Tiling with Transcendental Inflation Multiplier
An one-dimensional substitution rule that uses an infinite number of proto tiles and yields a transcendental inflation multiplier. The inflation factor is approximately $2.7899$. The substitution rules are given by: $T_{0}\rightarrow T_{0},T_{1}$ $T_{1}\rightarrow 3T_{0},T_{2}$ $T_{2}\rightarrow 2T_{0},T_{1},T_{3}$ $T_{3}\rightarrow T_{0},T_{2},T_{4}$ $T_{4}\rightarrow 2T_{0},T_{3},T_{5}$ $T_{5}\rightarrow T_{0},T_{4},T_{6}$ $T_{6}\rightarrow T_{0},T_{5},T_{7}$ $T_{k}\rightarrow (1+f\left(k\right))T_{0},T_{k-1},T_{k+1}$ with $f\left(k\right)$ as the Thue-Morse sequence. The corresponding substitution matrix can be written as: $1 3 2 1 2 1 1 2 2 …$

One Dimensional
Self Similar Substitution