## Kolakoski-(3,1), with dual

### Info

The substitution $a \rightarrow abc, b \rightarrow ab, c \rightarrow b$
is closely related to the Kolakoski-(3,1) sequence, and is one of the
examples whose windows (dual tiles,
Rauzy fractals) have been
analysed in detail [BaS04]
. It is
MLD to the
mirror symmetric variant
given by the palindromic substitution
$a \rightarrow aca, b \rightarrow a, c \rightarrow b$.
As a consequence, the Kolakoski-(3,1)
substitution is MLD to its mirror image, even though it is not mirror
symmetric itself. Its MLD class, however, is mirror symmetric. There are
two further variants,
A and
B,
which are in the same MLD class, along with their mirror images.
The scaling factor $\lambda \approx$ 2.20557 is the largest root of
$x^3-2x^2-1=0$.

This substitution has a simple dual,
with three mildly fractal tiles. The dual substitution
scales by about 1.485, and rotates clockwise by about 81.22°.

### Dual Substitution Rule

### Dual Patch

### References

[BaS04]

Baake, M and Sing, B

**Kolakoski-(3,1) is a (deformed) model set**

*Canad. Math. Bull.*
2004,
47,
168-190,
kol31.ps.gz