# Get Your Prox Off #3

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

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

Nor is this one: v = 4, d = (1,3,4)

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)

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) v = 4 + s1, d = (1,2,5) v = 4 + s1, d = (1,2,7) v = 4 + s1, d = (1,3,8) v = 4 + s1, d = (1,4,6) v = 4 + s1, d = (1,7,8) v = 4 + s1, d = (2,3,8) v = 4 + s1, d = (2,4,8)

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

Now here are some pentagonal fractals-by-distance: v = 5, d = (1,2,5), j = 1/2 v = 5 + s1, d = (1,2,7) v = 5 + s1, d = (1,2,8) v = 5 + s1, d = (1,2,9) v = 5 + s1, d = (1,9,10) v = 5 + s1, d = (1,10), j = 2/3 v = 5 + s1, d = (various), j = 2/3 (animated)

And now some hexagonal fractals-by-distance: v = 6, d = (1,2,4), j = 1/2 v = 6, d = (1,3,5) v = 6, d = (1,3,6) v = 6, d = (1,2,3,4) v = 6 + central point, d = (1,2,3,4) v = 6, d = (1,2,3,6) v = 6, d = (1,2,4,6) v = 6, d = (1,3,4,5) v = 6, d = (1,3,4,6)

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

Elsewhere other-accessible:

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

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