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.

What geometry could represent numbers better than polygons whose vertices correspond to the number? There is one that does even better: the geometry of the simplexes. Mathematically these are multidimensional tetrahedra. Drawn in the plane they are polygons in which every vertex is joined to every other. Star polygons arise inside them, and they describe geometrically how the number of vertices divides.

Numbers seem fleeting to us, signs on paper that have meaning only in our heads. In the simplexes they take on a fixed shape. That is why they can be called the solid state of number, or, with a wink, number made flesh. An example: 6 is divisible by 2 and by 3, and in the hexagon this shows as two triangles (2 × 3) and three lines through the centre (3 × 2).

This geometry gives orientation. Whoever studies numbers with digits is tied to a counting system. A strictly symmetric figure, by contrast, calculates in no counting system and frees one from its mental limits. All the more astonishing that the geometry of the simplexes is structured in steps of ten. Pursuing this is a main concern of this site.

How the game begins

One is a point in the centre of the circle.
With two a line arises whose ends lie on the circle.
Three is the first number to span an area with interior angles.
Four is the first number in whose polygon diagonals can be drawn. Continue the steps of dimension from point, line and surface and it is the tetrahedron, seen edge-on.

Every further vertex is a further step of dimension (see Multidimensional tetrahedra). There is a clear difference between even and odd numbers: only for even numbers do lines pass exactly through the centre. A line through the centre joins two points, it stands for two.

One number, all its divisors

The vertices lie on the circumference, they stand for the number itself. The connections stand for the possible divisors. Whoever draws a figure “every k-th point” computes the fraction n : k geometrically:

If n is divisible by k, the figure breaks 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 divisor g without k dividing n, the figure breaks into g equal stars.
If k is exactly half of n, all lines pass through the centre.

n-gon

PolygonCompound polygonStar polygonCompound starLine star

Choose a number and see which figures its simplex breaks into. The colour shows the divisibility, the number underneath the quotient n : k.

Building the 30-gon

At first sight a large simplex looks chaotic. But think of it from the outline inwards and a strict order appears. The 30-gon shows it well. Start at any vertex:

Join each vertex to the next: one round, the 30-gon itself.
Join every second vertex: you have to go round the circle twice before all vertices are reached. That is the first star figure.
Every third vertex: three rounds, the second star figure.

And so on, 15 figures in all. At the 15th, every 15th vertex, the lines pass through the centre, because 30 : 15 = 2. After that everything would repeat in reverse order. The many crossings of the star figures form small quadrilaterals, a coordinate system in the circle, as it were. In a simplex every vertex is indeed joined to every other, but to see the order behind it one must think of it this way, as a stack of symmetric figures.

Points

The 30-gon simplex. The buttons show and hide the 15 figures one by one, from the outline to the lines through the centre.

The four groups

These details matter for understanding the four groups of figures and their counterpart in number theory (see The key to the tetractys).

1. Polygon, drawn in one stroke: the polygons themselves, the n-gon for the number n. The first true polygon is the triangle, since point and line enclose no area.

2. Star polygon, drawn in one stroke, when number and divisor are coprime. Simplexes with a prime number of vertices consist only of such figures, apart from 2 and 3: line and triangle are also drawn in one stroke, but they are not stars. The first star is the pentagram, 5/2 = 2.5. The heptagon has two for the first time: 7/2 = 3.5 and 7/3. From seven on every number has such coprime stars, the 30-gon too. Join every eleventh vertex there and the construction is quite irregular step by step, yet the result is perfectly symmetric.

3. Compound polygon, a star of several polygons, when the number is divisible by the divisor. All simplexes from six vertices on whose number of vertices is not prime contain such figures. The first and best known is the hexagram, 6/2 = 3. Two points (2/2) or two lines (4/2) would be divisible too, but they are not polygons. The octagon holds the second figure of this kind, 8/2 = 4, two squares.

4. Compound star, a star of several stars, when number and divisor share a divisor without the divisor dividing the number. All composite numbers from ten on contain such figures, with a single exception, 12. The first is the double pentagram, 10/4 = 2.5. The next is in the 14-gon: 14/4 = 3.5, two seven-pointed stars. In the 30-gon every twelfth vertex gives 30/12 = 2.5, a star of six pentagrams.

n k

Try it yourself: 30/12 breaks into six pentagrams, 30/11 becomes a single star, 14/4 two seven-pointed stars. With ‘Draw’ the lines appear in the order in which you would draw them.

A word of warning about the names: the “compound polygon” is strictly speaking a star polygon too, just one made of polygons. To avoid confusion it is best to go by the colours.

Fanned out: the lambdoma

Now the figures of all simplexes can be fanned out in a coordinate system: the number at the top, the divisor on the left, in each cell the matching figure, from the outline to the lines through the centre.

Size

k = 1: the number itselfdivisiblecoprimecommon factor, not divisible

The figures of the simplexes 1 to 14, fanned out. Column = number, row = divisor, the colour shows the kind of figure.

Tilt this table by 45 degrees and you get exactly the lambdoma of harmonics, the most important diagram of music theory (see The lambdoma). Strikingly, only a quarter of the coordinate system is filled with figures. All even simplexes end with lines through the centre, and this line star always has the quotient 2, starting with the number 2 itself, a single line. One could fill the whole field, but then the figures would only repeat. The diagonal on which number and divisor are equal, 1/1, 2/2, 3/3 …, has its counterpart too: in harmonics it is called the generator-tone line, and all its cells have the value 1.

So it is the centre of the circle that sets the limit. The line through it divides the circle into two halves and corresponds in the table to the bound 2, which marks off exactly a quarter of the field. The diagonal halving the field describes a full turn of the circle, because after the bound 2 the same geometry runs once more in mirror image, up to that diagonal. Below the bound 2 divisibility no longer matters for the primes, since there no number can be split into at least two whole parts greater than 1. That circle geometry, with the number π, also plays a role in the Riemann zeta function has still not really been explained.

Two and a half

Above the bound 2 the third relation, “not divisible, but with a common divisor”, begins exactly at ten with the divisor 4: 10 : 4 = 2.5, two and a half. The pentagram and all stars made of pentagrams always have this quotient 2.5. Here the structure in steps of ten begins, which the section Number theory deals with.

The simplexes were drawn and studied up to the 100-gon. Numbers up to 100 are too few for a proof, one may object. But this geometry coincides completely with the sieve of Eratosthenes, and the sieve is known to hold into infinity. Tables up to 100 are therefore enough to show the principle.

Circle and circle number

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

Two large examples

An even and an odd simplex show the difference especially clearly:

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.