Ornaments in four-four time
Fill the simplexes with colour in alternation and ornaments appear whose centre is filled for four numbers in a row, then empty for four.
It starts with a chance find in a drawing program. Every possible star polygon is drawn into each polygon. Then all regions are filled in alternation: every region created by the crossings is either coloured or empty, depending on how many lines you have to cross from outside to reach it. The result is a series of ornaments, and their sequence holds a surprise.
The simplexes from 3 to 26 with alternating fill. Under each number is the total of steps to the centre. ● means the centre is filled, ○ means it is empty. The shaded blocks mark the rhythm of four.
The rhythm of four
The ornaments of the numbers 3 to 6 have a filled centre. For 7 to 10 the centre is empty. For 11 to 14 it is filled again, and so it goes on without end, always in blocks of four. Each ornament shows the character of its number, namely the numbers by which it is divisible.
The even numbers show something else: their last figure, joining every (n/2)-th point, gives only lines through the centre, a “line star” without area. It drops out when the regions are filled. That is why the centre is more open for even numbers than for odd ones.
Where the rhythm comes from
To understand it, fan out the figures of a simplex one by one. The polygon itself has one step: it is simply filled. For the star “every second point” you go round the centre twice. Two steps appear: filled points outside, an empty field inside. Each further figure skips one more point, circles the centre once more and has one more step.
Towards the centre the steps therefore add up to 1, 1 + 2, 1 + 2 + 3 … These are the triangular numbers again. The centre is filled when this sum is odd. But the triangular numbers run odd, odd, even, even, odd, odd … Since each figure stays the same for two consecutive numbers (the even number loses its line star), the rhythm of four emerges.
Compare it with the daisy oracle “she loves me, she loves me not”: whether you end on the first or the second phrase depends on whether the number of petals is even or odd. In the ornaments this alternation doubles, because two series of numbers interlock: the number of vertices and the number of steps. The double switch becomes a rhythm of four.
And the ten?
That a rhythm of four appears would not by itself be a tetractys. Two further observations lead to ten:
- The accumulated steps follow the triangular numbers, and 10 is the fourth of them.
- The decagon is the only simplex in which the number of vertices equals the number of steps. Smaller simplexes have fewer steps than vertices, larger ones more.
How the same shows up in the angle sums is described under Angle sums and full circles. Which four types of figure the decagon contains is explained in The key to the tetractys.