represents reflections.
3. For each field K, if the Clifford algebra n is of the form K and preceding Clifford algebra n-1 is of the form K+K, then the homotopy PI(n+1) has the order of spin structures on manifolds over the field K. For K = C, these are Z, rather than Z/2, because for n+1 = 1 and 5 these are complexifications of real Clifford algebras of type C and, since CxC = C+C, are complex Clifford algebras of type C+C as in pattern number 1 above. For K= R and K = Q, these are Z/2. Such a "second Z/2" for K = R and K = Q represents spinors.

Poincare Dodecahedral Space

WHAT ABOUT the 3-sphere S3, of dimension 2^k + 1 for k=1?

It is accurate to say that there is no exotic S3, in the sense that anything homeomorphic to normal S3 is also diffeomorphic to normal S3. 

HOWEVER, the 3-dimensional analogue S3# of the exotic Milnor sphere DOES exist, but it is not only not diffeomorphic to normal S3, it is NOT even homeomorphic to normal S3.  That is because S3# is NOT SIMPLY CONNECTED.

WHAT IS S3# ?

S3# is called the Poincare Dodecahedral Space

Poincare, because it would be a counterexample to the Poincare conjecture in 3 dimensions if it were simply connected.

Dodecahedral, because it has dodecahedral/icosahedral symmetry of the 120-element binary icosahedral group, which double covers the simple 60-element icosahedral group.

A reference is Topology and Geometry, by Glen Bredon, Springer (1993).

WHAT DOES S3# LOOK LIKE?

Here are some images from the WWW pages of Richard Hawkins, who calls S3# the Mayan Time Star.  His pages contain many more images and movies that help you understand how S3# looks, and also how a lot of other things look.

Five tetrahedra fit inside the dodecahedron.

How did Richard Hawkins find out about the Time Star? Krsanna Duran says: "... I wrote an article about what the Sirians told me about five interpenetrated tetrahedra embodying and unifying all prime geometries which was published in January, 1995. Richard Hawkins read the article and and sent an email to Gerald de Jong about it. Gerald de Jong constructed a computer model of the five interpenetrated tetrahedra to discover that it did all the things I said it did with extraordinary elegance. ...".

The Time Star is one of my favorite Archetypes.

Start with a dodedecahedron. Five tetrahedra fit inside the dodecahedron.

The alternating permutation group of the 5 tetrahedra is the 60-element icosahedral group.

Now, to see things clearly, look at just one tetrahedron. You can symmetries more clearly when you put an octhedron inside the tetrahedron and a cuboctahedron inside the octahedron.

Take the one tetrahedron and put it inside a cube, with one edge of the tetrahedron in each face of the cube.  Now rotate the cube around inside the dodecahedron, while you also rotate each of the 6 edges of the tetrahedron each of the 6 faces of the cube.

The tetrahedra edges now are parallel to the cube edges. 36 more degrees, after 72 degrees total rotation, the edges will have again formed a tetrahedron.  Keep rotating. After 360 degrees, you have made 5 tetrahedra (one each 72 degrees)

The cube is back like it was, BUT THE TETRAHEDRON IS ORIENTED OPPOSITELY with respect to the cube from its original position.

YOU HAVE TO ROTATE 720 degrees TO GET BACK LIKE YOU STARTED.

That means that, to make S3#,  instead taking the quotient of SO(3) by the 60-element icosahedral group, you should take the quotient of S3 = Spin(3) = SU(2), the double cover of SO(3), by the 120-element binary icosahedral group.

Therefore, S3# is a natural spinor space, and 5-fold Golden Ratio Icosahedral Symmetry is a manifestation in 3 and 4 dimensions of the  Milnor sphere structure of 7 and 8 dimensions.

John Baez, in a 2003-01-23 post to the sci.physics.research thread "The magic of 8", said:

"... "even unimodular lattices" and "invertible symmetric integer matrices with even entries on the diagonal" ...[are]... secretly the same thing as long as your matrix is positive definite ... a "lattice" is a subgroup of R^n that's isomorphic to Z^n. If we give R^n its usual inner product, an "even" lattice is one such that the inner product of any two vectors in the lattice is an even integer. A lattice is "unimodular" if the volume of each cell of the lattice is 1. To get from an even unimodular lattice to a matrix, pick a basis of vectors in the lattice and form the matrix of their inner products. This matrix will then be symmetric, have determinant +-1, and have even entries down the diagonal. ...[a famous 8 x 8 invertible integer matrix with even entries on the diagonal and signature +8 is]...

It's called E8, and it leads to huge wads of amazing mathematics. For example, suppose we take 8 dots and connect the ith and jth dots with an edge if there is a "-1" in the ij entry of the above matrix. 

Now, make a model with one ring for each dot in the above picture, where the rings link if the corresponding dots have an edge connecting them.

Next imagine this model as living in the 3-sphere. Hollow out all these rings: actually delete the portion of space that lies inside them! We now have a 3-manifold M whose boundary dM consists of 8 connected components, each a torus. Of course, a solid torus also has a torus as its boundary. So attach solid tori to each of these 8 components of dM, but do it via this attaching map: (x,y) -> (y,-x+2y) where x and y are the obvious coordinates on the torus, numbers between 0 and 2pi, and we do the arithmetic mod 2pi. We now have a new 3-manifold without boundary. This manifold is called the "Poincare homology sphere". Poincare invented it as a counterexample to his own conjecture that any 3-manifold with the same homology groups as a 3-sphere must *be* the 3-sphere. But he didn't invent it this way. Instead, he got it by taking a regular dodecahedron and identifying its opposite faces in the simplest possible way, namely by a 1/10th turn. So, we've gone from E8 to the dodecahedron!

Voila! Back to E8. ...".

... The fundamental group of the Poincare homology sphere has 120 elements. In fact, we can describe it as follows. The rotational symmetry group of the dodecahedron has 60 elements. Take the "double cover" of this 60-element group, namely the 120-element subgroup of SU(2) consisting of elements that map to rotational symmetries of the dodecahedron under the double cover p: SU(2) -> SO(3) This is the fundamental group of the Poincare homology ...[sphere]... Now, this 120-element group has finitely many irreducible representations. One of them just comes from restricting the 2-dimensional representation of SU(2) to this subgroup: call that R. There are 8 others: call them R(i) for i = 1,...,8. Draw a dot for each one, and draw a line from the ith dot to the jth dot if the tensor product of R and R(i) contains R(j) as a subrepresentation.

Spheres, Octonions, and Reflexivity

The Octonions are the only Division Algebra with Reflexivity

The Unit Spheres of the Division Algebras K are related to the Hopf Fibrations of the form  Sn = Sm / KP1.

For the Real Numbers, the Complex Numbers, and the Quaternions, their Unit Spheres directly form Lie groups.

For the Octonions, the Unit Sphere S7 does NOT form a Lie group. However, the Unit Sphere S7 does EXPAND to form the Lie group Spin(8).

If the process of expansion is continued upward from S7 to Sp(2) to SU(4) to Spin(8), and you note that Spin(8) is in the Clifford Algebra Cl(8), then you see that the upward expansion can continue indefinitely to Cl(8N) for arbitrary N.

If you go downward from Spin(8), based on Real Numbers, to SU(4), based on Complex Numbers, to Sp(2), based on Quaternions, to S7, based on Octonions, then you see that the downward expansion can continue indefinitely to the ZeroDivisor algebra ZD(2^N) for arbitrary N.

THE INTERPENETRATION OF THE UPWARD-BASED WEDGE AND THE DOWNWARD-BASED WEDGE SHOWS OCTONION REFLEXIVITY.

OCTONION REFLEXIVITY IS INHERITED BY THE SEDENIONS.

OCTONION REFLEXIVITY IS ALSO INHERITED BY THE LEECH LATTICE, BASED ON A FIBRATION OF S23 BY A MANIFOLD M7 THAT IS NOT A SPHERE.

AS OP2 IS THE HIGHEST DIMENSIONAL OCTONION PROJECTIVE SPACE, THERE ARE NO HIGHER DIMENSIONAL STRUCTURES OF THAT TYPE.

Sri Yantra

There are only 3 Pairs of Interpenetrating Triangles, corresponding to the Octonions. the Sedenions (pairs of Octonions) and the Leech Lattice (triples of Octonions). For each Pair of Interpenetrating Triangles, each Single Triangle corresponds to Mt. Meru.

Each Pair of Interpenetrating Triangles corresponds to a Mogan David.

A Mogan David expands to form a Sri Yantra.

This Sri Yantra is a more symmetrical modification of Sri Yantras from two different web pages. It has 4 Sides, corresponding to the 4 dimensions of physical spacetime of the D4-D5-E6-E7-E8 physics model and to Octonion coassociative squares. The 2 Border lines of the 4 Sides correspond to the 2 Quaternionic 4-dimensional Spaces that form the 8-dimensional Octonions, and so to the 8 Directions.

The Sri Yantra has a Center, which, combined with the 4 Sides, corresponds to the Five Elements.

The Sri Yantra has an Outer Lotus of 16 Petals, corresponding t o two half-spinor representations of the Spin(8) Lie algebra and to the first-generation fermions of the D4-D5-E6-E7-E8 physics model. The 3 Border Rings beyond the Outer Lotus Petals correspond to the 3 generation of fermions of the D4-D5-E6-E7-E8 physics model.

The Sri Yantra has an Inner Lotus of 8 Petals, corresponding to the vector representation of Spin(8) and to the 8-dimensional spacetime of the D4-D5-E6-E7-E8 physics model prior to dimensional reduction.

The Sri Yantra has 9 Triangles, with each triangle corresponding to an Octonion associative triangle.

The first 6 triangles, 3 pairs, correspond to:

• the Octonions O (red pair of reflexive interpenetrating triangles),

• the Sedenions OxO (red and green pairs of reflexive interpenetrating triangles), and

• the Leech Lattice OxOxO (red, green, and blue pairs of reflexive interpenetrating triangles),

which in turn correspond to the Lie algebras E6, E7, and E8, and to the 3 generations of fermions in the D4-D5-E6-E7-E8 physics model;

The remaining 3 triangles, one gold pair and one purple triangle, correspond not to pairs of reflexive interpenetrating triangles, but to Octonion associative triangles. Each triangle therefore represents an entire Octonion algebra containing that associative triangle, and the 3 triangles together represent how the 3 Octonions of OxOxO are related to one another.

The gold pair of triangles corresponds to the two mirror image Octonion half-spinor representations of Spin(8). The purple triangle corresponds to the Octonion vector representation of Spin(8). The Sri Yantra has a Central Vertex.

The 9 Triangles have 27 Vertices, corresponding to the 22 Hebrew letters plus 5 Finals.

The 27 Triangle Vertices correspond to the 27-line Configuration whose symmetry group is the Weyl Group of the 78-dimensional Lie algebra E6.

The 27 Triangle Vertices also correspond to the 27-complex-dimensional space E7 / (E6xU(1)).

The 27 Triangle Vertices plus the 1 Central Vertex correspond to the 28-dimensional adjoint representation of Spin(8), which in turn corresponds to the 12+16 = 28 gauge bosons, Higgs mechanism, and propagator phase of the D4-D5-E6-E7-E8 physics model. The 28 Vertices also correspond to the 28-quaternionic-dimensional space E8 / (E7xSU(2)).

The 28 Vertices plus the 16+8 = 24 Lotus Petals form the 52-dimensional Lie algebra F4, to which the Lie algebras E6, E7, and E8 are related by the Freudenthal-Tits Magic Square.

The construction of this Sri Yantra is based on the Golden Ratio of the Great Golden Pyramid.

It is so constructed that the distance from the lower corners of the 2x1 rectangle to the top and bottom vertices of the Sri Yantra is PHI = (1/2)(diagonal of rectangle + 1) = (1/2)(sqrt(5) + 1). Since PHI^2 = PHI + 1, the radius of the circle is sqrt(PHI^2 - 1^2) = sqrt ((PHI + 1) - 1) = sqrt(PHI).

Let D be the distance between the base lines of the two largest triangles. Then the highest and lowest base lines are at distance D above and below the base lines of the two largest triangles. Everthing else is constructed symmetrically from those basic elements.

Leech lattice

In mathematics, the Leech lattice is an even unimodular lattice Λ₂₄ in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by John Leech (1967). It may also have been discovered (but not published) by Ernst Witt in 1940.

Characterization

The Leech lattice Λ24 is the unique lattice in E24 with the following list of properties:

• It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1.
• It is even; i.e., the square of the length of each vector in Λ24 is an even integer.
• The length of every non-zero vector in Λ24 is at least 2.

The last condition is equivalent to the condition that unit balls centered at the points of Λ24 do not overlap. Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls that can simultaneously touch a single unit ball. This arrangement of 196,560 unit balls centred about another unit ball is so efficient that there is no room to move any of the balls; this configuration, together with its mirror-image, is the only 24-dimensional arrangement where 196,560 unit balls simultaneously touch another. This property is also true in 1, 2 and 8 dimensions, with 2, 6 and 240 unit balls, respectively, based on the integer lattice, hexagonal tiling and E8 lattice, respectively.

It has no root system and in fact is the first unimodular lattice with no roots (vectors of norm less than 4), and therefore has a centre density of 1. By multiplying this value by the volume of a unit ball in 24 dimensions, pi^12/12!, one can derive its absolute density.

Conway (1983) showed that the Leech lattice is isometric to the set of simple roots (or the Dynkin diagram) of the reflection group of the 26-dimensional even Lorentzian unimodular lattice II25,1. By comparison, the Dynkin diagrams of II9,1 and II17,1 are finite.

Applications

string theory

The vertex algebra of the two-dimensional conformal field theory describing bosonic string theory, compactified on the 24-dimensional quotient torus R24/Λ24 and orbifolded by a two-element reflection group, provides an explicit construction of the Griess algebra that has the monster group as its automorphism group. This monster vertex algebra was also used to prove the monstrous moonshine conjectures.

Constructions

Using the binary Golay code

Using the binary Golay code

Due to some of the symbols in this part of the Wikipedia page I have turned it into a picture which is on the right.

Using the Lorentzian lattice II_25,1

The Leech lattice can also be constructed as W^'/W where W is the Weyl vector: (0, 1, 2, 3,...,22, 23, 24; 70) in the 26-dimensional even Lorentzian unimodular lattice II25,1. The existence of such an integral vector of Lorentzian norm zero relies on the fact that 1² + 2² + ... + 24² is a perfect square (in fact 70²); the number 24 is the only integer bigger than 1 with this property. This was conjectured by Édouard Lucas, but the proof came much later, based on elliptic functions.

The vector (0, 1, 2, 3,...,22, 23, 24) in this construction is really the Weyl vector of the even sublattice D24 of the odd unimodular lattice I25. More generally, if L is any positive definite unimodular lattice of dimension 25 with at least 4 vectors of norm 1, then the Weyl vector of its norm 2 roots has integral length, and there is a similar construction of the Leech lattice using L and this Weyl vector.

Based on other lattices

Conway & Sloane (1982) described another 23 constructions for the Leech lattice, each based on a Niemeier lattice. It can also be constructed by using three copies of the E8 lattice, in the same way that the binary Golay code can be constructed using three copies of the extended Hamming code, H8. This construction is known as the Turyn construction of the Leech lattice.

As a laminated lattice

Starting with a single point, Λ₀, one can stack copies of the lattice Λ_n to form an (n + 1)-dimensional lattice, Λ_n+1, without reducing the minimal distance between points. Λ1 corresponds to the integer lattice, Λ2 is to the hexagonal lattice, and Λ3 is the face-centered cubic packing. Conway & Sloane (1982b) showed that the Leech lattice is the unique laminated lattice in 24 dimensions.

As a complex lattice

The Leech lattice is also a 12-dimensional lattice over the Eisenstein integers. This is known as the complex Leech lattice, and is isomorphic to the 24-dimensional real Leech lattice. In the complex construction of the Leech lattice, the binary Golay code is replaced with the ternary Golay code, and the Mathieu group M24 is replaced with the Mathieu group M12. The E₆ lattice, E₈ lattice and Coxeter–Todd lattice also have constructions as complex lattices, over either the Eisenstein or Gaussian integers.

Using the icosian ring

The Leech lattice can also be constructed using the ring of icosians. The icosian ring is abstractly isomorphic to the E8 lattice, three copies of which can be used to construct the Leech lattice using the Turyn construction.

Using octonions

Using octonions

Due to some of the symbols in this part of the Wikipedia page I have turned it into a picture which is on the right.

Symmetries

The Leech lattice is highly symmetrical. Its automorphism group is the Conway group Co₀, which is of order 8 315 553 613 086 720 000. The center of Co₀ has two elements, and the quotient of Co₀ by this center is the Conway group Co₁, a finite simple group. Many other sporadic groups, such as the remaining Conway groups and Mathieu groups, can be constructed as the stabilizers of various configurations of vectors in the Leech lattice.

Despite having such a high rotational symmetry group, the Leech lattice does not possess any hyperplanes of reflection symmetry. In other words, the Leech lattice is chiral. It also has far fewer symmetries than the 24-dimensional hypercube and simplex.

The automorphism group was first described by John Conway. The 398034000 vectors of norm 8 fall into 8292375 'crosses' of 48 vectors. Each cross contains 24 mutually orthogonal vectors and their negatives, and thus describe the vertices of a 24-dimensional orthoplex. Each of these crosses can be taken to be the coordinate system of the lattice, and has the same symmetry of the Golay code, namely 2¹² × |M₂₄|. Hence the full automorphism group of the Leech lattice has order 8292375 × 4096 × 244823040, or 8 315 553 613 086 720 000.

Geometry

Conway, Parker & Sloane (1982) showed that the covering radius of the Leech lattice is √2; in other words, if we put a closed ball of this radius around each lattice point, then these just cover Euclidean space. The points at distance at least √2 from all lattice points are called the deep holes of the Leech lattice. There are 23 orbits of them under the automorphism group of the Leech lattice, and these orbits correspond to the 23 Niemeier lattices other than the Leech lattice: the set of vertices of deep hole is isometric to the affine Dynkin diagram of the corresponding Niemeier lattice.

The Leech lattice has a density of pi^12/12!=0.001930.   Cohn & Kumar (2009) showed that it gives the densest lattice packing of balls in 24-dimensional space. Henry Cohn, Abhinav Kumar, and Stephen D. Miller et al. (2016) improved this by showing that it is the densest sphere packing, even among non-lattice packings.

The 196560 minimal vectors are of three different varieties, known as shapes:

• 1104 vectors of shape (4²,0²²), for all permutations and sign choices;
• 97152 vectors of shape (2⁸,0¹⁶), where the '2's correspond to octads in the Golay code, and there is an even number of minus signs;
• 98304 vectors of shape (-3,1²³), where the changes of signs come from the Golay code, and the '3' can appear in any position.

The ternary Golay code, binary Golay code and Leech lattice give very efficient 24-dimensional spherical codes of 729, 4096 and 196560 points, respectively. Spherical codes are higher-dimensional analogues of Tammes problem, which arose as an attempt to explain the distribution of pores on pollen grains. These are distributed as to maximise the minimal angle between them. In two dimensions, the problem is trivial, but in three dimensions and higher it is not. An example of a spherical code in three dimensions is the set of the 12 vertices of the regular icosahedron.

Spheres

In the mid-1960s, John Leech created a lattice that gives the tightest lattice packing of spheres in 24 dimensions. Then in the late-1960s, John Conway analysed the symmetry of this lattice and discovered three previously unknown sporadic groups (exceptional ‘symmetry atoms’), as described in my book Symmetry and the Monster.

The points of the Leech Lattice are the centres of spheres of equal diameter each of which touches 196,560 others — the maximum possible in 24 dimensions. Since there are 24 dimensions, each lattice point is specified using 24 coordinates, which can be determined using the Witt design, a pattern using a set of 24 symbols, in which certain subsets of 8 symbols, called ‘octads’, play an important role.

Take one sphere centred at the origin, so the coordinates of its centre are all zero. The centres of the 196,560 neighbouring spheres split naturally into three subsets of sizes 97,152 + 1,104 + 98,304 = 196,560.

The subset of size 97,152. This number is 2⁷×759. There are 759 octads in Witt’s design and for each one there are 2⁷ lattice points. The coordinates of each point are plus or minus 2 in the positions of an octad, and zero elsewhere; the number of minus signs is even.

The subset of size 1,104. This number is 2²×276. There are 276 ways of choosing two coordinates from twenty-four: each of these two coordinates is plus or minus 4, and the other twenty-two coordinates are zero.

The subset of size 98,304. This number is 2¹²×24. One coordinate is plus or minus 3, the others are plus or minus 1. There are 2¹² sign choices, all coming from the Golay code.

This shows that all these 196560 points are equidistant from the origin.

The distance of a point from the origin, when squared, is the sum of the squares of its coordinates—this is Pythagoras’s theorem generalized to n dimensions. For each of the 196,560 points specified above, the sum of the squares of its coordinates is 32.

E8 octonions TOE

The following information is from theoryofeverything.org so check out that website for more information.

Fibonacci/Pascals triangle patterns

I just saw an interesting post by Lucien Khan on some Theological numerology (gematria) as it relates to the Modulo 10 patterns of Fibonacci, I verified the pattern and added a few more as it relates to the repeating Modulo 9 and 10 patterns, Pascal Triangle and E8.

Numbers are highlighted when divisible by:

• 60 are in red
• 30 are in green
• 15 are in blue
• 12 are in magenta

This is all tied to the Pascal Triangle with its 2^n binary (Clifford Algebra) representation, E8 and the number 4!=24 and Octonions.

All of these structures, including the 3 dimensional geometry of the Platonic solids (and its big brother, the 4D non-crystallographic H4 group geometry) are related to E8 via a folding matrix I determined a few years ago.

Introducing the sedenion fano tesseract mnemonic

The Sedenion “Fano Tesseract” Mnemonic is an extension of the “Fano Cube” idea introduced by John Baez in his much cited blog post. The Fano Cube identifies each valid each triad by a hyper-plane which intersects the e_0 node.

Notice in this VisibLie_E8 output for the pane #3 “Fano Visualization Demonstration”, there are 35 sedenion triads, 7 of which are from the octonion used as an upper left quadrant base for a Cayley-Dickson doubling (highlighted in red).

The 16 vertices of the tesseract are sorted by the same “triad flattening” process used to construct a consistent Fano Plane Mnemonic for all 480 unique octonion multiplication tables.

As in the Fano Cube, the edges are highlighted in Cyan if they are selected in the n1-n3 buttons. Unlike the Fano Plane and Cube, the edges represented by the split octonion multiplication table columns/rows are not highlighted in red.

While there are 32 edges in the formal tesseract, each valid sedenion triad is identified by a hyper-plane which intersects the e_0 node, which are not necessarily those of the formal tesseract.

G2 as the automorphism group of the (split) octonion algebra

Here we show the output of getG2Null for the example non-Flipped E8 #164)

This post describes the derivation of G2 automorphisms for each of the 480 unique octonion multiplication tables, as well as each of the 7 split octonions (created from negating the 4 row/column entries which are not members of the split #, which is an index to one of the 7 triads that make up the octonion).

The Exceptional Lie Algebra/Group G2 is identified by its Dynkin diagram and/or associated Cartan Matrix (also shown here with its Hasse diagram)

The particular octonion multiplication table selected in the example referenced above is associated in my model as a “Non Flipped E8 #164”. The multiplication table in various formats (IJKL, e_n, and Numeric) is (the image on the right)

Each set of 21 upper triangle pairs of octonion elements (utOct) has 14 “derivations” which are null (alternatively, the NullSpace of the derivation vectors of utOct has 7 which are NOT null…) The 14 null pairs define G2 for a SPECIFIC octonion multiplication matrix.

For each of 7 non-null derivations there is a triple of D_{x1, y1}=D_{x2, y2}+D_{x3, y3} related derivations.

There is a 7×21 matrix describing the non-NullSpace for the octonion given in the Wolfram MathSource reference above along with the 7 non-null triples.

Emergent patterns:

1. For each of the 7 non-null derivation triples (the first column of the 7 rows), there are precisely 3 (of the 7) split octonions that don’t share the exact same non-null triple derivation pair as the parent (non-split) octonion (these deviations are shown in column 2 thru 4, along with the associated Fano plane diagrams for the parent and 7 split octonions of E8 #164).
2. Each non-null derivation triple contains 6 of 7 indices, and each row is missing a different index (highlighted in green before the column 1 triple). The sequence of missing element numbers in each row of the given example follows the row number.
3. It is only the equality relationship (signs) of the non-Null triple that change in 3 of the 7 splits, not the D_{x,y} itself.
4. There are always 2 pairs of 2 D_{x,y} which are negative (i.e. with a -1 entry in the NullSpace matrix located above the “==”) that occur across the row.
5. There is always a common positive D_{x,y} (i.e. with a +1 entry in the NullSpace matrix located below the “==”) in each entry of the row (colored magenta). The common positive entries in the 7 rows suggest a “distinguished” non-null indicator for the 14=21-7 G2 automorphism.
6. All octonion multiplication matrices have the first 3 rows of distinguished entries of {6,7},{5,7} and {5,6} in that order (i.e. the last 2 rows of utOct).
7. There are several possible choices for G2 automorphism sets of 14 elements within each of 480 octonions based on the 7 non-Null entries. Interestingly, there are the 4 sets of rows in utOct which sum to 14 elements, specifically rows {{1, 2, 4}, {1, 2, 5, 6}, {1, 2, 4, 6}, {2, 3, 4, 5}}). The distinguished entries of the flipped E8 #164 example below (as in the MathSource post referenced above) suggests a G2 created by rows {1,2,4} of utOct. Although, not all G2 sets must use complete rows as in this example.

(more information linked below)

Detail visible E8 demonstration description

Chaos

This demonstration combines several demonstrations related to chaos theory, fractals and Navier-Stokes computational fluid dynamics (CFD) as it relates to a theory of a Universe that has a superfluid space-time.

Fano

Octonions use a range of 1-7 numbered triplet subsets. A Fano plane is a collection of seven of these triplets (or triads) where no two share two numbers. A Fano plane is a configuration with 30 distinct numberings, when the numbers 1 to 7 (a flattened set of triads) are assigned to the seven points. For each of the 30 distinct numberings, there are eight possible reversals of triplet order, which are identified here as bitwise (hexadecimal) sign masks. These 240=30*8 triplets can also be inverted by inverting the eight sign masks, giving a total of 480 Fano planes. These are visualized here as either a directed 2D graph or a non-directed 3D cubic form. The product of two distinct units equals the third unit, such that the three form immediately connected vertices of the graph.

The octonions can be constructed from quaternions by means of the Cayley-Dickson construction. Like quaternions and the lower-dimensional complex numbers, octonions have one real number (e0=1). Octonions have seven imaginary units e1-e7, or in IJKL format {I,J,K=IJ,L,IL,JL,KL=IJL} whose multiplication tables can be encoded using the Fano plane mnemonic, shown here in both numeric and symbolic form.

For any three octonions a, b, and c, the associator is (a*b)*c – a*(b*c). The associator measures the non-associativity of those three octonions. Select triples of octonionic units to compute their associator. Observe that their associator vanishes on immediately connected triples of octonionic units. Such triples, together with the unit element, form the quaternionic sub-algebras of octonions.

The odd-4 graph is constructed by taking the 35 triplets that share no values. The graph is the odd-4 graph (black edge lines). If a 1-7 point Fano plane is removed from the odd-4 graph, the result is the Coxeter graph (cyan edge lines). If connections to the fifteen Fano planes of a single color are added instead, the result is the Hoffman-Singleton graph (yellow edge lines). The Fano plane index (fPi) driven by fp_gen and fp_color determines the type (cyan or yellow) of graph.

Select any of the 480 permutations for the Fano basis using the same bit-wise quantum particle selections (described in the “Particle” section below, with the addition of the “flip” check-box to select the second E8 linked octonion for the particle). Red or blue numbers in the flattened triads indicate it is one of 240 E8 related permutations needing those colored node positions flipped (or reversed) to create a proper Fano plane from it. Blue indicates it is a member of the 128-vertex C8 subgroup of E8 and red for subgroup D8 with 112 vertices. Node colors are changed from cyan to yellow for button bar selected associator dimensions {n1,n2,n3}. A selection to view the symbolic multiplication tables in IJKL form is also included. Octonion math examples are shown below the numeric and symbolic multiplication tables.

Each new selection of the 480 different bases changes the octonion multiply and commutator results and verifies the conjugate, norm, and division results are the same. This verifies proper octonion operation as shown in the octonion math examples.

The construction of the 480 octonion sets of 7 triples uses converted C source code from Donald Chesley of Davidson Laboratories, Stevens Institute of Technology.

The list of 30 Fano planes (made up of sets of 7 quaternion triads) can be determined algorithmically by brute force (checking each of the 35 possible unique triples against the 21 possible unique pairs and finding the 30 sets of 7 that don’t share more than one number).

The 16 sign mask (sm) sets can also then be algorithmically determined for each index into 30 sets of Fano plane quaternion triads (by taking each of the 30 Fano planes and brute force checking which of the 128=2^7 sign reversal bit patterns keep the associator=0). Each sign reversal constitutes an arrow direction reversal in the Fano plane mnemonic.

It turns out that all of the 128=2^7 possible bit patterns for reversing the Fano plane arrows is used. These are grouped into 2 sets of 64 with an “anti” (or 0< ->1 bit reversal) operation. The 64 are found to be grouped into 8 sets of 8 which get associated (or indexed) with the 30 Fano planes. The tally of these associations between the 8 sign mask sets and the 30 Fano plane sets is {{1,2},{2,2},{3,7},{4,3},{5,3},{6,7},{7,3},{8,3}}. That is, there are 4 of the 8 that associate with 3 Fano planes each. There are 2 of the 8 that associate with 7 Fano planes and 2 that associate with 2 of them.

A split of the 30 triad sets into 15=7+8 (or D8+C8) provides the indicator on where to assign the “excluded” permutations. They get the 0 color and 0 generation assignment – the same logic applied to the excluded E8 vertices or particles. It becomes trivial to apply a naive sequential assignment of 30 Fano planes into two sets of 7/8 assigned to the 4 bits associated with generation and color in the Lisi model.

The binary patterns for the 8 sign mask groups are easily seen and can also be constructed (rather than brute force determined) using right/left symmetry patterns. Interestingly, this construction uses a Clifford algebra Cl(0,8) primitive idempotent e{2,3,5,8} (and its inverse e{1,4,6,7}) as a base. These sets are assigned (R)ight and (L)eft based on the pattern of adding Reverse@Range@3 to 1 (or subtracting Range@3 from 8) respectively. More interestingly, these are also used in assigning the octonions to the E8 particle assignments – specifically, they index to the 4 spins (up/down, left/right) with the set assigned by the pType (neutrino/electron lepton or up/down quark split). These two groups of 4 are also used to index the 15=7+8 (or D8+C8) split associated with generation and color bits. This is a common pattern between the split of octonions into the Cayley-Dickson quaternions (the first 4 a-p & spin bits of the first quaternion e0-e3 and the second 4 color-generation bits of the second quaternion e4-e7).

The Fano plane indices also “pair up” when assigning the 480 octonion permutations into the 240 E8 permutations with D8Pairs={{1,2},{4,7},{8,10},{15,16},{17,19},{23,24},{25,26}} and C8Pairs={{3,5},{6,9},{11,12},{13,14},{18,20},{21,22},{27,29},{28,30}}. The pairs are always found together in the E8 particle assignment as “flipped” and “non-flpped” Fano planes (except the last of the C8 Pairs {28,30}, which have an anomalous flip pattern associated with the 3rd generation blue quarks).

To reach the doubling of the 512=480+32 “excluded” octonion permutations, I added what I call a 9th “flip” bit parameter. This correlates well with A Zee’s approach to grand unification using a 9D SO(18) group theory. It turns out that the pattern required for flipping works in unison with the a and p bits in a binary logic relationship. As the demonstration clearly shows, the p bit flips the two sets of four sm bits. This is the key to a eureka moment where I created the notion of a 3rd spin type out of the pType bit (in/out or z-x or yaw intrinsic rotation or spin). Also note: Charge-Parity-Time (CPT) conservation law (symmetry) considerations also strongly link these a, p, and spin bits.

The binary logic relationship with the “flip” bit and the sm bits clearly create two 4 bit quaternion sets. The first quaternion is the anti(e0=1 Real) bit combined with 3 spin bits {e1,e2,e3} or {I,J,K=IJ}. The second quaternion is 2 generation bits {e4,e5} or {L,IL} combined with 2 color bits {e6,e7} or {JL,KL=IJL}. While this double set of quaternions was already obvious to me from the E8 construction, it was good to confirm it with octonion integration! It seems that this pair of quaternions could be visualized as Hopf Fibrations on 7-manifolds.

There are 7 sets of split octonions for each of the 480 “parent” octonions. The 7 split octonions are identified by selecting a triad. The complement of {1,2,3,4,5,6,7} and the selected triad list leaves 4 elements which are the rows/colums corresponding to the negated elements in the multiplication table (highlighted with yellow background). The red arrows in the Fano Plane indicate the potential reversal due to this negation that defines the split octonions.

This demonstration now also includes the ability to generate the sedenions from octonions through the application of the Cayley-Dickson doubling procedure. As in the animated Fano Cube, the sedenion display includes the generation of an animated Fano Tesseract mnemonic visualization which steps through highlighting the vertices/edges of the 34 sedenion triads.

These allow for the simplification of Maxwell’s four equations which define electromagnetism (aka.light) into a single equation.

It correlates well with my theory relating time to octonions, charge to quaternions and 3-space to complex (imaginary) numbers. This idea also validates the dimensional considerations that started me down this path 15 years ago.

This demonstration combines and extends Wolfram demonstrations from Wolfram’s Ed Pegg Jr.

Dynkin

This demonstration lets you create and manipulate multiple Dynkin and Coxeter-Dynkin diagrams with some ability for topological pattern recognition of known Simple Lie Groups (up to rank 8 ) and its designated type, including finite, affine, hyperbolic, and very extended.

You can use the drop down “Dynkin diagrams” to get various diagrams and geometric permutations.

It also provides for manipulating, recognizing, and naming of the 2^rank (binary) Coxter-Dynkin geometric permutations on uniform polyhedron and their Wythoff construction operator naming. These permutations are indicated by filled nodes. When more than one diagram is created, they can be interpreted as a geometric Cartesian product, such as a DuoPrism.

The Cartan matrix which defines the Lie algebra (along with the Coxeter, and Schlafli matrices) are calculated directly from the last diagram entered. If there are more than one diagrams, all with the same rank, the diagram matrices are dotted together.

The main window is a clickable pane. Clicking more than 1.5 cells away from all nodes creates a new diagram. Clicking less than 1.5 cells away from all nodes creates a node and a single line to its nearest neighboring node. Clicking on a node in the last diagram created toggles the node fill on/off and changes the geometric permutation (unless it is an already filled node that is adjacent to an unlinked node, which implies the creation of an affine loop). Clicking on a line between nodes in the last diagram created toggles the symmetry angle and/or the directionality between its nodes. Please note that when more than one diagram is in the clickable pane, only the last diagram is active. Clicking on previous diagrams will create a new overlapping diagram. For better Dynkin diagram topology recognition, select the affine level early and create the linear (A type diagram) nodes before any off-linear nodes (for D and E diagrams).

E8 lie group

The first drop down list provides for the selection of Coxeter (and other) projection symmetries. Most names indicate Simple Lie groups. The Bn and Cn groups are the same as the ones for Dn+1. (e.g. the menu label “D+1=BC3” applies to D4, B3, and C3 simple Lie algebras or groups and will be projected to their Petrie projection with the basis vectors associated with this menu selection). The next set of projections indicate the number of perimeter vertices in parenthesis. The PASCAL# projections display the number of vertices in a column which is has the same number as that numbers row of the Pascal triangle.

The next two menus provide “Favorites” that are preset combinations of over 50 variables to produce beautiful projections of E8 and its subgroups. Their short names attempt to indicate what is in the resulting image. For example, the references to “n Cube Pascal” in menu 2 refers to a projection that creates columns of vertices (as in the PASCAL# in the Coxeter projection menu). The references to Petrie refer to the corresponding Petrie projection of that n dimensional cube. Favorite names using the word “binary” imply it uses a purely binary data set from the Pascal triangle instead of SRE E8.

The counts of selected vertices, edges, triality lines, and clicked vertex edge lines are shown. The check boxes select major elements of what gets displayed in the output window:

• axes selects whether to show the 8 axes (and the frame bounding box)

• edges selects whether to show the edges between each vertex (subsets of edge lengths is selected in the HMI pane 8)

• physics shows the assigned physics particle shape/color/size (vs. dots)

• labels shows the particle labels

• dimLocs checkbox is used to activate the dimension locators in 2D (deactivates particle interactions).

Checking the “axes locator” box forces it into 2D mode with axes shown. Instead of being a clickable pane, you can now drag the locators on the basis vectors to show new projections on the horizontal and vertical 2D plane using “H” and “V”. The controls to change the azimuthal (Z) basis vector are not available for manipulation in this Demonstration.

Viewing the static or animated 2D (or 3D) positions of selected vertices (or their theoretically assigned fundamental physics particle) is accomplished by projecting them using a dot product of 2 (or 3) 8D basis vectors. These can be manipulated with controls in the HMI pane 8. 3D displays use native Mathematica rotation and zoom options (cntrl or alt-click-drag)

The triality buttons “T1”, “T1p”, “T2”, “T2p”, “T4”, and “T4p” select different sets of triality rotations using one or more of the selected rotation matrices. These are named for the number of sixths of a unit circle involved in selected hyperplanes of a higher-dimensional rotation. The suffix of “p” indicates that it uses the “physics” data set, which is a rotation of the SRE E8 vertices that let A. G. Lisi calculate particle charges on a single generation of fermion particles. The physics data set is activated in some “Favorite” selections.

Animations are generated by increasing the number of (time) steps and selecting the type of 8D “flight path” (Translational or Rotational and Spin – controlled in the HMI pane 8).

It is interesting to note that the 8D vertex positions are always static. What is moving is the 2D (or 3D) perspective inside an 8D “information space”. The “information space” in this case is made up of the highly symmetric vertex positions of the Split Real Even (SRE) E8 Lie Group assigned to fundamental physics particles based on their quantum numbers. The specifics of each flight path are depend ant on the 3 basis vectors and dimensions selected for creating the hyper – planes about which they rotate. These are also manipulated with controls in the HMI pane 8.

Mouse over the shapes in the result to show the detail particle information relating to E8, Cellular Automata and physics (as shown in the “Particle” pane 4).

The output window is a clickable pane (if in 2D mode). If the physics box is not checked (showing vertex dots), clicking on or near a vertex calculates and shows the connecting vertex edges of the (nearest) clicked vertex. The edges are color coded by overlap count. If the physics box is checked (showing assigned particle shapes), clicking calculates triads of edges that include the particles whose E8 vertex values sum to the clicked particle. Clicking an already highlighted vertex now shows only that edge (or triad of edges).

Edge calculation is done with a modified code snippet from Eric W. Weisstein, “600-Cell.” 

Particle

This demonstration creates an interactive pane for identifying fundamental particles in physics through the selection of quantum parameters. The information displayed in the output window is also displayed in the ToolTip on mouse-over of the particles in the E8 window.

It includes the well known Standard Model particles (leptons, quarks, W and Z bosons) as well as a small group of theoretically predicted particles related to a sector of Higgs boson(s). There are also included in this demonstration a very small group of predicted particles which are loosely based on an E8 Lie group extension of the Standard Model originated by A.G. Lisi.

The quantum parameters used to identify individual fundamental particles based on an 8 bit pattern are as follows:

• 1 bit (a)nti-particle vs. the normal particle,
• 1 bit (p)article type identifying the Neutrino vs. Electron for Leptons and the Up vs. Down for 3 generations of Quarks,
• 2 bit (c)olor for the 4 colors w,r,g,b (where (w)hite is used to identify particles with no color),
• 2 bit (s)pin for the 4 combinations of Up/Dn and Left/Right,
• 2 bit (g)eneration for 4 generations (with 0 for Bosons and 1,2 and 3 for the {e, [Mu], [Tau]} generations of Standard Model (SM) Fermions).

This unique 8 bit pattern for each particle is ordered as {a, p, c2, c1, s2, s1, g2, g1} into a base 2 number from 0 to 255. It is also used to generate the display for Steven Wolfram’s New Kind of Science (NKS) Cellular Automata (CA) patterns, which may be related to the theory.

The 2^8=256 particles identify the known or predicted SM Fermions (192), Bosons (30 with 18 in the “Higgs Sector”), and the 8 dimension generator vertices (plus 8 excluded “anti-dimension” vertices). There are 18 remaining (Boson) particles added which are identified using an approach loosely connected with the theoretical work of A.G. Lisi in arxiv.org/pdf/0711.0770.

This approach to particle assignments allows for them to be associated with the 240 vertices of the split real even (SRE) E8 Lie group plus the 8 Orthoplex (dimensional) generator vertices (and 8 anti-dimensional generator vertices typically excluded from E8). The lexicographic ordering of the SRE E8 is consistent with a Cl(8) Clifford algebra, which has a familiar sequence of row 9 of the Pascal Triangle.

The particle families are split into 5 rows {two rows being the familiar 192 Fermion (l)eptons and (q)uarks, the 48 Bosons in the next two rows [Omega]g and [Phi][CapitalPhi], and a row for the 16 (Ex)ception dimension generator particles}. This demonstration applies a rigorous particle reference label for symbolic pattern matching to facilitate this identification. Each particle is given a unique 2D and 3D reference symbol. The symbol’s size, shape, color, and shade are used to uniquely identify the bitwise pattern. There are 2D particle shapes {circle, square, triangle} and their corresponding anti-shapes {pentagon, diamond, inverted triangle} and 3D shapes {sphere, cube, tetrahedron} and anti-3D shapes {dodecahedron, icosahedron, octahedron}. The particle’s SM generation alters the size in accordance to the tendency for mass to increase with increasing generation. The particle mass and lifetime (if known), are given by the ParticleData PDG curated data set for particles. The colors are, of course, used to identify the color content. The shade is modified with spin.

CKM

There are added capabilities to view both the PNMN and CKM unitary triangle matrices, print and reference my ToE Neutrino mass predictions, which now accomodate the Koide relationships in particle masses.

Hadron

There are added capabilities to view the 4 and 6 Quark Hadrons as well as list all composite particles made up of the same quark content, including buttons to access their decay modes. To this list is a menu selectable Tooltip display of any particle property, and with decay modes shown with a button click.

This also integrates with particle selection on pane 3.

ATOMic Elements

A 2D/3D/4D (with s-p-d-f colors) Stowe-Janet-Scerri Periodic Table. This is organized by the four quantum numbers {n,l,m, and s=spin +/- 1/2}. The legend for how this maps to the traditional periodic table is shown to the right.

The isotope selection is not used with this UI. There is a selection for the n quantum number. In 2D display mode, this shows elements in the layer with that n quantum number. If in 3D mode, it will show all layers up to that selected n quantum number. As in the older version, a dropdown menu is available to show the detial element information in the tooltip. A slider for exploding the 3D display helps seperate the element layers. There is also a radio button to switch between the Stowe periodic table (Z=n) and the Scerri version (Z=n+l) when in 3D mode. Also available is a selectable option for showing the 2D/3D spherical harmonics of the Shroedinger electron probability density |[Psi],n,l,+/-m]|^2, based on whether it is in 2D or 3D display mode.

Interestingly, this periodic table has 120 elements, which is the number of vertices in the 600 Cell or the positive half of the 240 E8 roots. It is integrated into VisibLie_E8 so clicking on an element adds the particle who’s ordered E8 algebra root is that atomic element number. The negative roots can be thought of as “anti-elements”. The physics checkbox enables showing the associated 2D or 3D physics particle for each element.

E-Infinity theory

E-infinity theory is based on Ultimate L, the K-Theory, Penrose fractal Tiling, E-Infinity, Hardy's Quantum Entanglement, Witten's M-Theory and Topological Quantum Field Theory.

Golden Geometry of E-Infinity Fractal Spacetime

The golden ratio is at the core of our fractal universe of infinite dimensions that looks and feels 4-dimensional. Dr. Mae-Wan Ho

Real processes do not happen at points in a spacetime continuum

Alfred North Whitehead (1861-1947) lived through what must have been a most exciting period in Western science. The very fabric of reality – the flat, smooth, and static absolute spacetime of the Newtonian universe – is being thoroughly ruffled, if not torn asunder, by Albert Einstein’s theories of special and general relativity at large scales and by quantum mechanics at the smallest.  Out goes the proverbial objective scientific observer outside nature, to be supplanted by the knower irreducibly quantum-entangled with the known, possibly with all entities in the entire universe.

These surprising and exhilarating lessons from nature became the basis of Whitehead’s perennial, universal philosophy (a cosmogony), initiating a new age of the organism that has inspired generations of scientists, including me. Most enlightening though least understood was his argument on why the mechanical laws of classical physics and differential calculus fail to describe real processes. Apart from the all-important knowing experiencing organism left out of account, real processes have characteristic times (durations) and spaces (volumes). Nothing, absolutely nothing happens at a point in an instant. In other words, spacetime is not a smooth continuum; it is discrete and discontinuous, as quantum physics already discovered at the smallest scale. Indeed spacetime can more easily be conceived as being created by action, by processes that naturally occur in quantum jumps.

Unfortunately, mathematics - a major tool of thought for western science, especially physical science - had lagged somewhat behind physics. Both relativity theory and quantum theory inherited the predominant mathematics of classical mechanics.

British theoretical physicist Roger Penrose’s monumental tome, The Road to Reality, A Complete Guide to the Laws of The Universe is indeed a “tour de force”, as advertised. It charts the heroic efforts of mathematical physicists to make sense of post Newtonian physics; their ingenuities and successes, as well as failures. More than a century later today, the dream of uniting the two great theories of quantum physics and general relativity has remained stubbornly beyond our grasp. And one main reason is that the two theories are divided by a common ground: a differentiable spacetime manifold.

Differentiable spacetime and differential calculus assume objects have instant point-like ‘simple locations’ in space and time. It is fundamentally at odds with physical reality, and may well be creating more problems than it can solve.  But there is still a tendency to put mathematics (and mathematical physics) on a pedestal as though it has an ‘objective’ independent existence to which physical reality must conform. This neo-Platonist view championed by Penrose is surely a case of misplaced concreteness. It has been more than 80 years since Austrian mathematician and logician Kurt Gödel (1908-1978) proved his incompleteness theorems, which essentially state that a set of axioms both complete and consistent is nevertheless unable to guarantee a complete and consistent theory, let alone one that applies to physical reality.

Nevertheless, mathematics is a wonderful, exquisite tool for thought, and I have come to appreciate it more and more in my continuing quest for the meaning of life and the universe. The mathematics of non-differentiable and discontinuous spaces, which has the potential to describe physical reality more authentically, is among the most significant discoveries/ inventions beginning in the latter half of the 19^th century though it did not really blossom until well into the 20^th century.

I shall describe an intuitive and ingenious theory of spacetime that makes full use of the newer mathematics. Before proceeding, please read Box 1 for a quick guide to mathematical terms that you will be bumping into a lot in the rest of this article; and return to it as often as is necessary.

mathematical terms

Set theory

The branch of mathematical logic about collections of mathematical objects. The modern study of set theory was initiated by German mathematicians Georg Cantor (1845-1918) and Richard Dedekind (1831-1916) in the 1870s.

Closed and open sets

A closed set contains its own boundary, its complement is an open set which does not contain its boundary.

Borel set

A Borel set is any set in a topological space that can be formed from open sets ( or equivalently from closed sets) through the operations of countable union, countable intersection, and relative complement. A countable set is one with the same number of elements as some subset of the set of natural numbers. The elements of a countable set can be counted one at a time, and although the counting may never finish, every element of the set will eventually be associated with a natural number. Union, denoted by ⋃ of a collection of sets is the set of all distinct elements in the collection. Intersection of sets, denoted by ⋂,  is the set that contains only elements belonging to all the sets. The relative complement of set A in B is the set of elements in B but not in A.

Bijection

Bijection is a mapping both one-to-one (an injection) and onto (a surjection); it is a function that relates each member of a set S (the domain) to a separate and distinct member of another set T (the range), where each member in T also has a corresponding member in S.

Cantor Set and Random Cantor Set

The classical triadic Cantor set (named after inventor Georg Cantor) is obtained by dividing the unit line into three  equal parts, discarding the middle part except for its end points, and repeating the operation with the two remaining parts ad infinitum. In the random version, it could be any of the three parts that is discarded at random after each division.

Space

A space is a set with some added structure.

Metric space

A metric space is a set where a notion of distance (a metric) between elements of the set is defined. The metric space that corresponds most closely to our intuitive understanding of space is the 3-dimensional flat Euclidian space.

Riemannian space

A topological space with metric properties that can be defined continuously from point to point (hence also called a Riemannian manifold) including standard non-Euclidean spaces, ie., spaces that are not flat.

Topological space

A topological space is a set of points and a set of neighbourhoods for each point that satisfy a set of axioms relating to points and neighbourhoods. The definition of a topological space relies only on set theory and is the most general notion of a mathematical space.

Topological dimension

Topological dimension is the dimension of a topological space. For example, a point has topological dimension 0 whereas a line has topological dimension of 1, closing up the line into a circle makes no difference; it still has a topological dimension of one. Similarly, a flat sheet has a topological dimension of 2, the same for the surface of a cylinder, a sphere or a doughnut.

Menger-Urysohn dimension

The Menger-Urysohn dimension is a generalized topological dimension of topological spaces, arrived at by mathematical induction. It is based on the observation that, in n-dimensional Euclidean space Rⁿ, (n−1)-dimensional spheres (that is, the boundaries of n-dimensional balls) have dimension n−1. Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets.

Hausdorff dimension

The Hausdorff dimension generalizes the notion of dimension to irregular sets such as fractals. For example, a Cantor set has a Hausdorff dimension of ln2/ln3, the ratio of the logarithm to the base 2 of the parts remaining to the whole after each iteration.

Fractal

A fractal is a mathematical set that typically displays self-similar patterns, and has fractional dimensions instead of the usual integer, 1, 2, 3, or 4. Geometric examples are branching trees, blood vessels, frond leaves etc.

Chaos theory

The study of dynamical systems with locally unpredictable behaviour that is highly sensitive to initial conditions, but are nevertheless globally determined, such that the trajectories are confined within a region of phase space called ‘strange attractors’.

Transfinite

A term coined by Cantor; it means beyond finite, but not necessarily the absolute infinite.

E infinity fractal spacetime looks and feels 4-dimensional

The idea that spacetime is fractal originated with Canadian mathematician Garnet Ord who coined the term ‘fractal spacetime’, using a model in which particles are confined to move on fractal trajectories.  Independently, French astrophysicist Laurent Nottale proposed a scale-relativity theory of fractal spacetime. As Ord reminds us, American quantum physicist Richard Feynman (1918-1988) had already pointed out that the paths of quantum mechanical particles look more like non-differentiable curves than straight lines when examined on a fine scale.

It is Egyptian-born Mohamed El Naschie, however, who has taken us furthest towards a coherent theory of physical reality that is also closest to our intuitive notion of organic spacetime. For precisely the same reasons, perhaps, El Naschie has attracted admiration and antagonism in equal measure.

As an aside, the journal Nature initiated a smear campaign against El Naschie in 2008 with an article written by its German correspondent that was filled with insinuations and innuendoes, if not outright lies.  It accuses El Naschie of “self-publishing” papers of “poor quality” without proper peer review in a theoretical physics journal of which he has been editor-in-chief. In fact, El Naschie has had hundreds of papers published in other journals; and his prolific output continued throughout the subsequent four-year period during which he brought a libel case single-handedly against Nature. El Naschie turned down all attempts by Nature to settle out of court, until the court ruled against him in 2012. By this time, Nature had spent £5 million in legal fees to defend itself. Given Nature’s shameful record in libeling myself and other scientists over the hazards of genetic modification, I became all the more determined to find out about El Naschie’s work; and have been suitably rewarded as a result.

El Naschie trained and practiced as an engineer while indulging in his hobby of cosmology, and produced a startling new theory of spacetime, which soon took over his life entirely. E-infinity, as he calls it, is a fractal spacetime with infinite dimensions. Yet its Hausdorff (fractal dimension) is 4.236067977... In other words, at ordinary resolution, it looks and feels 4 dimensional (three of space and one of time), with the rest of the dimensions ‘compacted’ in the remaining 0.236067977… “fuzzy tail”.

E infinity fractal spacetime represented as nested four dimensional hypercubes

Think of a four dimensional hypercube with further four dimensional hypercubes nested inside like Russian dolls [. In fact, the exact Hausdorff dimension is 4 + f³, where f = (√5-1)/2, the golden ratio.  Particularly suggestive is that the dimension 4 + f³ show the following self-similar continued fraction which sums to precisely 4 + f³ at infinity.

The 4-dimensional hypercube is the Euclidean representation of the E-infinity universe. It is a challenge to represent E-infinity it in its proper non-Euclidean form.

The E-infinity universe is mathematically a random Cantor set (the simplest fractal) extended to infinite dimensions, and the remarkable result is that the limit of this infinite extension is no larger than 4 + f³.

Fractals and dimensions in the derivation of E infinity spacetime

Figure 2 depicts some of the steps involved in deriving E infinity universe. We start from the 1-dimensional Cantor set, the unit line, which is divided into three equal parts; take out the middle, and carry on the same operation on the two parts remaining, up to an infinite number of steps. At the end, there should be nothing left but isolated points (Cantor dust). Its Menger-Urysohn dimension is 0, but amazingly, its Hausdorff (fractal) dimension is ln2/ln3, for the usual triadic Cantor set; but is f = (√5-1)/2 = 0. 61803398.. for the random Cantor set, in which the section removed is not necessarily the middle one, but c0uld be any of the three parts at random. This important result, proven by American mathematicians Daniel Mauldin and S C Williams in 1986, is what makes it possible to derive the E-infinity universe with all its remarkable properties, as we shall see. Another mathematical property of the Cantor set is that its cardinality (number of points or elements) is exactly the same as the original continuous line. Thus, the Cantor set is a perfect compromise between the discrete and the continuum; it is a discrete structure that has the same number of elements as the continuum.

From the 1-dimensional Cantor set, we can construct the higher dimensional spaces. The 2-dimensional version is the Sierpinski gasket, with random Hausdorff dimension of 1/f = 1.61803398..; the 3-dimensional version is the Menger sponge, with random Hausdorff dimension 2 + f = 2. 61803398.. The Sierpinski gasket and Menger sponge are both well-known geometric shapes. Not so the 4-dimensional version, with random Hausdorff dimension 4 + f ³ = 4.23606797…, where only an artist’s representation is given. The 4-dimensional version is the same as the E infinity universe constructed from an infinite number of random Cantor sets (as will be made clear later). Note that the diagram is space-filling: it consists of spheres of different sizes representing space-times at different scales so that the entire volume is completely packed. This space-filling is analogous to, if not the same property as quasi-periodic Penrose tiling in 2 dimensions of Euclidean (flat) space where the golden ratio is key (see The Story of Phi Part 1, SiS 62), as it is in E infinity universe. And incidentally, branching processes based on the golden ratio are also space-filling [12], as are spiral leave arrangement patterns with the golden angle between successive leaf primordial (see [13] Watching the Daisies Grow, SiS 62).

El Naschie has presented multiple formal derivations of E infinity spacetime, I shall give the one relying on the mathematical properties of Borel sets to which Cantor sets belong. It is so simple that even a non-mathematician can understand it.

The expectation value of the Hausdorff dimension of the Cantor set extended to infinity is simply a sum over n , for n = 0 to n = ∞, of n multiplied by the Hausdorff dimension of the random Cantor set raised to the power n. This is more concisely encapsulated in the following equation.

where the left hand side is the expectation value of the Hausdorff dimension of the Cantor set extended to infinity; the superscript in d_c⁽⁰⁾ refers to the Menger-Urysohn dimension of the random Cantor set, which is 0, while the corresponding Hausdorff dimension d_c⁽⁰⁾ is f. The summing up of the infinite number of terms gives the answer 4 + f³ exactly.

Now, the intersection rule of sets, also known as the bijection formula, which relates the Menger-Urysohn dimension to the Hausdorff dimension, shows that we can lift d_c⁽⁰⁾  to any Menger-Urysohn dimension n to arrive at the correct Hausdorff dimension d_c⁽ⁿ⁾ as follows:

d_c⁽ⁿ⁾ = (1/ d_c⁽⁰⁾) ^n-1 

Taking d_c⁽⁰⁾ =  f, and lifting to n = 4 dimensions gives

d_c⁽⁴⁾ = (1/ d_c⁽⁰⁾) ^4-1 = 4 + f³ = 1/f³ = 4.236067977… = ⟨Dim E ˗ ∞⟩_H

In other words, the expectation value of the Hausdorff dimension of E infinity universe is the same as that of a universe with a Menger-Urysohn dimension of 4. That is why E infinity is a hierarchical universe that looks and feels 4 dimensional.

E-infinity, Penrose tiling, and Fibonacci sequence

The E-infinity universe is intimately connected with Penrose tiling and the Fibonacci sequence through E-infinity algebra. In his important book Noncommutative Geometry, French mathematician Alain Connes identified Penrose’s fractal tiling as a mathematical quotient space (a space of points ‘glued together’ by an equivalence relationship), with the following dimensional function:

• D(a, b) = a + bf

Where a, b are integers (whole numbers) and f = (√5-1)/2. Writing D_n (a_n, b_n) with the Fibonacci sequence, it is easy to see that starting with D₀ = D (0, 1) and D₁ = D (1, 0), the following dimensional hierarchy is obtained:

• D₀ = D (0, 1) = 0 + f = f
• D₁ = D (1, 0) = 1 + (0)f = 1
• D₂ = D (0+1, 1+0) = 1 + f = 1/f
• D₃ = D (1+1, o+1) = 2 + f = (1/f)²
• D₄ = D (1+2, 1+1) = 3 + 2f = (1/f)³
• D₅ = D (2+3, 1+2) = 5 + 3f = (1/f)⁴

D_n(a_n, b_n) = D {(a_n-1, a_n-2) + (b_n-1 + b_n-2)}f = (1/f)^n-1 

Not that for D₄ (dimension 4), the Fibonacci number is (1/f)³ = 4 + f³, exactly the Hausdorff dimension of a Menger-Urysohn 4-dimensional space.

By induction, D_n = (1/f)^n-1 which is identical to the bijection formula from E-infinity algebra (see Eq (3) above):

d_c⁽ⁿ⁾ = (1/f) ^n-1

The summing of random Cantor sets to infinity is very suggestive of spacetime being created or constructed by actions over all scales, from submicroscopic to macroscopic and beyond, which is close to what I have envisaged, following Whitehead and German theoretical physicist Wolfram Schommers.

E-Infinity Spacetime, Quantum Paradoxes and Quantum Gravity

E-infinity fractal spacetime may resolve major quantum paradoxes and take us further towards the unification of quantum physics and general relativity.

E-infinity fractal spacetime is constructed from an infinite number of random Cantor sets, and is therefore infinite-dimensional. Nevertheless it has a Hausdorff dimension of 4 + f³ = 4.236067977..., the same as that of a space with 4 Menger-Urysohn dimensions, which is why E-infinity looks and feels like a 4-dimensional universe that we actually live in. More importantly, at least as far as physics is concerned, it may resolve some major quantum paradoxes and take us towards quantum gravity and the dream of uniting quantum physics with general (and special relativity).

El Naschie has used E-infinity algebra to solve quantum conundrums, presenting multiple derivations if not proofs. Let’s look at some of them.

Wave particle duality, the zero and the empty set

The two-slit experiment

The pattern on the photographic plate with one slit open (left) and two slits open (left)

The wave-particle duality of light and matter is among the greatest paradoxes in quantum mechanics, as clearly exhibited in the two-slit experiment. A source of light is placed in front of a screen with two narrow slits that allow light to pass through to strike a photographic plate behind the screen.

When only one slit is open, the light passes through the single slit and hits the photographic plate forming an image of the slit, which when examined closely, consists of microscopic grains of silver deposited as the individual photon particles impinge on the plate. When both slits are open, light passes through both slits and form an interference pattern on the photographic plate, behaving like typical waves up until the moment the wave strikes the plate.

One can reduce the light level to such an extent that only single photons are emitted one at a time; and if both slits are open, the single photon would still pass through both slits at once and create an interference pattern on the photographic plate, which is detected after a suitably long period of time when sufficient single photons have gone through. If one puts a detector behind a slit to try to find out which slit the single photon has gone through, then the pattern reverts to that of the single slit. It is as though the observer must remain ignorant as to which slit the photon passes through for the quantum wave to survive intact. Any observation or attempt to gain information will ‘collapse’ the wave function, so only the particle remains.

The way this paradox can be resolved is through the E-infinity algebra of the zero and the empty set.

Remember that a one-dimensional Cantor set has both a Menger-Urysohn dimension and a Hausdorff dimension. The random Cantor set has a Menger-Urysohn dimension of 0 while its Hausdorff dimension is f. It is called the zero set, and El Naschie proposes to identify this zero set with the particle:

dim (particle set) =  P(d_MU, d_H) = P (o, f)

where d_MU is the Menger-Urysohn dimension and d_H is the corresponding Hausdorff  dimension.

The quantum wave on the other hand, is identified with the empty set, as it is devoid of matter and momentum as it is spread out ultimately over the entire universe:

dim (wave set) =  W(d_MU, d_H) = W (-1, f²)

How does one get to this empty set? You get there by a process of induction as follows.

What is the dimension of a 3D cube boundary? It is clearly an area, i.e., a surface of 2D. That means 3D(cube) – 1 = 2D (surface) 

Next, what is the dimension of the boundary of a 2D surface? It is obviously a one-dimensional line. 2D(surface) -1 = 1D (line)

Finally, what is the dimension of the boundary of a line? This is evidently a zero dimensional point 1D(line) – 1 = 0D (point)

By induction, one could write a general expression for the above in the form of: D(boundary) = n – 1 

where n is the dimension of the geometrical object for which we would like to know the dimension of its boundary. By induction, using this formula, we can derive the boundary of a point:

D(boundary) = D(point) – 1 = 0 – 1 = -1

This is the dimension of the classical empty set as deduced for the first time by Soviet Russian mathematician Pavel Urysohn (1898-1924) and studied by Austrian American mathematician Karl Menger (1902-1985), which is the origin of the Menger-Urysohn topological dimension. Its Hausdorff dimension is obtained by the bijection formula you have already come across earlier as f².

d_c⁽ⁿ⁾ = (1/ d_c⁽⁰⁾) ^n-1

d_c^(-1) = (1/ d_c⁽⁰⁾) ^n-1

= (1/f)^-2 = f²

The quantum wave is thus identified as the boundary of the particle, which is completely empty, with a Menger-Urysohn dimension -1, but nevertheless possesses a Hasudorff fractal dimension f². The empty set is de facto two identical things at the very same time, the surface or the topological neighbourhood of the zero set as well as being the guiding quantum wave. The zero set is a Cantorian fractal point as well as the quantum particle guided by the ‘ghost’ wave. This may be understood in a very elementary manner, according to El Naschie, by recalling that the wave is the surface of the particle and it is evident that the smaller, say a sphere, the larger is the ratio between its surface area  and its volume. When the volume tends to zero, the ratio will tend to infinity.

Now, on taking measurement on this particle-wave packet, we inevitably enter into the wave and consequently into the domain of the empty set. So the empty set becomes non-empty and “practically reduced or jumps to at best, a zero set.”

Wave particle duality and dark energy

Continuing in the same vein, El Naschie proposes that Einstein’s famous formula E = mc2 consists of two parts. The first part is the positive energy of the quantum particle modeled by the topology of the zero set. The second is the absolute value of the negative energy of the quantum Schrödinger wave modeled by the topology of the empty set (see above). The latter is the missing dark energy (actually dark energy and dark matter) of the universe accounting for 95.45 % of the total energy-matter in agreement with the findings from the Wilkinson Microwave Anisotropy Probe and the supernova cosmic measurement awarded the 2011 Nobel Prize in Physics. The dark energy of the quantum wave cannot be detected in the normal way because measurement collapses the quantum wave. Several recent attempts to detect dark matter with sophisticated detectors have failed No Dark Matter Detected Yet (SiS 62), which is potentially devastating for the standard model of cosmology that depends on postulates of dark matter and dark energy.

The Menger-Urysohn dimension and Hausdorff dimension of a random Cantor set are [o, f]. The dimensions of the complement (gaps) are [-1, and  1 – f = f²], as established above.

Raising both the f (points) and f² (gaps) set to the Kaluza-Klein 5 dimensional spacetime gives f⁵ (volume) and 5f² (boundary) and respectively equal to 4.5 % and 95.5 % of Einstein’s energy, the latter corresponding to dark energy/matter.

(Different estimates of dark matter and dark energy vary somewhat. According to the latest figure, dark energy plus dark matter constitute 95.1 % of the total content of the universe)

The Kaluza-Klein 5 dimensional spacetime attempted to unify gravity with electromagnetism. It originated with German mathematician physicist Theodor Kaluza (1885-1954) who extended general relativity to a five-dimensional spacetime to include the electromagnetic field. Swedish theoretical physicist Oskar Klein (1894-1977) later propose that the extra fourth spatial dimension is curled up, or compactified,  in a circle of very small radius, so that a particle moving a short distance along that axis would return to where it began. This compactification of dimensions is now widely employed in string theories that attempt to give a realist explanation of why the universe looks and feels 4 dimensional (see Box). There may well be a more intimate relationship between the Kaluza-Klein spacetime and E-infinity spacetime, which El Naschie has not pointed to, though it is implicit in his work. Recall that the embedding dimension for E-infinity space-time of Hausdorff dimension 4 + f³ = 4. 236067977 is also 5, and the fuzzy tail of 0. 236067977 is the compactified (∞ - 4) dimensions. If we equate 0.236067977 = f³ with the sum of ordinary matter and energy that we can detect, and ask what percentage it is of the transfinite dimension of the Kaluza-Klein 5 dimensional spacetime, 5+ f³, we get 4.5 %. In other words, f³/(5+ f³) = .04721359…. ≈ 4.5 % which gives 95.5 % dark matter/energy.

Thus, the compactified spacetime dimensions is likened to the super-particle of the universe, while the rest is the empty-set ‘halo’ and quantum wave that collapses as soon as it is measured. If that is the case, neither dark matter nor dark energy would be detected.

Quantum entanglement and E-infinity

Quantum entanglement is another quantum paradox where the quantum states of different particles are inseparable even when they are far apart, so that measuring the state of one instantly determines the state of the other. This has been demonstrated, not just for particle/antiparticle pairs that are created together and so expected to be correlated, but also for particles independently generated, that when made to interact, will become entangled. This is an extremely active field of research that has implications for quantum communication and computation, and even teleportation.

In 1992, theoretical physicist Lucien Hardy, then at Oxford University in the UK, now at Perimeter Institute for Theoretical Physics Waterloo, Canada, proposed a thought experiment that predicted a probability of quantum entanglement at ~ 9 %. This was re- derived and interpreted by American solid state physicist David Mermin at Cornell University, New Haven, Connecticut who first noticed the connection to the golden ratio, and also American theoretical physicist Daniel Styer at Oberlin College, Ohio.

It turns out that this probability of quantum entanglement can be derived using pure logic and E-infinity theory, as El Naschie showed.

Consider two particles; the probability to be at point 1 is d₁, while the probability of being at point 2 is d₂. Consequently, the probability of not being at 1 is 1 - d₁, and not being at two is 1 - d₂. The total probability of being all of the above at the same time - a distinctive quantum property of entanglement - is:

P₁ = d₁(1 - d₁)d₂(1 - d₂)

The simplest local realism (classical, non-quantum property) is the negation of being at 1 and 2 at the same time. This non-entangled state is,

P₂ = 1 - d₁d₂ 

A relative probability is defined as

P = P₁/P₂

P = [d₁(1 - d₁)d₂(1 - d₂)]/(1 – d₁d₂) 

Next, find an extremum (maximum or minimum) for P where P is a maximum, i.e., when the change of P with d₁ and d₂ are both zero:

∂P/d₁ = ∂P/d₂ = 0

This results in a cubic algebraic equation with three solutions for d₁ = d₂ = 1, -1/f and f. The third solution f is a confirmation of E-infinity theory where d_c⁽⁰⁾ = f is both the Hausdorff dimension of a random triadic Cantor set as well as  the topological probability of finding a Cantorian point in this set. To obtain Hardy’s result explicitly, we insert d₁=d₂= f into Eq (7) for P and find P = f⁵ ≈ .0909829…

This probability can also be derived directly from E-infinity theory as follows.  The probability of finding a point in the E infinity space is f³, the inverse of 4+ f³, the Hausdorff dimension of E-infinity spacetime.

The general formula for the dimension is

⟨n⟩ = (1 + d_c⁽⁰⁾)/(1 - d_c⁽⁰⁾) 

Consequently the probability is

1/⟨n⟩ = (1 - d_c⁽⁰⁾)/(1 + d_c⁽⁰⁾) 

The probability of two entangled points in this space is (d_c⁽⁰⁾)², i.e., the Hausdorff dimension of the points multiplied together.

Consequently, the total probability of entanglement is the product of 1/⟨n⟩ and (d_c⁽⁰⁾)².

P = (d_c⁽⁰⁾)²[(1 - d_c⁽⁰⁾)/(1 + d_c⁽⁰⁾)]

On maximizing P or from E-infinity theory, d_c⁽⁰⁾ = f and 1/f. Inserting d_c⁽⁰⁾ = f, we get Hardy’s result again P = f⁵

To obtain the result P = 0 of the classical (no entanglement) expectation, we set d_c⁽⁰⁾ = d_c⁽¹⁾ = 1.

From the preceding derivations, it seems clear that Hardy’s result is geometrically and topologically rooted in the Cantorian nature of microquantum spacetime.  This is in line with Einstein’s general relativity representation of mass and energy as geometry (curvature) of spacetime.

Quantum entanglement and dark energy

Perhaps the most audacious feat with E-infinity algebra is a new quantum gravity formula predicting the measured cosmic energy content of the universe, by fusing the probability of quantum entanglement with Einstein’s E = mc² formula, thereby unifying relativity and quantum mechanics.

Essentially, Einstein’s celebrated equation is multiplied by a scaling parameter g = f⁵/2, where f⁵ is half the probability of quantum entanglement.

Einstein’s equation is replaced by an effective quantum gravity formula:

E_QR = g_QR mc² = [1/2(1 – b)/(1 + b)] mc² 

which recovers Newton’s kinetic energy E = ½ mv², when b is set to 0 or 3; while setting b = 4 + f³ or b = f results in Einstein’s non quantum but relativistic formula E = mc².

In particle physics, it is proposed that at the Planck energy scale of around 1.22 x10¹⁹ GeV, which corresponds by mass energy equivalence E = mc² to the Planck mass 2.17659 x 10^-8 kg, the quantum effects of gravity becomes strong and expected to be comparable to other forces; and it is theorized that all the fundamental forces are unified at that scale, though the exact mechanism of this unification remains unknown.

At the other extreme of huge distances, quantum corrections accumulate and relativity cannot ignore quantum effects. Hence Einstein’s E = mc² is a candidate for major modification when the quantum mechanical effect of entanglement is taken into account. The new equation is E_QR = (f⁵/2)mc² representing a synthesis of Newton, Einstein and quantum mechanics.

Quantum relativity theory is seen as the intersection of three fundamental theories of physics, hence predicts only 4.5 % of the mass-energy in Einstein’s E = mc² equation, in accordance with cosmological measurement, with the rest being dark matter-energy.

There are four steps in generalizing E = mc² of special relativity to quantum relativity or effective quantum gravity formula E_QR =g_QR = (mc²)/22.1803989.. (the relevance for the denominator will be explained later).

The first step transforms space, time and mass to a probabilistic space, time and mass using quantum mechanics, leading to E_p = (P/2)mc² where P is the quantum entanglement probability. Second, a special form of E_R = gmc² is derived where g is a function of a special relativity correction for the unit interval b. Third, E_p is equated to E_R to find the exact value of b for which E becomes a maximum.

The probability of quantum entanglement of two quantum particles in Lucien Hardy’s thought experiment as derived in Eq (7) above is:

P = [p₁ (1 –p1)p₂(1 – p₂)]/(1 - p₁p₂)

For p₁ = p₂ = d, the expression simplifies to:

P = d² (1 –d)/(1 + d)

Now, introducing the probabilistic transformation:

• Space (X) →  xp

• Time (T) → tp

• Mass (M) → mp

Inserting into Newton’s kinetic energy, one finds the following probabilistic energy for v → c (velocity approaching the speed of light)

E_p = ½ mp (xp/tp)² = ½ mp (v→ c)²

Substituting  p in Equation (3) for the probability of entanglement in Eq (2) gives:

E_p = ½ d²[(1 –d)/(1 + d)] mc² 

From relativity theory three phenomenological effects are well known: time dilation, length contraction and increase in mass when velocity approaches the speed of light. This is represented by some unspecified correction factor b.

• x → x(1 – b)
• t → t(1 + b)
• m → m(1 + b)

Consequently, Newton’s relativistic kinetic energy becomes

E_R = ½ mc² (1 + b) [(1 – b)/(1 + b)]²  = ½ [(1 – b)²/(1 + b)] mc² 

The next step to arrive at an effective quantum gravity E is to require

E_P  =  E_R 

Therefore,

d²[(1 –d)/(1 + d)] = [(1 – b)²/(1 + b)]

This is only possible for d = b and

b² [(1 – b)/(1 + b)] = [(1 – b)²/(1 + b)] 

which leads to the simple quadratic equation

b²  + b  - 1 = 0

and the solutions: b₁ = f = (√5-1)/2 the golden ratio, and b₂ = -1/f.

In the fourth and final step, the correct expression for the quantum gravity formula is given by setting d = b = f into Eq.

E_QR = ½ f ²[(1 – f)/(1 + f)] mc²

= ½ f ²f ³ mc²  = (f⁵/2) mc²  =  mc²/22.18033989

= (probability of quantum entanglement)/2) mc²

Eq (19) E_QR = mc²/22.18033989 approximates E_QR = mc²/22, and can be given different simple interpretations. First, the factor 22 can be regarded as what remains of the 26 dimensions of string spacetime of the original bosonic strong interactions theory (see Box) after subtracting Einstein’s 4 dimensions. Then, the 26-4 = 22 dimensions “dilute” the energy content of the cosmos and reduce it from 100/22 ~4.5 %, in agreement with the cosmological measurement of the three 2011 Nobel Laureates.

This interpretation of energy content is not quite the same as that in the previous sections. Intriguingly, the link to quantum entanglement not only combines quantum theory with gravity, but suggests that gravity may be a form of quantum entanglement (though El Naschie himself has not said so).

Golden ratio and transfinite corrections to integer dimensional symmetry groups of string theories

The significance of the golden ratio lies in the extension of the concept dimensions, originally applying only to integers, to non-integers. All high energy particle physics theories are currently constrained to Lie algebra groups with integer dimensions that are also a differentiable manifold. Just as the concept of dimension has been expanded from integer topological to a non-integer Hausdorff dimension, the same could be done for a Lie manifold. In fact, irrational small corrections to integer Lie group dimensions would be a blessing for the overall model. Gauge anomalies (anomalies under transformation) in the standard model arise from the clash of integer symmetries (invariant properties under transformation). With transfinite (beyond the finite, but not absolute infinite) irrational corrections, i.e., fuzzy tail added or subtracted from the original integer dimension, we get fuzzy symmetry group dimensions that fit together harmoniously and eliminate gauge anomalies.

A simple example is the vital role of the golden ratio in Penrose tiling of the plane, without which, nothing would fit, and we end up with gaps or overlaps. Similarly, of the five Platonic solids, two of the most important, the icosahedron and dodecahedron, depend on the golden ratio. British-born Canadian geometer Harold Scott MacDonald Coxeter (1907-2003) extended those to four dimensional ‘polytopes’, which could not be constructed without the golden ratio. Now, the skeleton upon which the most important symmetry group used in superstring theory, the E8 exceptional E group, could be constructed from two 600 cells Coxeter polytopes by sliding a smaller one inside a larger one. The result is the E8 Gosset, which is once again based on the golden ratio.

For example, the dimension of the special orthogonal group SO(n) is

Dim (SO(n)) = n (n – 1)/2. For n = 4, the dimension is 6. If we take n to be the Hausdorff dimension of the E-infinity manifold, i.e., 4 + f³, the dimension is 6.854102.., i.e., 0.852102.. larger. This transfinite tail makes all the difference. Take 20 copies of the Lie group dimension for n = 4 + f³, then the total dimension will be 137.082039. This is the exact transfinite version of the inverse electromagnetic fine-structure constant, a, a fundamental dimensionless physical constant that characterizes the strength of the electromagnetic interaction, the currently accepted value is 7.29734257 x 10^-3 

In a series of detailed calculations which is being the scope of this article, El Naschie shows that the inverse electromagnetic fine-structure constant and the golden ratio are embedded in the mass spectrum of the elementary high energy particles. To El Naschie “the mass spectrum of high energy particles resembles a non-linear dynamical symphony where everything fits with everything else. We could start virtually anywhere and derive everything from everything else.”

Infinite-dimensional lie group

Lie groups are often defined to be finite-dimensional, but there are many groups that resemble Lie groups, except for being infinite-dimensional. The simplest way to define infinite-dimensional Lie groups is to model them locally on Banach spaces (as opposed to Euclidean space in the finite-dimensional case), and in this case much of the basic theory is similar to that of finite-dimensional Lie groups. However this is inadequate for many applications, because many natural examples of infinite-dimensional Lie groups are not Banach manifolds. Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite-dimensional Lie groups no longer hold.

The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix Lie in Lie group. On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix Lie in Lie algebra are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined.

Some of the examples that have been studied include:

• The group of diffeomorphisms of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, whose central extension the Virasoro algebra (see Virasoro algebra from Witt algebra for a derivation of this fact) is the symmetry algebra of two-dimensional conformal field theory. Diffeomorphism groups of compact manifolds of larger dimension are regular Fréchet Lie groups; very little about their structure is known.
• The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity.
• The group of smooth maps from a manifold to a finite-dimensional Lie group is an example of a gauge group (with operation of pointwise multiplication), and is used in quantum field theory and Donaldson theory. If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are (more or less) Kac–Moody algebras.
• There are infinite-dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have simpler topological properties: see for example Kuiper's theorem. In M-theory, for example, a 10 dimensional SU(N) gauge theory becomes an 11 dimensional theory when N becomes infinite.

Planck Spherical Units(PSU)

Sphere packing: The planck electromagnetic waveforms are omnitriangulated spheres due to the tetrahedral packing of the 64 Tetrahedron (/Isotropic vector) matrix/grid.

The universe is being created from energy in the form of little spherical photons or Planck scale units (PSU)s AKA Planck spherical units (Morphogenetic field/Partiki grid spinning), the PSUs oscillation creates the torus effect.

Space is completely full of energy but appears to us to be completely empty because the geometry of the space itself is so perfectly balanced and polarized in every way possible, that it forms what Buckminster Fuller called a "vector equilibrium". In this kind of geometry, no matter where a force comes from, there is an equal and opposite structure in the space itself that counter-balances that force so that in the end, all forces cancel-out and it seems like there is nothing there, creating a "zero point field" or "vacuum", otherwise know an aether, plenum, quantum foam, source field, Mana, Prana, Chi, space, etc..

The only 3D geometry where every line (or vector) is the same length, including vectors from the exact middle point of the geometry (singularity) to the vertexes of each of the edges: cuboctahedron (in light blue in the center followed by dark blue, purple, red and orange).

The first octave of cuboctahedron is formed when a total of 64 tetrahedrons come together, the next octave of cuboctahedron forms at 512 tetrahedrons, followed by 1024 and continues to infinity forming the scalar and holofractal structure of the fabric of spacetime itself at the sub-quantum scale being visualized in this image.

What are 'Planck Spherical Units'?

What are 'Planck Spherical Units'? Nassim Haramein, December 6th, 2013:

In 1899 Max Planck, father of quantum theory, defined a Planck unit as a result of the renormalization of the electromagnetic spectrum of a black body radiation by the utilization of a quantum of action.(angular momentum), which was confirmed by experimental results and led to the quantum orbitals and the Bohr atom.

Planck quantities are natural units, free of any arbitrary anthropocentric measurements, are based on fundamental physical constants and can be defined as, for example, the time it takes a photon to travel one Planck length, which is the Planck time. In other words, Planck units are the smallest vibrations of the electromagnetic field.

In quantum field theory it was found that the structure of spacetime (the vacuum) itself vibrates with significant levels of energy initially calculated to be infinitely dense and renormalized by using the Planck units as a cutoff wavelength to define its smallest possible vibration.

In the words of John Wheeler, the physicist that popularized the term black hole and collaborator of Einstein:."The vision of quantum gravity is a vision of turbulence, turbulent space, turbulent time, turbulent spacetime... spacetime in small enough regions should not be merely 'bumpy', not merely erratic in its curvature; it should fractionate into ever-changing, multiply-connected geometries. For the very small and the very quick, wormholes should be as much a part of the landscape as those dancing virtual particles that give to the electron its slightly altered energy and magnetism."

In Quantum Gravity and the Holographic Mass, Haramein defines the fluctuation of the vacuum in terms of the Planck Spherical Unit.(PSU), a unit of vacuum oscillations with the diameter of a Planck's length and a radius of a half wavelength. In his generalized approach to the Holographic Principle, he tiles the surface area and volume of an object with intersecting spheres rather than squares or cubes, to better represent natural structures. In other words, he calculates how many Planck spheres.(as energy nodes).would fit on the surface horizon and interior of the black hole if they were space filling.(which means that the little spheres intersect with each other). This intersecting wave pattern is reminiscent of holographic interference wave patterns.

As an analogy, one could think of stones hitting the calm surface of a pond, each generating a little ripple and all the ripples intersecting in interference patterns. In this analogy, Haramein calculates how many little stones are hitting the surface or are inside the volume making a little ripple and calculates the ratio relationship between the two.

His result appropriately outputs the correct mass for any black hole, the mass of protons, the gravitational coupling constant, the interaction time of the action producing the so-called strong force and its range. All of these results support the PSU approach to be the correct definition of the discrete structure of vacuum fluctuations.

The Oscillating Planck Spherical Unit

This is from an article written by Matt Lorusso: Either there is 1 physical unit that fills space and connects all points within it or there is not. It’s one or the other and nothing in between. Here I’ll make the case that there is and it has intrinsic features that are evident in the atomic particles and the natural structures they form.

In 2012, Nassim Haramein published a paper titled “Quantum Gravity and the Holographic Mass.” There he defines a unit called the Planck Spherical Unit (PSU). Its diameter is equal to the Planck length, the singular unit of space derived from combining the fundamental physical constants for gravity G, the speed of light c, and Planck’s constant h. Its mass is equal to the Planck mass which is derived from the same set of constants.

The reduced form of the holographic mass solution for the proton mass (mp) and proton radius (rp) in terms of the Planck mass (mP) and Planck length (ℓP). This leads to rp = 0.84123 fm.

With the PSU filling space as a “quantum foam,” he devised a formula for computing the holographic mass of a spherical object by counting the number of (red) PSUs in its volume and dividing that by the number of (white) PSU equatorial disks on the object’s spherical surface area. Multiplying this quantity by the Planck mass gives the object’s holographic mass. He then applied this to the proton mass, which is measured with high precision, and calculated the less precisely measured proton charge radius. The value obtained through this method has been corroborated by the most precise measurements of the proton radius while the Standard Model gives a value that is about 4% too big, which is a significant deviation.

The reduced form of the holographic mass solution for the electron mass (me) and Bohr radius (a0) in terms of the fine-structure constant (α), the Planck mass (mP), and Planck length (ℓP).

The formula was then applied to the electron mass in a subsequent paper titled “The Electron and the Holographic Mass Solution.” At the distance known as the Bohr radius a0, which is the minimum stable distance between a single electron and a single proton, the electron’s holographic mass emerges in proportion with the fine-structure constant α.

So at a distance of a0 from the proton, α is the ratio of the electron’s expected velocity cα to the PSU’s surface velocity c. But this dimensionless constant α, which is equal to about 1/137, appears in multiple contexts related to the atomic and subatomic domains. To take just 2 examples: it is the coupling constant that determines the strength of the interaction between electrons and protons, and it determines in part the electron’s anomalous magnetic moment. Yet despite its ubiquity, α is a quantity of unknown provenance. If space is indeed composed of a fundamental unit, then the successful application of the holographic mass ratio to both the proton radius and Bohr radius establishes the PSU as a legitimate candidate, and we should expect further investigation to shed light on this mysterious quantity of 137.

rp = 0.84153 fm

Although a static PSU is sufficient to account for the holographic mass solutions described above, Haramein often refers to the unit as the “oscillating PSU” or “PSU oscillator” without specifying the nature of this action. As we will see, there is substantial evidence that the oscillation of the PSU occurs within well-defined proportional limits. Let us now hypothesize that the PSU oscillates by a factor of 1/20 the size of its radius, expanding and contracting with 1 oscillation cycle per half rotation of the PSU. PSUs naturally rotate at the speed of photon transmission c. So the regular oscillation of the PSU by a factor of 1/20 not only generates a natural unit of time, but also implies a linear subdivision of space throughout. This is certainly evident in the proportional relationship between the 2 essential distances for which the holographic mass solution applies: the proton radius rp and Bohr radius a0.

mp = 0.93827209 GeV/c² ; me = 0.00051099895 GeV/c²

Dividing the proton radius into 20 subunits, we find exactly (135)(136)(137) of them in the diameter of hydrogen in its ground state. Of course, the numbers 135 and 136 are consecutive with 137, which is the integer approximation of the inverse fine-structure constant. This intrinsic order in the simplest atom is quite striking, yet we might dismiss it as an amusing coincidence if not for the fact that the same proportion of (135)(136)/20 also appears directly in the proton-electron mass ratio (mp/me).

So the PSU provides the physical basis for the mathematical relationships in the holographic mass solution of the proton mass with the proton radius, and the electron mass with the Bohr radius. The ratios of the 2 complementary masses (mp/me) present in the simplest atom and the 2 distances (a0/rp) that define its core spatial dimensions share a common proportion of (135)(136)/20, which may reasonably be inferred to have some physical connection to the PSU. Furthermore, to obtain more precise values for the proton and electron masses we can apply a version of this ratio as follows.

Let’s leave aside the quantity of 22 for now. The ratio of (135)(136)/20² resembles a perfect square. This is important because both masses here have units of GeV/c², in which c² refers to the squared speed of light — a squared unit of space divided by a squared unit of time. As the primary participants in photon exchange, protons and electrons have an integral relationship with light, so perhaps the ratio that defines this universal speed limit is connected physically via the PSU to the numerical ratio that relates these 2 complementary masses.

Since the PSU oscillation factor of 1/20 occurs with perfect regularity, 20² may function here as the squared unit of time. However, despite its evident association with the fine-structure constant, the multiple of (135)(136) is not quite a perfect square nor do we yet understand its connection to the PSU. If the ratio of (135)(136)/20² is functionally equivalent to the squared speed of light in this relationship, converting our arbitrary units of space and time (i.e. meters²/second²) into proportionate terms, then we must be able to attribute a physical basis to the multiple of (135)(136), just as the quantity of 20 corresponds with the PSU oscillation. Let’s continue with the analysis and then return to this idea.

These 2 relations demonstrate the narrow and well-ordered constraint on the proton mass in its natural units. It is extremely improbable that the first 7 digits of the (2*mp) proton mass, a supposedly random number, would match the series 2–0.123456 by chance. But given that fact, it is unfathomable for the electron mass to also reduce the ratio of (135)(136)/20² proportionally to about 0.02345 without any coordination between the two. For both proton and electron masses to be so clearly associated with this series demonstrates their inherent complementarity and hints at the fundamental nature of mass in relation to the PSU.

mμ = 0.1056583755 GeV/c²

Absolute proof this is no mere numerical coincidence is found in the muon-electron mass ratio. The electron has an unstable form known as a muon which has a mass measured with a precision on par with the atomic particles. Due to their lack of composite particles and nearly identical magnetic moments, the difference in mass is the only significant difference between electron and muon. The muon mass (mμ) can be accurately expressed in terms of the electron mass (me) as the surface area of a torus (π²).

Like the proton mass, we again obtain the first 7 digits of the muon mass through an irreducibly simple relation. And because 19/20 is equal to half of 2–0.1, the muon-electron mass ratio has a direct link to both the upper and lower bounds on the proton mass. Furthermore, the emergence of the rational number 20+19/20 in this limit strongly supports the notion that PSUs oscillate by a factor of 1/20, as the PSU radius divided into 20 subunits would have a maximum extent of 21, exactly 1/20 greater than 20+19/20. And the factor of π² is a clear indication that 20+19/20 indeed refers to a radius as the surface area of a quarter torus is equal to π²*R*r.

So with respect to the rest masses of the proton, electron, and electron-related muon, we reject the view of the Standard Model that these are independent physical constants with no meaningful proportionality between them. This conclusion is based not only on the exceptional precision and the economy of terms in the individual equations, but their interrelation and the ability to associate the proportional terms with physical processes in a logical manner.

The highly precise equations above demonstrate an intrinsic order that places specific, proportional limits on the rest masses expressed by electrons and protons. Now, we know that electrons behave as point particles while protons are composite particles. With the universe fundamentally composed of the PSU, we may deduce that an electron conveys a disturbance of a single PSU that travels to the next PSU along its QED-defined trajectory at a rate always less than the speed of light. Naturally, the PSU disturbance represented by a single electron may be collectively present in the proton. Because the ratio of (135)(136)/20 reduces to 918, the proton mass may be expressed in terms of the electron mass as follows.

This formulation presents an intriguing possibility for the composition of a proton. 6 spheres aligned in a plane, all surrounding a central sphere, form a hexagon with 1 sphere per side. A hexagon with 17 spheres per side contains 918 spheres, excluding the central sphere.

mn = 0.93956542052 GeV/c²

The proton mass is obtained when each sphere is a PSU with a disturbance representing approximately double the electron rest mass (2*me). An elegant consequence of this model is that when a proton captures an electron it transforms into a neutron and its new mass is obtained simply by including the central sphere and marginally increasing the 2*me multiplier.

What might be the “disturbance” in the PSUs that would express mass as either me or ~2*me to signify the presence of a massive particle? To transmit photons in accordance with observations the PSUs must spin at 1 rate: the speed of light. A disturbance to this natural rotational rate may thus produce what we perceive to be a massive particle. For instance, a neutrino, which has an immeasurably small mass and typically travels very near the speed of light, would represent a minuscule decrease in the rotational rate that moves from 1 PSU to the next in accordance with the trajectory of the particle; an electron would represent a similar but more substantial decrease; and the nucleons would be hexagonal collections of PSUs with each PSU slowed by about double the amount of an electron.

But this is somewhat backwards. Since the 2–0.123456 relation for the (2*mp) proton mass obtains 7 of its 9 well-established digits — an extraordinary 99.99999% match — we may reasonably infer that, as a first order effect, the overall proton mass mp aligns with this value, yielding its total diminished PSU rotation. Consequently, the 918 PSUs that compose the proton each have a diminished rotation of mp/918; the PSUs that compose a neutron each have a slightly more diminished rotation than those of a proton; and the diminished rotation of a PSU transmitting an electron is about halfway between that of a proton’s component PSU and a PSU with undiminished rotation. The fact that it is not exactly half is analogous to the anomalous magnetic moment in the electron’s g-factor (~-2.002319). It has an expected absolute value of 2, however due to higher order effects it is slightly greater than 2. And the electron’s anomalous magnetic moment is actually greater than its anomalous mass differential with respect to both proton (~2.000166) and neutron (~2.000744). Because the electron’s anomalous magnetic moment can be calculated using corrections from higher order effects, a future area of investigation would be to see if a similar approach can be applied to the anomalous mass differential in protons and neutrons.

Let’s review: The holographic mass solution posits the physical existence of the oscillating PSU to explain the relationship between the proton radius and proton mass, as well as the Bohr radius and electron mass. The ratios of the 2 distances and the 2 masses share a common proportion of (135)(136)/20 which reduces to 918. A perfect hexagon is formed from 918 spheres surrounding a central sphere in a plane. The ratio of (135)(136)/20² that yields exceptionally precise values for the electron and proton masses reduces to 918/20, where both terms have a natural physical basis — the numerator refers to the hexagonal arrangement of PSUs that form a proton and the denominator refers to the oscillation factor of PSUs.

918 is the number of PSUs in the hexagonal proton and 20 is the PSU oscillation factor.

ge = -2.002319304362 ; gn = -3.82608545 ; gp = 5.5856946893

Now if we divide the PSU radius into 20 subunits (like we did for the proton radius), then its regular oscillation produces a photon pathway proportional with a distance of 22π from one end of the PSU to the other. So if a particle of light (photon) is represented by a point on the sphere along the plane of rotation, then for every half revolution around the main radius (20π) and full oscillation cycle (2π), the photon has moved to the opposite end of the sphere and traversed a distance of 22π. This conjecture is supported not only by this equation, modified from earlier, involving the masses of both electron and proton — the primary participants in photon exchange.

But also by the fact that the nucleon masses and g-factors closely align with 7π, which is approximately equal to 22.

Other than the electron g-factor, the Standard Model offers no means of calculating or otherwise understanding the proportional relations between these parameters. Yet here they are in another irreducibly simple and succinct relation that also happens to be dimensionless. It is an elegant expression of the intrinsic order contained within every PSU and, by extension, the atoms that give form to natural structures.

Beyond offering insight into the proportional constraints on the fundamental parameters that define the atomic particles, the projection of the PSU oscillation and its 22π photon pathway appears to produce a stabilizing resonance at increasingly larger scales:

1. The Bohr radius, which we know from its proportionality with the proton radius, is also equal to the electron Compton wavelength scaled by a factor of 1/(2πα), which is about 21.81. So in the lowest energy and most stable state of the simplest atom, the expected length of space separating proton from electron is nearly 22 times greater than the photon wavelength with energy equivalent to the electron’s rest mass energy.

1. 22 protons inside an atomic nucleus form titanium, the metal with the highest strength-to-weight ratio. So of the 94 naturally occurring elements, this 1 forms the strongest molecular bonds per unit of mass.

1. 22 protons also appear in carbon dioxide. Of the prominent atmospheric gases — including hydrogen, helium, nitrogen, and oxygen — carbon dioxide is the heaviest. Yet it’s so stable that the atmospheres of Venus and Mars are about 95% CO2. More importantly though, CO2 provides the carbon necessary to create simple sugars during photosynthesis. It is the only gas absorbed by all plants and produced by all animals on Earth. So to breathe is to break apart and dissolve into the surrounding space 22 protons at a time.

1. The conversion of adenosine diphosphate (ADP) into adenosine triphosphate (ATP) and back again carries an electric potential that powers most cellular activity. The additional phosphate group consists of 40 protons and releases a discrete amount of energy as it breaks away, converting ATP to its more stable form as ADP. This molecule serves as the basic unit of energy transfer in every living cell, and is therefore essential to the metabolic function of all life on Earth. It consists of 220 protons.

1. Energy from the Sun is transformed into food on Earth through the action of chlorophyll. Certain wavelengths of red and blue light excite this molecule and cause it to split water into electrons, protons, and oxygen gas during the light dependent reactions of photosynthesis. At the core of every molecule of chlorophyll is a lone magnesium ion surrounded by a chlorin ring. The structure contains 1 magnesium ion, 20 carbon atoms, 16 hydrogen atoms, and 4 nitrogen atoms totaling 176 protons, which is exactly 8*22 protons. Furthermore, different types of chlorophyll molecules are associated with different side chains attached to the central ring, which alter the spectrum of light absorbed by the molecule. Chlorophyll a is the only type found in all photosynthetic organisms. It contains 2 protons less than 22², a 99.6% equivalence.

Proton number is the most important factor in determining the characteristics of an atom. For molecules it is less determinative, though even if proton number has a weak effect we should expect molecules to resonate with the natural harmonics of the PSU oscillators. Indeed, the pattern is unambiguous and the evidence is overwhelming.

The 5 examples above demonstrate how multiples of 22 convey stability from the simplest atom to successively larger structures, especially as it relates to their ability to store or transfer energy. Biologically, there are 3 molecules that are essential to the flow of energy through the global ecosystem. Chlorophyll absorbs sunlight, ADP receives and transports that energy within the cell, and CO2 provides the basic material to form nutrients that retain this vital energy. These nutrients are then consumed either directly or indirectly by every organism on Earth. The 3 molecules involved in this global process to store and transport energy all contain discrete multiples of 22 protons, with the largest consisting of just 2 protons less than 22². This unusual pattern of stability is evidence at the molecular scale of the oscillating PSU and its 22π photon pathway linking the particles that compose all these structures.

Fixing the problem with the PSU

In the article Matt Lorusso present Haramein’s formulation of the PSU which sets the diameter equal to the Planck length. Although it has no real bearing on the PSU oscillation, which is the main focus of the article, it’s important to note the flaw in his approach. It requires different coefficients for both holographic mass solutions, as the actual proton mass differs by a factor of 2 and the electron mass differs by a factor of 1/2. The issue is easily resolved for the proton mass when the radius, not the diameter, of the PSU is equal to the Planck length. This however leads to the actual electron mass differing by a factor of 1/4. The error seems to arise from applying the same holographic mass solution to the proton with respect to its charge radius and the electron with respect to the Bohr radius, which have one very important difference. While the proton’s center of mass resides in the center of its charge radius, the same is not true of the electron with respect to the Bohr radius. The fact that we must assign a velocity factor of ~1/137 to the electron’s motion is a clear indication that the relationship between me and a0 is different phenomenologically from the relationship between mp and rp. So to resolve the factor of 1/4 we can simply apply a different method of computation. Rather than count the number of PSU equatorial disks on the spherical surface area of the Bohr atom in the calculation of the electron’s holographic mass, we must count the number of PSU equatorial disks in the circular area defined by the Bohr radius. Adopting this approach and setting the PSU radius equal to the Planck length thus eliminates the issue with the unexplained coefficients.

Tachyonic field and complex numbers(2D numbers)

The equation 1+2+3+4+...=-1/12 doesn't make any sense when you think about it but the Riemann zeta function of -1 is equal to 1+2+3+4+..., when you visulise the Riemann zeta function the -1 moves to -0.083 which is -1/12 which shows us how we get 1+2+3+4+...=-1/12

The prime number cross(PNC) when you think about it is an outward spiraling positive number line which is shaped like a torus because it is a torus because it can be turned into a torus anyway we can extend the number line to also have the negative numbers and the negative numbers spiral inward with 0 being a barrier between the positive and negatives, the number line is 1-dimensional so lets extend the number line to make it 2D, the 2D number line contains the complex numbers so lets call it the complex number grid, it is made up of two number lines, both the negative number lines spiral inward whilst the positive number lines spiral outward and they squish the imaginary numbers inbetween, the 0 could be thought of as a circle and it contains the negative numbers so it is a circle with a negative area so it is actually a point so you could think of the (PNC) torus with a singularity in its center in an imaginary field.

The UPA geometric torus wave superstring is basically this PNC torus lets go over the UPA's structure: The 0 is the zero-point field and the 0 is the boundry of the negative volume point/singularity, the 0 is like an energy bubble point which is a wormhole, the negative numbers which are also negative mass can be accessed by squeezing the singularity which is the black (w)hole made out of light anyway the 0 could be seen as a mirror reflecting the PNC inward and the inner PNC which is comprised of negative numbers is -1 and as we know -1 forms the infinite tetrahedron grid since it forms -1/12 through the Riemann Zeta function which is talked about on the picture on the side. The torus superstring as we know expands out of the singularity since it is the actual singularity so the 0 is actually reflecting the PNC torus outward.

Tachyonic fields or particles which have imaginary mass (field) are important in my work because "One curious effect is that, unlike ordinary particles, the speed of a tachyon increases as its energy decreases." this is important because a decrease/loss of energy leads to something cooling and I suggested the energy in the zero-point field/singularity is a Bose-Einstein condensate and the cause of this is because it exists in an imaginary (number) field as shown above in the geometry. But where does the energy go when its lost? well it becomes the energy bubble which expands (like a sphere) to become the geometric torus wave in the infinite sea of energy, the central energy bubble is apart of the singularity/zero-point field and as I have said these singularities all merge to become the singularity/zero-point/god source and when these bubbles merge they become the membrane. The energy bubbles are the infinite tetrahedron grid negative volume singularity they are like the mandelbulb.

As we know the powers of 2 form pascals triangle and the tetrahedron grid fractal and the tetrahedron grid fractal forms the cosmic tree of life as the powers of 3 and the powers of 2 and 3 form Platos lambda so the tetrahedron grid fractal expands out of binary-trinary and as we know the cosmic tree of life forms the dimensions so each level that is formed in the tetrahedron grid fractal corresponds to a higher dimensional sphere(geometric torus wave).

Bose-Einstein condensate black hole simulation

• Bose-Einstein condensates can be used to simulate black holes on quantum scales

Bose-Einstein condensates Kugelblitz black hole

• Xen energy(consciousness (electromagnetic) energy) vibrating at different frequencies is light and when it fractals it forms the universe(the universe is made out of xen energy) and as we know light is a photon which is the particle that carries the electromagnatism, electromagnatism has always existed so how can it be a superstring? Well energy=geometry as I have shown and the photon look like a prime number cross so they are a torus so the energy already took on a torus shape so its a 1-dimensional torus already, when this photon which is basically just a quanta of energy is condensed enough it forms a black hole so it can then form matter.
• the xen energy Bose-Einstein condensate's geometry as we know is the tetrahedron grid fractal this fractal causes the smallest portions of the fractal to become points which are singularities black holes with a negative volume and they are all entangled because all the black holes singularities merge as I have explained, this entanglement is actually a wormhole network, the entanglement forms the tetrahedron grid structure and is formed out of the energy loops merging to form quantum loop fields. The xen energy Bose-Einstein condensate basically allows light to be condensed into an infinitely dense point forming a black hole and condensed light causes the light to switch between antimatter and matter which explains the partiki grids.

Spherical wave PSU

Above the PSU was explained, "Haramein defines the fluctuation of the vacuum in terms of the Planck Spherical Unit.(PSU)" we could think of the spheres being spherical waves.

1-dimensional torus UPA

The singularity of a black hole is an infinitely (con)dense(d) energy point and I suggest that the black hole is a negative volume kugelblitz black hole that spins and because of this the black hole itself is a 0-dimensional point and its actual singularity is a ringularity because the black hole is spinning and because this singularity is an energy point that means it becomes a ring/loop of energy and this loop of energy is therefore 1 dimensional and it can act like a string so it can be a superstring and this ring/loop is actually a torus so, therefore, its a 1-dimensional torus which is more sphere-shaped. The UPA superstring is a 1-dimensional sphere-shaped torus.

1-dimensional torus/sphere wave

Above I have explained how the 1-dimensional torus UPA superstring is also sphere-shaped and when it vibrates it becomes a superstring so a vibrating torus/sphere which is basically a spherical wave.

Extra information

The I Ching constructs the 64 tetrahedron grid and is based on the number 8 showing it could be related to base 9 also the I Ching corresponds to Tzolkin which relates it to the cubistic matrix

The fractaling star tetrahedron(64 tetrahedron grid/Sri Yantra)

√2, doubling sequence and Fibonacci spiral

Phi (based geometry) encoded in the fingerprint of God geometry

The following information will go over extra information such as extra info about the second paragraph in this post, the four forces and base 9:

• The eight dimensions of the octonions aren't the only interesting thing about the number eight, however. Baez highlights the number eight as one of his three favorite numbers. (The other two? Five and 24.) In 2008 Baez gave a series of lectures explaining what makes five, eight and 24 such unique and mysterious entities. The lectures, which are intended for a general interest audience, live on the Internet as both pdfs of the slides he used and video recordings. Watching them, you can learn not only a lot more about what makes octonions special, but also sphere stacking, the golden ratio, Islamic tiles, and why the sum of all integers equals –1/12. ( www.scientificamerican.com/article/octonions-web-exclusive/ )
• As we know all forces can be unified in the grand unified theory or theory of everything(TOE) the forces could be an example of polar opposite, the strong and weak forces could be opposites electromagnetism could be its own opposite which makes sense but what about gravity? Well I believe dark matter/dark energy is the opposite of gravity which makes sense. I also believe the strong/weak force and dark matter-energy/gravity are opposites which makes sense in my opinion. To solve quantum gravity we can treat gravity like electromagnetism and have gravity as waves which has basically already been proven because gravitational waves have been proven, light could produce the gravitron particle. All the particles and forces correspond to the 4/5 elements.
• The fingerprint of God could be base 9 as we will explain next anyway the fingerprint of God is based on the 3, 6 and 9 triangle which forms the base 3 trinary/powers of 3, quantum computers are actually using base 3, the universe deals with information like a quantum computer in base 3 because the universe runs on quantum physics. So the base nine 8 dimensional numbers 0, 1, 2, 4, 8, 7, 5 run on base three 3, 6, 9. The 1, 2, 4, 8, 7, 5 and 3, 6, 9 surround a central 0 which is a singularity zero-point.
• The fingerprint of God could be base 9 when you think about it since it only uses 9 numbers and as I have shown 1.68 is phi in base 9 and phi constructs the geometry of the fingerprint of God, lets talk about base 9 now:
• this post could possibly link into base 9 as I have talked about before in the following post which overall links into this:

Chaos theory, numbers and more

As I talked about above this post links into the fingerprint of God which links into

Plato's lambda

as shown in this post and this overall links into:

X and Y equation

My X and Y equation is symbolic (and doesn't really mean anything I just created it to encode geometries within each other through a fractaling fraction) so the value of Y doesn't matter.

√2, doubling sequence and Fibonacci spiral

Information about this section is in the picture on the right.

Sources

• www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/
• www.smphillips.mysite.com/article-15.html (This article can also be viewed in PDF form on the linked articles in the Stephen M. Phillips category of this wiki)
• www.quantamagazine.org/universal-math-solutions-in-dimensions-8-and-24-20190513
• golem.ph.utexas.edu/category/2014/10/mtheory_octonions_and_tricateg.html
• www.valdostamuseum.com/hamsmith/PDS3.html
• en.wikipedia.org/wiki/Leech_lattice
• www.markronan.com/mathematics/symmetry-corner/leech-lattice/
• theoryofeverything.org/TOE/JGM/Integrated%20E8,%20Binary,%20Octonion-web_files/frame.htm
• theoryofeverything.org/theToE/2016/03/10/more-fibonacci-pascal-triangle-patterns/
• theoryofeverything.org/theToE/2015/12/23/g2-as-the-automorphism-group-of-the-split-octonion-algebra/
• theoryofeverything.org/theToE/2016/01/08/introducing-the-sedenion-fano-tesseract-mnemonic/
• theoryofeverything.org/theToE/2015/03/10/detail-visiblie_e8-demonstration-description/
• eternalkeys.ca/Creation52d.html
• medium.com/@matt.lorusso/the-oscillating-planck-spherical-unit-7caf07310573
• www.i-sis.org.uk/Golden_Geometry_of_E_infinity_fractal_spacetime.php
• www.i-sis.org.uk/E_infinity_spacetime_quantum_paradoxes.php