Suppose a point traces all possible routes jumping half-way towards the three vertices of an equilateral triangle. A special kind of shape appears — a fractal called the Sierpiński triangle that contains copies of itself at smaller and smaller scales:

Sierpiński triangle, jump = 1/2

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And what if the point jumps 2/3rds of the way towards the vertices as it traces all possible routes? You get this dull fractal:

Triangle, jump = 2/3

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But if you add targets midway along each side of the triangle, you get this fractal with the 2/3rds jump:

Triangle, jump = 2/3, side-targets

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Now try the 1/2-jump triangle with a target at the center of the triangle:

Triangle, jump = 1/2, central target

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And the 2/3-jump triangle with side-targets and a central target:

Triangle, jump = 2/3, side-targets, central target

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But why stop at simple jumps like 1/2 and 2/3? Let’s take the distance to the target, td, and use the function 1-(sqrt(td/7r)), where sqrt() is the square-root and 7r is 7 times the radius of the circumscribing circle:

Triangle, jump = 1-(sqrt(td/7r))

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Here’s the same jump with a central target:

Triangle, jump = 1-(sqrt(td/7r)), central target

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Now let’s try squares with various kinds of jump. A square with a 1/2-jump fills evenly with points:

Square, jump = 1/2 (animated)

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The 2/3-jump does better with a central target:

Square, jump = 2/3, central target

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Or with side-targets:

Square, jump = 2/3, side-targets

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Now try some more complicated jumps:

Square, jump = 1-sqrt(td/7r)

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Square, jump = 1-sqrt(td/15r), side-targets

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And what if you ban the point from jumping twice or more towards the same target? You get this fractal:

Square, jump = 1-sqrt(td/6r), ban = prev+0

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Now try a ban on jumping towards the target two places clockwise of the previous target:

Square, jump = 1-sqrt(td/6r), ban = prev+2

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And the two-place ban with a central target:

Square, jump = 1-sqrt(td/6r), ban = prev+2, central target

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And so on:

Square, jump = 1-sqrt(td/6.93r), ban = prev+2, central target

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Square, jump = 1-sqrt(td/7r), ban = prev+2, central target

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These fractals take account of the previous jump and the pre-previous jump:

Square, jump = 1-sqrt(td/4r), ban = prev+2,2, central target

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Square, jump = 1-sqrt(td/5r), ban = prev+2,2, central target

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Square, jump = 1-sqrt(td/6r), ban = prev+2,2, central target

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Elsewhere other-accessible

• Boole(b)an #2 — fractals created in similar ways