# Holey Trimmetry

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 Fractal cross #2 Fractal cross #3 Fractal cross #4 Fractal cross #5 Fractal cross #6 Fractal cross (animated) Fractal cross (static)

If you trim the copies so that six are left, you have another symmetrical seed for a symmetrical fractal: Fractal Hex-Ring #1 Fractal Hex-Ring #2 Fractal Hex-Ring #3 Fractal Hex-Ring #4 Fractal Hex-Ring #5 Fractal Hex-Ring #6 Fractal Hex-Ring (animated) Fractal Hex-Ring (static)

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

It can be divided into twelve smaller copies of itself, like this: Rep-12 Nonomino (discovered by Erich Friedman)

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 Fractal cross #2 Fractal cross #3 Fractal cross #4 Fractal cross #5 Fractal cross #6 Fractal cross (animated) Fractal cross (static)

If you trim the copies so that six are left, you have an asymmetrical seed for another familiar symmetrical fractal: Fractal Hex-Ring #1 Fractal Hex-Ring #2 Fractal Hex-Ring #3 Fractal Hex-Ring #4 Fractal Hex-Ring #5 Fractal Hex-Ring (animated) Fractal Hex-Ring (static)

Elsewhere other-available:

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