Symmetry arising from symmetry isn’t surprising. But what about symmetry arising from asymmetry? You can find both among the rep-tiles, which are geometrical shapes that can be completely replaced by smaller copies of themselves. A square is a symmetrical rep-tile. It can be replaced by nine smaller copies of itself:

Rep-9 Square

If you trim the copies so that only five are left, you have a symmetrical seed for a symmetrical fractal:

Fractal cross stage #1

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Fractal cross #2

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Fractal cross #3

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Fractal cross #4

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Fractal cross #5

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Fractal cross #6

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Fractal cross (animated)

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Fractal cross (static)

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If you trim the copies so that six are left, you have another symmetrical seed for a symmetrical fractal:

Fractal Hex-Ring #1

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Fractal Hex-Ring #2

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Fractal Hex-Ring #3

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Fractal Hex-Ring #4

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Fractal Hex-Ring #5

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Fractal Hex-Ring #6

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Fractal Hex-Ring (animated)

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Fractal Hex-Ring (static)

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Now here’s an asymmetrical rep-tile, a nonomino or shape created from nine squares joined edge-to-edge:

Nonomino

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It can be divided into twelve smaller copies of itself, like this:

Rep-12 Nonomino (discovered by Erich Friedman)

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If you trim the copies so that only five are left, you have an asymmetrical seed for a familiar symmetrical fractal:

Fractal cross stage #1

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Fractal cross #2

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Fractal cross #3

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Fractal cross #4

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Fractal cross #5

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Fractal cross #6

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Fractal cross (animated)

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Fractal cross (static)

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If you trim the copies so that six are left, you have an asymmetrical seed for another familiar symmetrical fractal:

Fractal Hex-Ring #1

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Fractal Hex-Ring #2

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Fractal Hex-Ring #3

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Fractal Hex-Ring #4

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Fractal Hex-Ring #5

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Fractal Hex-Ring (animated)

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Fractal Hex-Ring (static)

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Elsewhere other-available:

• Square Routes Re-Re-Visited