The tetractys in the fractal polygons
The Sierpiński procedure applied to n-gons. From the pentagon on the parts overlap, and in the decagon the overlaps arrange themselves as 1 + 2 + 3 + 4 = 10.
The Sierpiński triangle is the best-known fractal. Through Pascal’s triangle it is closely tied to number theory: the odd entries there form exactly this pattern (see Multidimensional tetrahedra).
The Sierpiński triangle: every triangle is replaced by three half-sized ones at its corners.
The chaos game
There is an astonishingly simple way to create the Sierpiński triangle. Start at any point, pick a corner of the triangle at random, go half the distance towards it and set a point. Repeat. After a few hundred steps the fractal can be glimpsed, after tens of thousands it is clear. For other polygons you must go a little less than half the way so that the parts just touch. For the pentagon this gives a beautiful, clearly fractal pattern.
The chaos game. Use the slider to change the number of vertices.
The strict rules
If instead the rules of the Sierpiński triangle are applied strictly to other polygons, the result looks unremarkable at first:
- The sub-figures lie within the polygon itself.
- There are as many sub-figures as the polygon has vertices.
- The sides are halved at every step.
With the triangle, gaps appear between the parts. Not with the square: four half-sized squares fill the large one without gaps, because the square tiles with itself. From the pentagon on, the parts must overlap and cluster around a common centre. The square is the last fractal without overlaps.
The strict version. The second slider shows further levels. In the pentagon a hole remains at the centre, and every further level punches an inverted pentagon out of each part.
In the pentagon a hole appears at the centre, just as in the Sierpiński triangle, and every further level punches a hole out of each of the five parts again, an upside-down pentagon. Such holes appear only in odd polygons. From the heptagon on, the multiple overlaps allow no more holes in the further levels. Only the hole at the centre of the first level remains.
With this, the core of the tetractys is already visible: only the triangle and the square divide cleanly into themselves, just as they alone fill the plane and space without gaps and form the lattices of sphere packings. From the pentagon on, the polygons overlap. Five is the crosser of the boundaries of space.
The decagon
Just as the Sierpiński triangle divides into three triangles, the first-level fractal decagon divides into ten decagons, which overlap. The overlaps of neighbouring decagons create 40 quadrilaterals in the whole fractal. Each of the ten sub-decagons contains exactly ten quadrilaterals, produced by the overlaps with six neighbouring decagons. Smaller polygons contain fewer quadrilaterals than vertices, larger ones more.
And again they are arranged like the tetractys: from the outside towards the centre in rows of 1, 2, 3 and 4.
The first-level fractal decagon. The alternating fill makes the small quadrilaterals in the overlaps visible.
Three and five
This geometry holds as much information about the primes as the simplexes do. The polygons whose appearance sets them apart from all others, the triangle and the pentagon, also lead the way in the question of the primes: three produces the rhythm of six of the twin primes in the sieve of Eratosthenes, five chops this order up again. See The order of the primes.