Geometry

The simplex – the solid form of number

Every simplex shows the complete divisibility of its number. That makes geometry a second state of matter for numbers.

Numbers seem fleeting to us: signs on paper that only mean something in the head. In the geometry of the simplexes they take on a solid shape. An n-gon with all its connections drawn in shows at a glance how the number n stands to every smaller number. That is why the simplexes can be called the solid state of number.

One number, all its divisors

The vertices lie on the circumference. They stand for the number itself. The connections stand for the possible divisors. Drawing the figure “every k-th point” computes the fraction n : k geometrically:

• If n is divisible by k, the figure falls apart into k equal polygons.
• If n and k are coprime, a single star appears that can be drawn in one stroke.
• If they share a factor g without k dividing n, the figure falls apart into g equal stars.
• If k is exactly half of n, every line passes through the centre.

n-gon

PolygonCompound polygonStar polygonCompound starLine star

Choose a number and see which figures its simplex falls apart into. The colour shows the divisibility, the number beneath it the quotient n : k.

Two large examples

An even and an odd simplex show the difference especially clearly. You can generate them here and show or hide each star figure:

Points

The 20-gon simplex as an example of an even number. The slider goes up to the 45-gon, the example of an odd number.

Circle and π

Because all figures lie in the same circle, the simplex links two series of numbers: the circumference carries the number, the chords through the interior carry its divisors. Circumference to diameter is the ratio described by π. Hence the thesis: in their essence the simplexes correspond to π, for the ratio of a natural number to its divisors mirrors the ratio of circumference to diameter. More under Simplexes and π.

Only up to two

One restriction matters. Star figures exist only as long as the quotient n : k is greater than 2. At k = n/2 only the line star remains. In the division table the geometry therefore matches the numbers in only a quarter of the field. But that quarter is enough, because the question of how the primes are distributed only arises above this bound.