As I’ve shown pre-previously on Overlord-in-terms-of-issues-around-the-Über-Feral, you can create interesting fractals by placing restrictions on a point jumping inside a fractal towards a randomly chosen vertex. For example, the point can be banned from jumping towards the same vertex twice in a row, and so on.

But you can use other restrictions. For example, suppose that the point can jump only once or twice towards any vertex, that is, (j = 1,2). It can then jump towards the same vertex again, but not the same number of times as it previously jumped. So if it jumps once, it has to jump twice next time; and vice versa. If you use this rule on a pentagon, this fractal appears:

v = 5, j = 1,2 (black-and-white)

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v = 5, j = 1,2 (colour)

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If the point can also jump towards the centre of the pentagon, this fractal appears:

v = 5, j = 1,2 (with centre)

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And if the point can also jump towards the midpoints of the sides:

v = 5, j = 1,2 (with midpoints)

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v = 5, j = 1,2 (with midpoints and centre)

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And here the point can jump 1, 2 or 3 times, but not once in a row, twice in a row or thrice in a row:

v = 5, j = 1,2,3

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v = 5, j = 1,2,3 (with centre)

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Here the point remembers its previous two moves, rather than just its previous move:

v = 5, j = 1,2,3, hist = 2 (black-and-white)

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v = 5, j = 1,2,3, hist = 2

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v = 5, j = 1,2,3, hist = 2 (with center)

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v = 5, j = 1,2,3, hist = 2 (with midpoints)

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v = 5, j = 1,2,3, hist = 2 (with midpoints and centre)

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And here are hexagons using the same rules:

v = 6, j = 1,2 (black-and-white)

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v = 6, j = 1,2

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v = 6, j = 1,2 (with centre)

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And octagons:

v = 8, j = 1,2

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v = 8, j = 1,2 (with centre)

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v = 8, j = 1,2,3, hist = 2

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v = 8, j = 1,2,3, hist = 2

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v = 8, j = 1,2,3,4 hist = 3

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v = 8, j = 1,2,3,4 hist = 3 (with center)

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