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 (green) repeat a fraction already seen.
So green and yellow mark fractions that occurred earlier. Red always marks a new constellation.
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: 25 is the only number at which the interior angle sum (in full circles) and the quotient agree, namely 2.5 at divisor 10. 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
Anyone who takes the time to draw simplexes and check their types of star polygon will understand what this is about. Anyone who does not check the positions has to trust the illustrations. In the end only one’s own checking can convince.
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.