Square Routes Re-Re-Re-Revisited

Discovering something that’s new to you in recreational maths is good. But so is re-discovering it by a different route. I’ve long been passionate about what happens when a point is allowed to jump repeatedly halfway towards the randomly chosen vertices of a square. If the point can choose any vertex any number of times, the interior of the square fills slowly and completely with points, like this:

Point jumping at random halfway towards vertices of a square


However, if the point is banned from jumping towards the same vertex twice or more in a row, an interesting fractal appears:

Fractal #1 — ban on jumping towards vertex vi twice or more


If the point can’t jump towards the vertex one place clockwise of the vertex it’s just jumped towards, this fractal appears:

Fractal #2 — ban on jumping towards vertex vi+1


If the point can’t jump towards the vertex two places clockwise of the vertex it’s just jumped towards, this fractal appears (two places clockwise is also two places anticlockwise, i.e. the banned vertex is diagonally opposite):

Fractal #3 — ban on jumping towards vertex vi+2


Now I’ve discovered a new way to create these fractals. You take a filled square, divide it into smaller squares, then remove some of them in a systematic way. Then you do the same to the smaller squares that remain. For fractal #1, you do this:

Fractal #1, stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #1 (animated)


For fractal #2, you do this:

Fractal #2, stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #2 (animated)


For fractal #3, you do this:

Fractal #3, stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #3 (animated)


If the sub-squares are coloured, it’s easier to understand how, say, fractal #1 is created:

Fractal #1 (coloured), stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #1 (coloured and animated)


The fractal is actually being created in quarters, with one quarter rotated to form the second, third and fourth quarters:

Fractal #1, quarter









Here’s an animation of the same process for fractal #3:

Fractal #3 (coloured and animated)


So you can create these fractals either with a jumping point or by subdividing a square. But in fact I discovered the subdivided-square route by looking at a variant of the jumping-point route. I wondered what would happen if you took a point inside a square, allowed it to trace all possible routes towards the vertices without marking its position, then imposed the restriction for Fractal #1 on its final jump, namely, that it couldn’t jump towards the vertex it jumped towards on its previous jump. If the point is marked after its final jump, this is what appears (if the routes chosen had been truly random, the image would be similar but messier):

Fractal #1, restriction on final jump


Then I imposed the same restriction on the point’s final two jumps:

Fractal #1, restriction on final 2 jumps


And final three jumps:

Fractal #1, restriction on final 3 jumps


And so on:

Fractal #1, restriction on final 4 jumps


Fractal #1, restriction on final 5 jumps


Fractal #1, restriction on final 6 jumps


Fractal #1, restriction on final 7 jumps


Here are animations of the same process applied to fractals #2 and #3:

Fractal #2, restrictions on final 1, 2, 3… jumps


Fractal #3, restrictions on final 1, 2, 3… jumps


The longer the points are allowed to jump before the final restriction is imposed on their n final jumps, the more densely packed the marked points will be:

Fractal #1, packed points #1


Packed points #2


Packed points #3


Eventually, the individual points will form a solid mass, like this:

Fractal #1, solid mass of points


Fractal #1, packed points (animated)


Previously pre-posted (please peruse):

Square Routes
Square Routes Revisited
Square Routes Re-Revisited
Square Routes Re-Re-Revisited

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