A prime number is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole numbers that can be divided evenly into another number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Numbers that have more than two factors are called composite numbers.
The Riemann Hypothesis
The German mathematician Bernhard Riemann worked mainly in the field of the geometry of non-Euclidean (or curved) spaces, in which he laid the basis for what will become the mathematics of Einstein’s General Relativity. Nonetheless, he’s also famous for a conjecture that to this date is still unproven (or is it?). As a matter of fact, his only paper on number theory is also probably the most important of all times in the field! In this work, Riemann discovered the exactformula for counting the number of primes below N, i.e. π(N) ), which can be expressed as follows:
The function ζ(s) is a complex function usually referred to as Riemann Zeta function. The Riemann hypothesis states that the zeros of ζ(s) (i.e. the numbers s for which ζ(s) = 0) are all either negative even numbers, or complex numbers with real part equal to 1/2. Riemann discovered that the exact expression of π(N) involves the roots of the zeta function. The Riemann conjecture is part of the 23 Hilbert’s problems and its proof is still up for grabs!
In the 1970s, some physicists such as Freeman Dyson and Hugh Lowell Montgomery speculated that the zeros of ζ(s) may play some role in quantum mechanics. The Primon gas is an example of a quantum field theory of a set of non-interacting particles whose states depend on prime numbers. It can be shown that the Hamiltonian function describing the energy of such a system is related to the Riemann Zeta function. In some sense, there may be a chance that Nature has found a way by itself to prove the Riemann Hypothesis!
The Prime Number Cross
According to Dr Peter Plichta a German chemist, the ancient Egyptians were aware of a hidden pattern buried away within the prime number sequence. By placing the numbers from 1 to 24 into a circle,as we did previously with the 24 reduced Fibonacci numbers and moving in a clockwise direction, then placing the next 24 numbers of the sequence running concentrically around it, repeating this manouvre, we discover that the prime numbers fall on the diagonals which in turn appear to form the image of a Templar cross.
While it is likely that Dr Plichta is not the first person to uncover and understand the Prime Number Cross, his efforts are by no means diminished by this possibility. The priests of ancient Egypt held the number 8 to be sacred, and indeed its role in obscuring the truth about the number 81 and the infinite sequence of all numbers has been very significant. Plichta himself notes the striking similarity between the cross of the order of Christian knights, that persists to this day as the emblem of the St John Ambulance brigade, and the structure of the Prime Number Cross that he discovered through his study of chemistry and mathematics. He also makes it clear that he does not claim to have invented the PNC but rather discovered it. In his view the design of such a perfect and elegant model can only be of divine origin.
centrality of the Prime Number Cross in ordering the structure of everything we perceive. There is not sufficient space to explain all the details.
- If you refer to the above PNC diagram you will see that it is based on the infinite series of natural numbers;0123456789(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)….∞
- There are 24 rays that radiate from the notional centre of the structure, which is really the frontier between the whole numbers and the reciprocal numbers. The latter sequence starts with 1/1, ½. 1/3, ¼, 1/5 …. and never ends. The reciprocal sequence approaches zero, but never reaches it. So, on the outer 'edge’ of the PNC there are increasingly large whole numbers that grow towards infinity, and in the centre of the PNC is a space in which the reciprocal numbers diminish towards zero. The PNC has no outer 'edge’, nor does it have a tangible 'centre’. At the outer boundary is endlessness and at the inner boundary is nothingness. Both are difficult concepts to grasp, but quite real.
8 fold structure
The prime number cross has an 8-fold structure.
- There are 8 numbers with the form 6n ± 1. These are 1, 5, 7, 11, 13, 17, 19, 23.
- There are 8 multiples of two. These are 2, 4, 8, 10, 14, 16, 20, 22. The number 2 is a prime number because the series starts with it.
- There are 8 multiples of three. These are 3, 6, 9, 12, 15, 18, 21, 24. The number 3<( is a prime number because the series starts with it..
The speed of light
Dr Plichta went on to show a direct correlation between the PNC’s expansion constant of 3 and the speed of light. He was also able to explain how the numerical structure of the PNC is the basis for the accurate transmission of information via light, sound and radio waves.
Photon and the prime number cross
The photon is a superstring(Torus/UPA/Xen particle)
The photon looks like a prime number cross!
Tetraktys model
Plichta relates the cross to many aspects of the chemical periodic table, the structure of DNA, fractal geometry, pi, tonal music, and universal constants to the spatial organization of numbers. I would like to offer his prime number cross as another example of the type of organizational principle that I believe is the essence of the Pythagorean tetraktys model for tonal music, which in turn is basically a form of the enneagram(1/7). Although Plichta covers many different aspects of the physical world, I will try to confine my discussion to his ideas about music. Plichta has organized his model around a property of prime numbers, that is, a prime number in this arrangement is located on either side of multiples of six:
prime number = 6n-1 and 6n+1 (n is an integer)
This property holds true for all the known prime numbers with the exception of the numbers 2 and 3. In the model there are eight rays emanating from the first circle aligned with prime numbers that lie in the first circle (5,7,11,13,17,19,23). A multiple of six lies in the rays between the prime number rays. The pattern for the prime numbers works for the pairs 5-7, 11-13, 17-19 but at 24, the prime pair is 23 and the non-prime number 25 (the square of the prime number 5). This is one reason why Plichta restarts the cycle such that 25 lines up with 1.
In Plichta’s analysis of the prime number cross he found that the circle of numbers contains only three basic categories of numbers derived from the first three whole numbers 1,2, and 3 This is because each quadrant of the circle is an isomorphic structure with 3 and its multiples as axes of symmetry.
This means that these three numbers are the basis of the entire structure of the number system.
- 1-5 7 11 13 17 19 23…
- 2- 4 8 10 14 16 20 22…
- 3- 6 9 12 15 18 21 24…
The number six is what Plichta calls the scaffold of the prime number cross since the prime numbers cluster around it. Six is also the sum and product of the first three numbers: 6 = 1+2+3 and 1*2*3. Six is also the first perfect number because of the fact that its sum and product are equal (28 is the next perfect number). Eight is a structural number because of the eight prime number rays based on 1. In addition, there are eight rays for the numbers based on both two and three.
In an earlier drawing of the cross, the numbers –1, 0, and 1 are a part of the pairs of prime numbers. This group, -1,0,1 fit into the prime number property of six shown above leading Plichta to conclude that they belong in the cross and that –1,1 should be included as a prime number pair. The prime numbers belong with this –1,1 group and both are then prime numbers. Plichta removes this group and places it in what he terms an inner shell, evoking a model very similar to the electron shells of atomic theory.
The number twenty-four is a link to the tetraktys from the relationship mentioned above. (1*2*3*4, 4! = 24). Plichta points out that the twenty-four hour clock was an invention of the Egyptians who also introduced the decimal system to humankind, implying that they may have a common source of derivation, although he does not elaborate this point further.
Another central concept in his model is that the prime number cross is a model for matter, which exists in four-dimensional space. By his logic, three-dimensional shapes must exist in the context of a fourth dimension. If the prime number cross exists in two dimensions it is defined by an area which is the square of a number, Number space exists as two intersecting prime number crosses and is therefore defined as the square of the square of the area of a number, hence there are four dimensions to number space. Having established a four-dimensional space, he formulates the relationship between energy and time by stating that: "If matter did not exist…there would also be no space. But if only the two exist at the same time, then the one would have to be the other, only reversed. Where there is no movement, there also no time. Energy can therefore be nothing else but the reverse of time"
Plichta makes the claim that central to his model’s organization is the relationship of number to space. He emphasizes that there are two aspects to four-dimensional space, one of which, governed by the whole numbers, increases toward infinity, the other of which, governed by and simultaneous with the reciprocal of every whole number, decreases infinitely toward zero. If this is true, then the "infinity of space will exist reciprocally as points of matter" This is an important point in understanding how music is organized. In a sense then the organization of nature by the whole numbers in the form of the prime number cross creates a simultaneous correspondence that follows the organization of the reciprocal of the whole numbers. Both types of space depend on the base numbers 1,2, and 3 and the "structural number" 8. The prime number cross spherically expands toward infinity as its reciprocal simultaneously contracts as an equilateral triangle toward zero.
Plichta’s thinking here definitely relates back to Pythagoras and his followers. The Pythagoreans understood the enigma of infinity and its reciprocal relationship to 0. The problem here is that the whole numbers increase indefinitely to infinity and, at the same time, their reciprocals can infinitely approach 0, yet logically, the distance between 0 and 1 must be the same as the distance between 1 and 2:
- 1 2 3 4…….infinity 0...
- 1/infinity 1/2 1/3 1/4 ... 1
The number 1 then is a point of balance between the infinity of numbers (the macrocosm) and the infinity of zero (the microcosm). This relationship has a very simple, seemingly trivial, mathematical expression:
- 1=1/x*x
In mathematics, the hyberbola expresses the graph of the function of 1/X. Plichta points out that the shape of the hyperbola results from "the inversion of infinitely small sections of increasingly large circles" It follows from the above expression (when we multiply both sides of it by X) that a number, X is related to its reciprocal by its square, the two-dimensional area.
- x=1/x*x2 x: 1/x*x2
This equation, again seemingly trivial, is a link between a number and its reciprocal and it is a reflection of the relationship between 0 and infinity. If a number implies a space around itself, then the square of the number is meaningful in describing this space. This aspect of number theory relates to the Pythagorean fascination with the squares of numbers, reciprocals, and proportion. The latter two are central to the study of the monochord and to the theory of musical intervals. Starting with the open string of a given length, measurements that yield the various intervals are proportions of the string length. The ratios arise from the number of divisions made for a given string length. For example, 1:2 yields an octave relationship. Exactly one-half of the string length is twice the frequency of that of the total string length and the string divides into two equal parts. When you divide the string into three parts, the relation of the fifth (1/3 or 2/3), creates three equidistant increments in a 1:3 proportion to the original string length. We can see from this that at a very basic level there is a relationship between number, its reciprocal, and spatial increment created by the relationship. One can array numbers in a straight line or on a circle and the same incremental distribution almost seems to be an inherent property of the number system. In a sense, this seems true merely because numbers are just abstractions created by thought. Nonetheless, when we conceive of numbers there is an implicit evenness of value and distribution in the concept such that the distance from 0 to1 is equal to the distance from 1 to 2. The important conclusion that Plichta reaches is that "for each whole number the infinite reciprocal value also exists"
The Tetraktys Model and the Overlap of Three
After examining the enneagram and the prime number cross, I found that both circular arrangements have one interesting common feature. Each arrangement lines up the numbers so that there is an overlap of three beginning numbers with the three numbers that start with the penultimate number. This is clearly demonstrated in Plechta’s model in which he actually puts three numbers in the center, -1, 0,1. These numbers correspond to 23, 24, and 12. I believe that the enneagram reflects the same overlapping procedure although it is not as apparent and the overlap seems to be accomplished differently. I think that in the enneagram the same overlap occurs, but it simply involves the last three numbers with the ten left out as it is subsumed with the number one in the mod 9 arithmetic. Furthermore, I believe that Plichta’s inner shell numbers do correspond to the first numbers in the Greek model, so that when the Greek start the tonal scale they are really starting with a third number. This suggests that the number group –1,0,1 descends from some original conception of the first three numbers that were not counted as actual numbers. I do not know whether the Pythagoreans thought of the inner circle in the same way that Plichta does, but the overall result is the same.
Going back to the idea that a group of the first three numbers combine in the Pythagorean model, I offer the following model as the possible organizational principle of the tonal system, which shows how the tonal system is a model derived from the tetraktys:
The numbers 8, 9 and 10 are in parentheses to reflect the dual nature that they acquire by virtue of the system that allows for only seven different pitch names. The nine here therefore has two possible positions: it is the same as the ninth which corresponds to the compound second, just as the octave (8) is both a –1 and also corresponds to the compound prime. Comparing this model with the enneagram reveals that they are essentially similar, suggesting a common link to the tetraktys model. I show the "inner shell" numbers –1,0,1 to imply that they bear relationship to the three numbers greater than seven in the cycle. The "inner shell" numbers correspond to last three numbers of the spatial arrangement created when a group of numbers is arrayed in a circular pattern in a specific way:
- –1 corresponds to the number tied to the clock face mirror image of the first number (here 1=8: octave equivalence)
- 0 corresponds to the modulus of the circular arrangement (here of course 9)
- 1 corresponds to the last number of the group (here of course, 10)
The three points above may simply seem to be self-fulfilling and circular (if you will excuse the term) reasoning, because I have found that this arrangement works for any sequence of numbers, that is to say that any clock face or circular organization of numbers seems to be based on this overlapping of three principle. This observation may have relevance to the relationship between rhythm and melody, a consideration I plan to study further. To illustrate the overlap of three principle in a related system, I now demonstrate the organization of the Circle of Fifths as another model constructed with this principle.
Seven
The tetraktys model tells us that, when Pythagoras (or whoever it was who first described the tonal system) was thinking about the number seven; it was in relation to the number ten. The overlap of three principle is an easily demonstrable fact that automatically creates an organization of number in space. It is also such a simple notion that it is not so surprising that it was discovered a long time ago and that it seems to have been forgotten. I think that the most important concept that this circular model teaches us is that music is circular in nature. The circularity of music is not just metaphorical. It is to me a breath of fresh air to look at music from this viewpoint. Plichtas’ theory that time is the reverse of energy seems particularly applicable to describing the way music happens in time. His statement that "the human ear is specially constructed so that it registers fractal information transported through the air" is a thought provoking one that deserves further investigation. Music is the some of the best evidence there is that we have many new secrets to discover or rediscover about nature. The tonal system is an excellent example of the genius of the human spirit in its search to understand our relationship to this world.
Phi, Fibonacci and Vortex maths
Contained within his prime number cross are the numbers from 1 to 144 Within this series of the first 144 numbers there are 34 prime numbers. On Plichta’s diagram we note that only 32 primes fall into the cross. Taking these primes which fall within the cross
5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83, 89,97,101,103,107,109,113,127,131,137,139
Their total value is 2124 which reduces to 9. The numbers 2 and 3 are the excluded exceptions 2+3=5
Taking the full prime sequence up to 144 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79, 83,89,97,101,103,107,109,113,127,131,137,139
Their total value is 2129 which in turn reduces to 5 (14=5-PHIve)
3-6-9
Reducing the numbers on the prime cross to a single digit by mod 9 reveals the following array and it is a familiar one also-all the numbers fall into 3 distinct sequences alternating as follows:
- 1-4-7
- 2-5-8
- 3-6-9
Which in turn break down further to a single digit pattern: 3-6-9
This is exactly the same patterning as we have witnessed so far within the repeating 24 Fibonacci sub-code and the 3 groupings of the prime numbers divisable by 1,2 and 3. Or even by adding the single numbers together:
- 1,4,7=1-10-28(1-1-1)=3
- 2,5,8=3,15,36(3-6-9)=9
- 3,6,9=6,21,45(6-3-9)=9
- 3,9,9=21(3)
- 21
Counting down:
- 1,3,6=10(1)
- 1,6,3=10(1)
- 1,9,9=19(1)
- 1-1-1
1-1-1 times 8 (The pattern occurs 8 times in the prime cross) 8-8-8 (8-8-8=24-Indicative of time perhaps?)
Applying the same rule as above:
- 36-36-36=9-9-9
- 36+36+36=108=9
- 108
- 3 nines
- 27
- 9-8
Torus
Toroidal ‘S’ curves are also visibly present within the prime number cross sequence. (Starting from the 1 position in the cross) Note; all chains in this progression break down by mod 9 to the 3-9-6 sequence.
The last two numbers of a chain are always the first two numbers of the second (highlighted in red) The commencing number in the first line (18) is the same as the final number in the last line. (Highlighted in blue) This pattern repeats throughout the cross,turning on each 9th chain.
Also there would appear to be another sequence occuring in the last two numbers of each chain:
The first four seem to follow a 2,4,6,8 sequence, The next five follow a 1,3,5,7,9 pattern. Patterns are also visible within the array of numbers if we read them in straight lines from inner to outer: eg: 714714, this continues over in the opposite line with a chirality of 471471 The same patterning occurs within all the number stands throughout the prime number cross. There are six concentric circles in the cross within which the numbers are distributed. (Inner to Outer)
- 2,3,5,7,11,13,17,19,23.
(9 units;value-100=1)
- 29,31,37,41,43,47.
(6 units;value-228-12=3)
- 53,59,61,65,67,71
(6 units;value-376-16=7)
- 73,79,83,89,
(4 units;value-324=9)
- 97,101,103,107,109,113
(6 units;value-620=8)
- 127,137,139
(3 units;value-403=7)
1 3 7 9 8 7 value=35=8
Square prime numbers
The overall idea is that, apart from two exception on the first ring (integers 2 and 3), all the other prime numbers lay on 8 radii, but not all numbers that appear on those lines are primes. Also, the radius stemming from 1 contains the squares of prime numbers larger than 5, e.g. 5*5=25, 7*7=49, 11*11=121, 13*13=169, 17*17=289 and so on…
Prime Numbers and the Cuboctahedron(vector equilibrium)
If the numbers are arranged in a radial pattern around its twelve points, starting at 1, all the prime numbers (except 2 & 3) line up along two of the six axes as seen below. The primes above 3 are shown in purple.
