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.
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.
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.