I’ve looked at lot at the fractals created when you randomly (or quasi-randomly) choose a vertex of a square, then jump half of the distance towards it. You can ban jumps towards the same vertex twice in a row, or jumps towards the vertex clockwise or anticlockwise from the vertex you’ve just chosen, and so on.

But you don’t have to choose vertices directly: you can also choose them by distance or proximity (see “Get Your Prox Off” for an earlier look at fractals-by-distance). For example, this fractal appears when you can jump half-way towards the nearest vertex, the second-nearest vertex, and the third-nearest vertex (i.e., you can’t jump towards the fourth-nearest or most distant vertex):

vertices = 4, distance = (1,2,3), jump = 1/2

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It’s actually the same fractal as you get when you choose vertices directly and ban jumps towards the vertex diagonally opposite from the one you’ve just chosen. But this fractal-by-distance isn’t easy to match with a fractal-by-vertex:

v = 4, d = (1,2,4), j = 1/2

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Nor is this one:

v = 4, d = (1,3,4)

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This one, however, is the same as the fractal-by-vertex created by banning a jump towards the same vertex twice in a row:

v = 4, d = (2,3,4)

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The point can jump towards second-nearest, third-nearest and fourth-nearest vertices, but not towards the nearest. And the nearest vertex will be the one chosen previously.

Now let’s try squares with an additional point-for-jumping-towards on each side (the points are numbered 1 to 8, with points 1, 3, 5, 7 being the true vertices):

v = 4 + s1 point on each side, d = (1,2,3)

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v = 4 + s1, d = (1,2,5)

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v = 4 + s1, d = (1,2,7)

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v = 4 + s1, d = (1,3,8)

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v = 4 + s1, d = (1,4,6)

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v = 4 + s1, d = (1,7,8)

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v = 4 + s1, d = (2,3,8)

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v = 4 + s1, d = (2,4,8)

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And here are squares where the jump is 2/3, not 1/2, and you can choose only the nearest or third-nearest jump-point:

v = 4, d = (1,3), j = 2/3

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v = 4 + s1, d = (1,3), j = 2/3

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Now here are some pentagonal fractals-by-distance:

v = 5, d = (1,2,5), j = 1/2

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v = 5 + s1, d = (1,2,7)

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v = 5 + s1, d = (1,2,8)

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v = 5 + s1, d = (1,2,9)

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v = 5 + s1, d = (1,9,10)

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v = 5 + s1, d = (1,10), j = 2/3

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v = 5 + s1, d = (various), j = 2/3 (animated)

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And now some hexagonal fractals-by-distance:

v = 6, d = (1,2,4), j = 1/2

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v = 6, d = (1,3,5)

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v = 6, d = (1,3,6)

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v = 6, d = (1,2,3,4)

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v = 6 + central point, d = (1,2,3,4)

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v = 6, d = (1,2,3,6)

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v = 6, d = (1,2,4,6)

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v = 6, d = (1,3,4,5)

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v = 6, d = (1,3,4,6)

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v = 6, d = (1,4,5,6)

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Elsewhere other-accessible:

• Get Your Prox Off — an earlier look at fractals-by-distance
• Get Your Prox Off # 2 — and another