Simplexes and star polygons
When every point is joined to every other, the lines fall apart into stars, and the number of lines is again a triangular number.
A simplex is the simplest figure that can fill a dimension: a point in zero dimensions, a segment in one, a triangle in two, a tetrahedron in three. The rule of construction is always the same: add a new point and join it to all the previous ones.
From four dimensions on, simplexes can no longer be built, but they can be projected onto the plane. This is clearest when the vertices are spaced evenly around a circle and every one is joined to every other.
The buttons show and hide the individual star polygons. With five points the simplex consists of a pentagon and a pentagram.
Numbers in the circle
The natural numbers can be read as figures in a circle. One is the centre. Two is the diameter. Three gives the first true polygon that can be drawn in the circle. Four is the first polygon in which connecting lines pass exactly through the centre, and also the only one whose angles together make exactly one full circle (see Angle sums and full circles). With five a star can be drawn for the first time, the pentagram, and for the first time the lines also cross away from the centre. The larger the number, the more crossings appear.
The numbers 2 to 13 as simplexes in the circle: diameter, triangle, square with diagonals through the centre, pentagon with pentagram and so on. One would be the centre alone.
The edges arrange themselves into stars
At first glance it is a tangle of lines. But sort the lines by how many places they skip on the circle, and the picture falls apart into regular figures:
- Lines to the immediate neighbour form the outline, a regular polygon.
- Lines to the next-but-one point form a star, with five points the pentagram.
- Larger jumps give further, sharper stars.
Mathematicians write such stars as {n/k}: n points, each joined to its k-th neighbour. Every simplex is thus nothing but a set of nested star polygons.
The triangular number returns
Count the lines and something familiar appears. With n points, the first point has n − 1 connections, the second n − 2 new ones, and so on down to the last, which adds none. The sum
(n − 1) + (n − 2) + … + 1 = T(n − 1)
is a triangular number. The simplex with five vertices therefore has exactly ten edges, as many as the tetractys has points. The figure of ten points and the figure of ten lines count the same thing.
Two number lines in the circle
The crossing points of the lines are not random. From the edge towards the centre they arrange themselves by the numbers that divide the number of vertices. The circle thus carries two series of numbers: the circumference counts the natural numbers, the radius counts their divisors. The many crossings form a net of small quadrilaterals, a coordinate system in the circle. It corresponds to the lambdoma of harmonics, and conversely the star polygons of the simplexes can be fanned out into the lambdoma (see Tetractys and the Cartesian coordinate system and The lambdoma).
In this net, through the doubling of even and odd numbers, the tetractys shows itself. The geometry of the simplexes is multidimensional and only one way, if an all-embracing one, of making it visible. It can also be recognised in three-dimensional space. Beyond question: the geometry of the circle is linked to the divisibility of the natural numbers. Research on the Riemann hypothesis, too, confirms the key role of the number π in the distribution of the primes.
What matters is that all this works without the decimal system. That we count on ten fingers must not be the reason for the ten of the tetractys. The value of this geometry is precisely that it frees us from the mental limits of the decimal system. One could call the simplexes another state of matter of numbers.
Self-similarity
A second pattern lies in the triangle itself. Join the midpoints of a triangle’s sides and it splits into four smaller ones. Leave out the middle one and repeat with the other three, and you get the Sierpiński triangle. Every part looks like the whole.
With every step the number of triangles triples. The area tends to zero, yet the figure remains a triangle.
The pentagram has the same property of repeating itself within itself: at its centre lies a smaller pentagon, in which another pentagram can be drawn, without end. The ratios in which its lines cut each other lead to the golden ratio.