## Kidney and its dual

### Info

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 Patch