Kolakoski-(3,1), with dual
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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