The Prime Hexagon is a mathematical structure developed by mathematician Tad Gallion. A Prime Hexagon is formed when integers are sequentially added to a field of tessellating equilateral triangles, where the path of the integers is changed whenever a prime number is encountered (GitHub: kaustubhcs/prime-hexagon).

Each time a prime number is encountered, the spin or ‘wall preference’ is switched. So, from the first cell, exit from 2’s left side. This sets the spin to left and the next cell is 3, a prime, so switches to right. 4 is not prime and continues right. 5 is prime, so switch to left and so on. (HexSpin)

Surprisingly, the 24-cell hexagon confines all natural numbers. The reason: no prime numbers occupy a cell with a right or left wall on the t-hexagon’s outer boundary, other than 2 and 3, the initial primes that forced the number line into this complex coil. Without a prime number in the outer set of triangles, the number line does not change to an outward course and remains forever contained in the 24 cells. (HexSpin)

Cell types are interesting, but they simply reflect a modulo 6 view of numbers. More interesting are the six internal hexagons within the Prime Hexagon. Like the Prime Hexagon, they are newly discovered. The minor hexagons form solely from the order, and type, of primes along the number line (HexSpin).

Similarly, I have a six colored dice in the form of the hexagon. If I take a known, logical sequence of numbers, say 10, 100, 1000, 10000, and look at their spins in the hexagon, the resulting colors associated with each number should appear random – unless the sequence I’m investigating is linked to the nature of the prime numbers.

That is, if the powers of 10 all returned with blue spin, or as a series of rainbows, or evenly alternating colors or other non-random results, then I’d say prime numbers appear to have a linkage to 10. I may not know what the the linkage is, just that it appears to exist (HexSpin).

I wondered if that property might hold for the incremental powers of phi as well. For this reason I chose to see numbers in the hexagon as quantum, and truncate off the decimal values to determine which integer cell they land in. That is what I found. Phi and its members have a pisano period if the resulting fractional numbers are truncated. (HexSpin).

Proceeding, the number line begins to coil upon itself; 20 lands on 2’s cell, 21 on 3’s cell. Prime number 23 sends the number line left to form the fourth (4th) hexagon, purple. As it is not a twin, the clockwise progression (rotation) reverses itself. Twin primes 29 and 31 define the fifth (5th) hexagon, cyan. Finally, 37, again not a twin, reverses the rotation of the system, so 47 can define the yellow hexagon (HexSpin).

All perfect squares within our domain (numbers not divisible by 2, 3 or 5) possess a digital root of 1, 4 or 7 and are congruent to either {1} or {19} modulo 30.

• When the digital root of perfect squares is sequenced within a modulo 30 x 3 = modulo 90 horizon, beautiful symmetries in the form of period-24 palindromes are revealed. Here’s one modulo 90 spin on perfect squares.
• parsing the squares by their mod 90 congruence reveals that there are 96 perfect squares generated with each 4 * 90 = 360 degree cycle,
• which distribute 16 squares to each of 6 mod 90 congruence sub-sets defined as n congruent to {1, 19, 31, 49, 61, 79} forming 4 bilateral 80 sums.
• each of the 6 columns has 8 bilateral 360 sums, tor a total of 48 * 360 = 40 * 432 (much more on the significance of number 432, elsewhere on this site).

There’s another hidden dimension of our domain worth noting involving multiples of 360, i.e., when framed as n ≌ {1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53 59, 61, 67, 71, 73, 77, 79, 83, 89} modulo 90, and taking ‘bipolar’ differentials of perfect squares (PrimesDemystified)

Each of the digital root multiplication matrices produced by the six channels consists of what are known in mathematics as ‘Orthogonal Latin Squares’ (defined in Wikipedia as “an n x n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column” … in our case every row and column of the repeating 6x6 matrices possesses the six elements: 1, 2, 4, 5, 7, 8 in some order). Also, the sum of the multiplicative digital roots = 108 x 24 = 2592 = 432 x 6.

• Note: Channels A, D, E and F combined represent the set of natural numbers not divisible by 2, 3 and 5, the first 24 elements of which form the basis of the Magic Mirror Matrix.
• The graphic below illustrates the transformative relationships between the matrices employing their primary building blocks (one of the sixteen identical 6 x 6 (36 element) Latin Squares that constitute each matrix)
• When you rotate either the {1,4,7} or {2,5,8} magic square around its horizontal axis, i.e. columns {A,B,C} become {C,B,A}, then add the {1,4,7} {2,5,8} magic squares together, you produce a square with nine 9’s. For example, adding the first rows of each gives us: {2,8,5} + {7,1,4} = {9,9,9}.
• Triangles and magic squares similar–or identical–to those shown above can be derived from the digital root sequence cycles of all three twin prime distribution channels (namely numbers ≌ to {11,13}, {17,19} and {1,29} modulo 30).
• This is also true of dyads formed by paired radii of the Prime Spiral Sieve that sum to 30, i.e., numbers ≌ to {1,29}, {7,23}, {11,19}, or {13,17} modulo 30, as well as dyads formed when {n, n + 10} are ≌ to {1, 11}, {7, 17}, {13, 23} or {19, 29} modulo 30 (note their pairing by terminating digits). One example relating to twin primes: The first three candidate pairs in the twin prime distribution channel ≌ to {11,13} modulo 30 (all three of which are indeed twin primes) sequence their digital roots as follows:
  • {11,13} = digital roots 2 & 4
  • {41,43} = digital roots 5 & 7
  • {71,73} = digital roots 8 & 1.
• As you can see, this is the same digital root sequence illustrated above. It appears that the triangulations and magic squares structuring the distribution of twin primes (and as it turns out, all prime numbers) have a genesis in universal principles involving symmetry groups rotated by the 8-dimensional algorithms discussed at length on this site.
• You can see this universal principle at work, for example, with regard to the Fibonacci digital root sequence when coupled to a pair of dyads that follow certain incremental rules. As we illustrated above, the initializing dyad of the period-24 Fibonacci digital root sequence is {1,1, …}.

We can generate triangles and magic squares by tiering the Fibonacci digital root sequence with two pairs of terms that are + 3 or + 6 from the initial terms {1,1}. The values of the 2nd and 3rd tiers, or rows, must differ, or symmetry is lost. In other words, the first two columns should read either {1,4,7 + 1,7,4, or vice versa} but not {1,4,7 + 1,4,7, or 1,7,4, + 1,7,4}. (PrimesDemystified)

When these 9 squares are combined and segregated to create a 6 x 6 (36 element) square, and this square is compared to the Vedic Square minus its 3’s, 6’s and 9’s (the result dubbed “Imaginary Square”), you’ll discover that they share identical vertical and horizontal secquences, though in a different order (alternating +2 and -2 from each other), and that these can be easily made to match exactly by applying a simple function multiplier, as described and illustrated later below. (PrimesDemystified)

When we additively sum the three period-24 digital root cycles these dyads produce, then tier them, we create six 3 x 3 matrices (each containing values 1 thru 9) separated by repetitive number tiers in the following order: {1,1,1} {5,5,5} {7,7,7} {8,8,8} {4,4,4} {2,2,2}.

• The six (6) matrices these tiers demarcate are the source of triangular coordinates when translated into vertices of a modulo 9 circle (which by definition has 9 equidistant points around its circumference, each separated by 40°).
• The series of diagrams below show the six geometric stages culminating in a complex polygon of extraordinary beauty. We’ve dubbed this object a ‘palindromagon’ given that the coordinates of the 18 triangulations produced by the digital root dyadic cycles in the order sequenced sum to a palindrome: 639 693 963 369 396 936.

• Remarkably, this periodic palindrome, with additive sum of 108, sequences the 6 possible permutations of values {3,6,9}. Interesting to consider a geometric object with a hidden palindromic dimension. But that’s not all: When the six triadic permutations forming the palindrome are labeled A, B, C, D, E, F in the order generated, ACE and BDF form 3 x 3 Latin squares. In both cases all rows, columns and principal diagonals sum to 18:

  • ACE … BDF
  • 693 … 639
  • 369 … 963
  • 936 … 396
• The output of these algorithmically sequenced triangulations is fundamentally a geometric representation of the twin prime distribution channels (and, as we noted above, the same geometry is expressed in factorization sequencing, albeit the vertices may be ordered differently.
• This is because each set of three generator dyads roots to the same six elements: 1, 2, 4, 5, 7, 8. Thus, for example, dyad sets ({1,2} {4,5} {7,8}) and ({2,4} {5,7} {8,1}) will generate identical complex polygons, despite their vertices being sequenced in different orders.).

It’s remarkable that objects consisting of star polygons, spiraling irregular pentagons, and possessing nonagon perimeters and centers, can be constructed from only 27 coordinates pointing to 9 triangles in 3 variations. Each period-24 cycle produces two ‘palindromagons’. (PrimesDemystified)

In our approach a 3-form is not an object that exist in addition to the metric, it is the only object that exist and in particular the 4D metric, is defined by the 3-form.

• We would like to say that our present use of G2 structures (3-forms in 7D) is different from whatone can find in the literature on Kaluza–Klein compactifications of supergravity.
• We show that the resulting 4D theory is (Riemannian) General Relativity (GR) in Plebanski formulation, modulo corrections that are negligible for curvatures smaller than Planckian.
• Possibly the most interesting point of this construction is that the dimensionally reduced theory is GR with a non-zero cosmological constant, and the value of the cosmological constant is directly related to the size of . Realistic values of Λ correspond to of Planck size.

Also, in the supergravity context a 7D manifold with a G2 structure is used for compactifying the 11D supergravity down to 4D. In contrast, we compactify from 7D to 4D. (General relativity from three-forms in seven dimensions - pdf)

This is a necessary but not sufficient condition for N to be a prime as noted, for example, by N= 6(4)+1= 25, which is clearly composite. We note that each turn of the spiral equals an increase of six units. This means that we have a mod(6) situation allowing us to write: N mod(6)=6n+1 or N mod(6)=6n-1 (equivalent to 6n+5). (HexSpiral-Pdf)

Focusing on just the twin prime distribution channels, we see the relationships shown below [and, directly above, we show that two of the channels (B & C) transform bi-directionally by rotating 180° around one of their principal (lower-left to upper-right) diagonal axes]:

You likely noticed I began with 2 rather than 1 or 0 when I first constructed the hexagon. Why? Because they do not fit inside — they stick off the hexagon like a tail. Perhaps that’s where they belong. However, if one makes a significant and interesting assumption, then 1 and 0 fall in their logical locations – in the 1 and 0 cells, respectively. _(HexSpin)

Because the value 30 is the first (common) product of the first 3 primes. And this 30th order repeats itself to infinity. Even in the first 30s system, therefore, the positions are fixed in which the number information positions itself to infinity. We call it the first member of the MEC 30.

• The numbers not divisible by 2, 3 or 5 are highlighted. We call them prime positions, hence 1, 7, 11, 13, 17, 19, 23, 29. Important for our work is that in the following the term prime refers only to prime numbers that are in the prime positions. So primes 2, 3 and 5 are always excluded.
• These positions: 1 7 11 13 17 19 23 29. We refer to this basic system as MEC 30 - “Mathematical Elementary Cell 30”. By repeating the positions we show the function of the basic system in the next step. If we extend the 30th order of the MEC, for example, to the number 120, the result is 4 times a 30th order and thus 4 × 8 = 32 prime positions.
• Hypothetical assumption: If the product of the primes (except 2, 3, 5,) would not fall into the prime positions, thus be divided by 2, 3 or 5, the information would have 120 = 32 primes in 32 prime positions: 1, 7, 11, 13, 17, 19, 23, 29, / 31, 37, 41, 43, 47, 49, 53, 59, / 61, 67, 71, 73, 77, 79, 83, 89, / 91, 97, 101, 103, 107, 109, 113, 119
• These forms gives prime positions: 1, 7, 11, 13, 17, 19, 23, 29, / 1, 7, 11, 13, 17, 19, 23, 29, / 1, 7, 11, 13, 17 , 19, 23, 29, / 1, 7, 11, 13, 17, 19, 23, 29. The 30th order is repeated in the number space 120 = 4 times, 4 × 8 = 32 prime positions, thus 4 terms.

From our consideration we can conclude that the distribution of prime numbers must have a static base structure, which is also confirmed logically in the further course. This static structure is altered by the products of the primes themselves, since these products must fall into the prime positions since they are not divisible by 2, 3 and 5. (Google Patent DE102011101032A9)

Speaking of iterative digital division–a powerful tool for exposing structure–we get this astonishing equation: iteratively dividing the digital roots of the first 12 Fibonacci numbers times the divisively iterated 1000th prime, 7919, times 3604 gives us 1000. Keep in mind that the first two and last two digits of the Fibo sequence below, 11 and 89, sum to 100; that 89 is the 11th Fibo number; that there are 1000 primes between 1 and 892; and that 89 has the Fibonacci sequence embedded in its decimal expansion

The four faces of our pyramid additively cascade 32 four-times triangular numbers (oeis.org/A046092: a(n) = 2(n+1) …).

• These include Fibo1-3 equivalent 112 (rooted in T7 = 28; 28 x 4 = 112), which creates a pyramidion or capstone in our model, and 2112 (rooted in T32 = 528; 528 x 4 = 2112), which is the index number of the 1000th prime within our domain, and equals the total number of ‘elements’ used to construct the pyramid.
• Or, using the textbook way to visualize triangular numbers, 2112 = the total number of billiard balls filling the four faces, which in our case will be dually populated with natural numbers 1, 2, 3, … and their associated numbers not divisible by 2, 3, or 5 in a 4-fold progression of perfect squares descending the faces of the pyramid.

The table below shows the telescopic progressions of triangular, 4-times triangular numbers and cascade of perfect squares that populate the pyramid’s faces.

Using Euler’s method to find p(40): A ruler with plus and minus signs (grey box) is slid downwards, the relevant terms added or subtracted. The positions of the signs are given by differences of alternating natural (blue) and odd (orange) numbers. In the SVG file, hover over the image to move the ruler (Wikipedia).