Simplexes and primes in the division table
The figures of the simplexes can be fanned out into a table of all fractions. The four start figures form a grid in it, and the coprime cells have the density 6/π².
Geometry and number theory cannot really be separated. Each depends on the other and they are best seen as one. This is clearest in the division table: a table in which every column carries a number n and every row a divisor k. In harmonics the same table is called the lambdoma (see The lambdoma).
Fanning out the figures
Every cell (n, k) has a figure: the n-gon with every k-th point joined. Coloured by divisibility, the table shows how the four start figures are distributed.
The first cells of the division table with their figures. Column = number n, row = divisor k. Figures exist only while n : k is at least 2.
On a large scale the colouring shows a strict pattern:
The division table up to 60 × 60 (up to 200 × 200 with the slider). Pale cells lie beyond the bound n : k = 2, where there are no more star figures.
- The row k = 1 contains the numbers themselves, the polygons.
- Divisible cells (yellow) lie on rays starting from the origin.
- Coprime cells (red) are new ratios that cannot be reduced.
- Cells with a common factor but not divisible (teal) repeat a fraction already seen.
So teal and yellow mark fractions that occurred earlier. Red always marks a new constellation.
The pattern between the primes
In a novel by Mark Haddon the young narrator says, in substance, that prime numbers are what is left when all the patterns have been taken away. That is exactly the point here: only someone who examines the whole pattern between the primes, with all the fractions and above all with the geometry behind them, understands the system behind the primes. The site divisorplot.com shows a simplified version in which only the divisors are entered row by row instead of the fractions, an elegant illustration of the sieve of Eratosthenes (see Links).
This table is nothing other than the lambdoma of music theory and harmonics (see The lambdoma and Everything is frequency). In mathematics the arrangement is known from Georg Cantor, who used it to show that the rational numbers are countable. One should not be deceived by its simple arrangement: it is a many-layered matrix. Whoever does number theory without it is at a disadvantage, and whoever knows it but not the geometry behind it cannot gauge what follows from it.
The one in the middle
Tilted by 45°, the left axis shows the fractions 1/1, 1/2, 1/3 …, which tend to zero, the right one the fractions 1/1, 2/1, 3/1 …, which grow to infinity. Where the infinitely small and the infinitely large meet, in the middle, every fraction equals 1: 1/1, 2/2, 3/3 … The table is balanced around the value 1, one is its axis of symmetry, and the reciprocals to its left and right cancel in pairs.
The axes with the values 1/2 and 2/1 divide the lambdoma once more. That seems trivial, since any region could be divided further at will. But only the fractions above 2/1 describe the divisibility of whole numbers with figures, for below the bound 2 there are no more star polygons (see The key to the tetractys). 1/2 and 2/1 divide the lambdoma into four regions, and in the fourth, the quarter above 2/1, one is right in the middle of the tetractys and the primes. Only this quarter agrees completely with the geometry of the simplexes (see Simplex – the solid form of number).
Repetitions and the rhythm of six
Equal values lie most densely on the central axis, where every fraction equals 1. On the axes 1/2 and 2/1 it is only every second one, and so on. Mark all values that repeat and a perfectly symmetrical pattern arises, somewhat reminiscent of Pascal’s triangle and thus of the Sierpinski triangle (see Fractal polygons). In it one recognises the rhythm of six of the twin primes, including the false primes.
A prime can be recognised by the fact that its row and column contain only new values, except on the central axis: 2/2, 3/3, 5/5, 7/7 … Apart from 2 and 3, all primes lie in the rhythm of six 6n ± 1 after Leibniz, and the composite numbers on these positions, 25, 35, 49, 55 …, also fit into this grid. They should not be overlooked by anyone who wants to get to the bottom of the distribution of primes, for they assign each lone prime its missing twin.
There are exactly three ways a number can relate to a divisor, the three colours of the table. The coprime fields are also the fractions that appear for the first time, “prime” in the literal sense, and they are linked with twin formation on both axes. Together the other two constellations mark the repeated fractions, and this pattern is mirror-symmetric: swap numerator and denominator and “divisible” becomes “common divisor”. Only the yellow fields, “divisible”, give the whole-number results with which the sieve of Eratosthenes breaks every non-prime into prime factors. Striking in the large table: the teal fields above quotient 2 run in a rhythm of four and begin at the number 10 with divisor 4. The twin primes to the left and right of the numbers divisible by 6 give twelvefold symmetries. Harder to see through are the threefold structures, which at first still follow the rhythm of six before the patterns overlap. The irregularities that result are chaotic only at first glance.
The density 6/π²
A well-known fact of number theory: pick two natural numbers at random, and the probability that they are coprime is
6/π² ≈ 60.8 %
That is exactly the share of red cells in the division table, the more precisely the larger the table. The readout under the graphic shows the value converging.
Here lies a bridge: in a table about divisibility, π appears. This fits the thesis that the simplexes, as figures in a circle, express the essence of π (see Simplexes and π).
The pentagrams in the table
The disorder of the primes hinges on the coordinates with quotient 2.5. In the division table these are the cells 5/2, 10/4, 15/6, 20/8, 25/10 … The corresponding figures are made up entirely of pentagrams:
- 25 : 10 = 2.5 gives 5 pentagrams,
- 35 : 14 = 2.5 gives 7 pentagrams,
- 45 : 18 = 2.5 gives 9 pentagrams,
- 55 : 22 = 2.5 gives 11 pentagrams.
The 25-gon simplex. The figure 25/10 consists of five pentagrams. 25 is the first number at which a twin position in the rhythm of six holds no prime.
The numerators are always 10 apart, the denominators always 4. The pentagram with its crossings is thus the “crosser of boundaries” of the tetractys and becomes the leading figure in the ranges of numbers that follow. Its square, 25, opens the disorder of the overlapping grids.
A further finding: Interior angle sum (in full circles) and quotient agree for only four figures: 16/4, 18/3, 18/6 and 25/10. The last of them is the only one with a fractional value, 2.5. This recalls that 10 is the only number whose simplex has as many full circles as vertices (see Angle sums and full circles).
Draw it yourself
This geometry is compelling and not arbitrary, just like the number π. That is why the sieve of Eratosthenes and the tetractys exist independently of any counting system, even though the geometry itself is structured in steps of ten. This also gives a factual background to seemingly speculative teachings such as the Kabbalah with its ten sephirot in four worlds. A tree of divine emanations would make little sense if the ten meant nothing more than counting on ten fingers.
The royal road to understanding is to draw simplexes yourself and count which star types they consist of. Whoever does not check the positions of the star polygons personally has to trust the account given here. Only what one has worked out oneself is convincing.
Noncommutative geometry
Finally, an outlook: in the 1980s Alain Connes founded noncommutative geometry, a branch of mathematics that connects number theory with particle physics. Recent research suspects that the question of the primes has a counterpart in physics. It would follow that the division table and the circle geometry of the simplexes are directly related to particle physics and so to the world we experience. That would match the world view of the Pythagoreans 2,500 years ago. More under All is number – all is frequency.