Kidney and its dual

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The substitution $a \rightarrow ab, b \rightarrow cb, c \rightarrow a$ is the composition of the one with the smallest PV scaling factor, $a \rightarrow bc, b \rightarrow a, c \rightarrow b$, and its mirror image, $a \rightarrow cb, b \rightarrow a, c \rightarrow b$. As such, it is MLD to its own mirror image, $a \rightarrow ba, b \rightarrow bc, c \rightarrow a$. The scaling factor $\lambda \approx$ 1.7549 is the largest root of $x^3-2x^2+x-1=0$.

This substitution has a surprisingly simple dual, with three mildly fractal tiles similar to each other. The dual substitution scales by about 1.3247, and rotates clockwise by about 80.656°.

Dual Substitution Rule

Dual Rule Kidney and its dual

Dual Patch

Dual Patch Kidney and its dual