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. It was discovered with a vector drawing program: fill the polygon and the stars, then run the command “combine objects”, and filled and empty areas alternate. You need not understand what follows at once. At first it is enough to let the ornaments work on you and look for the fourness in their sequence yourself.
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
A rhythm born of a designer’s whim and a computer’s arbitrariness need mean nothing. Yet none of at least four mathematicians asked could say why the change comes after exactly four ornaments. The answer lies in a simple count.
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: a change occurs when both the horizontal and the vertical number are odd. What gives pause for thought is that each ornament thus carries a coordinate system of its own that produces this fourness.
Why the rhythm doubles
The doubling has a geometric reason. Lines through a polygon, seen from one vertex, run only as far as the centre. Once the centre is passed, the pattern repeats and the same star figures merely overlay one another. That is why a coordinate system of number and divisor is filled with star figures in exactly one quarter only (see The simplex – the fixed form of number). Moreover, all star figures of even simplexes have angle sums of whole full circles, whereas those of odd simplexes always have half a circle more.
Pascal’s triangle supports this reading. Its dark cells contain only odd numbers, the light ones only even numbers (see Multidimensional tetrahedra). And the steps of the ornaments reappear in it: the number of steps per round follows the series 1, 2, 3, 4, 5 … on the second diagonal, the total number of steps the triangular numbers 1, 3, 6, 10, 15 … on the third. The odd triangular numbers 1 and 3 belong to the four numbers 3 to 6, the even ones 6 and 10 to the next four, 7 to 10, and so on, exactly in the rhythm of four of the ornaments.
One would not need to attach any meaning to this if other symmetries did not show the same (see The triangle of points) and if it were not evidently connected with the distribution of the primes. So this one example touches on three typically Pythagorean themes at once:
- the tetractys, the numbers one to ten
- the character of numbers: the Pythagoreans saw in the odd numbers the male-active and in the even ones the female-passive principle, an idea dismissed today as “mystical” that gains a solid meaning here
- the pentagram, their sign of recognition, since the fourth star figure of the decagon consists of two pentagrams, and precisely these two pentagrams are what the question of the primes is about
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. Up to the octagon the simplexes have fewer steps than vertices, from the hendecagon on more, and the nonagon lies just above with ten steps.
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.