: mathematics :

a simple fractal - the Peano curve


A fractal is a structure which exhibits self-similarity on a number of scales.

cauliflower A cauliflower is one simple example from everyday life. You can see in the picture that each floret of the cauliflower is like a miniature of the whole cauliflower. Also each floret is composed of tiny sub-florets, and this pattern continues on at least another level below that.

The fractal at the top of this page, and that also forms the background, is called a Peano curve, discovered by Giuseppe Peano in 1890 and incorporated into fractal theory by Benoit Mandelbrot in 1977.

peanoTo make this fractal, take a straight line between two fixed points (represented in the diagram by dots) and replace the part between the dots with the zigzag path illustrated (illustration taken from Mandelbrot's book p.62).

Then repeat this process for each of the straight line segments in the path (called the curve even though it's not curvy - don't ask me why). You will then end up with something resembling a chequerboard. To make it look like the pattern I have made above you simply shave off the corners.

My original was made using a computer program in Basic or Logo (I forget which) on my old BBC model B computer. I then used the printout to produce a cutout in coloured paper (colour printers were not invented then). I found the cutout among some old papers the other day and scanned it in to produce the image at the top of this page. I suppose it would be simpler just to write a routine to make the final image, but modern computers don't encourage you to write your own routines in the way we all had to when the only computer normal folks could afford was a BBC model B or a Sinclair-ZX81.

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