When you replace a square with four smaller squares, each a quarter the size of the original, the smaller squares occupy the same area, because 4 * ¼ = 1. If you discard one sub-square, then divide each of the three remaining sub-squares into four sub-sub-square, discard one sub-sub-quare and repeat, you create fractals like those I looked at in Squaring and Paring. The fractals stay within a fixed boundary.

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Square replaced with four smaller squares, each ¼th the size of the original

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Animated fractal

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Static fractal

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This time I want to look at a slightly different process. Replace a square with nine smaller squares each a quarter the size of the original. Now the sub-squares occupy a larger area than the original, because 9 * ¼ = 2¼. If you discard — or cull — sub-squares and repeat, the resultant fractal grows beyond the original boundary. Indeed, sub-squares start to overlap, so you can use colours to represent how often a particular pixel has been covered with a square. Here is an example of this process in action:

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Square replaced with nine smaller squares, each ¼th the size of the original

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Animated fractal

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Static fractal #1

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Static fractal #2

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Here are the individual stages of a more complex fractal that uses the second process:

Stage 1

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Stage 2

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Stage 3

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Stage 4

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Stage 5

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Stage 6

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Stage 7

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Stage 8

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Stage 9 (compare Fingering the Frigit and Performativizing the Polygonic)

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Stage 10

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Animated version

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Static version #1

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Static version #2

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And here are some more of the fractals you can create in a similar way:

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Static version #1

Static version #2

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Static version #2

Static version #2

Static version #3

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Various fractals in an animated gif

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