Boole(b)an

Suppose you allow a point to jump at random half-way towards one of the four vertices of a square. But not entirely at random — you ban the point from jumping towards the same vertex twice (or more) in a row. You get this pattern:

ban on v(i) + 0


It’s a fractal, that is, a shape that contains smaller and smaller copies of itself. Next you ban the point from jumping towards the vertex one place clockwise of the vertex it last jumped towards (i.e., it can jump towards, say, vertex 2 as many times as it likes, but it can’t jump towards vertex 2+1 = 3, and so on). You get this fractal:

ban on v(i) + 1


Now ban it from jumping towards the vertex two places clockwise of the vertex it last jumped towards (i.e., it can’t jump towards the diagonally opposite vertex). You get this fractal:

ban on v(i) + 2


And if you ban the point from jumping towards the vertex three places clockwise of the last vertex, you get a mirror-image of the v(i)+1 fractal (see above):

ban on v(i) + 3


The fractals above have a memory one vertex into the past: the previous vertex. Let’s try some fractals with a memory two vertices into the past: the previous vertex and the pre-previous vertex (and even the pre-pre-previous vertex).

But this time, let’s suppose that sometimes the point can’t jump if the previous or pre-previous isn’t equal to v(i) + n. So sometimes the jump is banned when the test is true, sometimes when the test is false — you might call it a boolean ban or boole(b)an. Using boole(b)ans, you can get this set of fractals:
















With these fractals, the boolean test sends the point back to the center of the square:











Posteriously post-posted

Boole(b)an #2 — a later look at ban-fractals

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