Tri-Way to L

The name is more complicated than the shape: L-triomino. The shape is simply three squares forming an L. And it’s a rep-tile — it can be divided into four smaller copies of itself.

l-triomino

An L-triomino — three squares forming an L


l-triomino_anim

L-triomino as rep-tile


That means it can also be turned into a fractal, as I’ve shown in Rep-Tiles Revisited and Get Your Prox Off #2. First you divide an L-triomino into four sub-copies, then discard one sub-copy, then repeat. Here are the standard L-triomino fractals produced by this technique:

l-triomino_123_134

Fractal from L-triomino — divide and discard


l-triomino_234


l-triomino_124


l-triomino_124_upright


l-triomino_124_upright_static1

(Static image)


l-triomino_124_upright_static2

(Static image)


But those fractals don’t exhaust the possibilities of this very simple shape. The standard L-triomino doesn’t have true chirality. That is, it doesn’t come in left- and right-handed forms related by mirror-reflection. But if you number its corners for the purposes of sub-division, you can treat it as though it comes in two distinct orientations. And when the orientations are different in the different sub-copies, new fractals appear. You can also delay the stage at which you discard the first sub-copy. For example, you can divide the L-triomino into four sub-copies, then divide each sub-copy into four more sub-copies, and only then begin discarding.

Here are the new fractals that appear when you apply these techniques:

l-triomino_124_exp

Delay before discarding


l-triomino_124_exp_static

(Static image)


l-triomino_124_tst2_static1

(Static image)


l-triomino_124_tst2_static2

(Static image)


l-triomino_124_tst1


l-triomino_124_tst1_static1

(Static image)


l-triomino_124_tst1_static2

(Static image)


l-triomino_134_adj1

Adjust orientation


l-triomino_134_adj2


l-triomino_134_adj3


l-triomino_134_adj3_tst3

(Static image)


l-triomino_134_adj4


l-triomino_134_exp_static

(Static image)


l-triomino_234_exp

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