A pentomino is one of the shapes created by laying five squares edge-to-edge. There are twelve of them (not counting reflections) and this is the P-pentomino:

But it’s not just a pentomino, it’s also a rep-tile, or a shape that can divided into smaller copies of itself. There are two ways of doing this (I’ve rotated the pentomino 90° to make the images look better):

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Once you’ve divided the shape into four copies, you can divide the copies, then the copies of the copies, and the copies of the copies of the copies, and so on for ever:

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And if you’ve got a reptile, you can turn it into a fractal. Simply divide the shape, discard one or more copies, and continue:

Pentomino-based fractal stage 1

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Pentomino-based fractal stage 2

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Pentomino-based fractal stage 3

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Stage 4

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Stage 5

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Stage 6

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Stage 7

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Stage 8

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Stage 9

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Stage 10

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Here are some more fractals created using the same divide-and-discard process:

Animated version

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Animated version

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You can also use variants on a standard rep-tile dissection, like rotating the copies or trying different patterns of dissection at different levels to see what new shapes appear:

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