Performativizing the Polygonic

Maths is a mountain: you can start climbing in different places and reach the same destination. There are many ways of proving the irrationality of √2 or the infinitude of the primes, for example. But you can also arrive at the same destination by accident. I’ve found that when I use different methods of creating fractals. The same fractals appear, because apparently different algorithms are actually the same underneath.

But different methods can create unique fractals too. I’ve found some new ones by using what might be called point-to-point recursion. For example, there are ten ways to select three vertices from the five vertices of a pentagon: (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 3, 4), (1, 3, 5), (1, 4, 5), (2, 3, 4), (2, 3, 5), (2, 4, 5), (3, 4, 5). Find the midpoint of the first three-point set, (1, 2, 3). Then select two vertices to go with this midpoint, creating a new three-point set, and find the midpoint again. And so on. The process looks like this, with the midpoints shown for all the three-point sets found at each stage: vertices = 5, choose sets of 3 points, find mid-point of each  At stage 5, the fractal looks like this: v = 5, p = 3

Note that when pixels are used again, the colour changes. That’s another interesting thing about maths: limits can sometimes produce deeper results. If these fractals were drawn at very high resolution, pixels would only be used once and the colour would never change. As it is, low resolution means that pixels are used again and again. But some are used more than others, which is why interesting colour effects appear.

If the formation of the fractal is animated, it looks like this (with close-ups of even deeper stages): Here are some more examples: v = 4 + central point, p = 2 (cf. Fingering the Frigit) v = 4c, p = 2 (animated) v = 4, p = 3  v = 5, p = 4  v = 5 + central point, p = 3  v = 5c, p = 4 v = 5c, p = 5 v = 6 + 1 point between each pair of vertices, p = 6 v = 6, p = 2 v = 6, p = 3  v = 6, p = 4 v = 6c, p = 2 (cf. Fingering the Frigit)  v = 6c, p = 3  v = 6c, p = 4 v = 7, p = 3 v = 7, p = 4  v = ,7 p = 5  v = 7c, p = 4 v = 3+1, p = 2 v = 3+1, p = 3 v = 3+1, p = 4 v = 3+2, p = 5 v = 3c+1, p = 2 v = 3c+1, p = 4 v = 3c, p = 2 v = 3c, p = 3 v = 4+1, p = 3 v = 4+1, p = 4 v = 4+1, p = 6 v = 4+1, p = 2 v = 4c+1, p = 4 v = 4c, p = 3  v = 5+1, p = 4 (and more) v = 5, p = 2

Fingering the Frigit

Fingers are fractal. Where a tree has a trunk, branches and twigs, a human being has a torso, arms and fingers. And human beings move in fractal ways. We use our legs to move large distances, then reach out with our arms over smaller distances, then move our fingers over smaller distances still. We’re fractal beings, inside and out, brains and blood-vessels, fingers and toes.

But fingers are fractal are in another way. A digit – digitus in Latin – is literally a finger, because we once counted on our fingers. And digits behave like fractals. If you look at numbers, you’ll see that they contain patterns that echo each other and, in a sense, recur on smaller and smaller scales. The simplest pattern in base 10 is (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). It occurs again and again at almost very point of a number, like a ten-hour clock that starts at zero-hour:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9…
10, 11, 12, 13, 14, 15, 16, 17, 18, 19…
200… 210… 220… 230… 240… 250… 260… 270… 280… 290…

These fractal patterns become visible if you turn numbers into images. Suppose you set up a square with four fixed points on its corners and a fixed point at its centre. Let the five points correspond to the digits (1, 2, 3, 4, 5) of numbers in base 6 (not using 0, to simplify matters):

1, 2, 3, 4, 5, 11, 12, 13, 14, 15, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 41, 42, 43, 44, 45, 51, 52, 53, 54, 55, 61, 62, 63, 64, 65… 2431, 2432, 2433, 2434, 2435, 2441, 2442, 2443, 2444, 2445, 2451, 2452…

Move between the five points of the square by stepping through the individual digits of the numbers in the sequence. For example, if the number is 2451, the first set of successive digits is (2, 4), so you move to a point half-way between point 2 and point 4. Next come the successive digits (4, 5), so you move to a point half-way between point 4 and point 5. Then come (5, 1), so you move to a point half-way between point 5 and point 1.

When you’ve exhausted the digits (or frigits) of a number, mark the final point you moved to (changing the colour of the pixel if the point has been occupied before). If you follow this procedure using a five-point square, you will create a fractal something like this:  A pentagon without a central point using numbers in a zero-less base 7 looks like this:    But the images above are just the beginning. If you use a fixed base while varying the polygon and so on, you can create images like these (here is the program I used):   