An one-dimensional substitution rule that uses an infinite number of proto tiles.
The inflation factor is $2.5$
.
The substitution rules are given by:
$T_{0}\rightarrow T_{0},T_{1}$
$T_{1}\rightarrow T_{0},T_{0},T_{2}$
$T_{2}\rightarrow T_{0},T_{1},T_{3}$
$T_{k}\rightarrow T_{0},T_{k-1},T_{k+1}$
$T_{\infty}\rightarrow T_{0},T_{\infty},T_{\infty}$
The lengths of the proto tiles are given by:
$length(T_{0})=1$
$length(T_{1})=\frac{3}{2}$
$length(T_{2})=\frac{7}{4}$
$length(T_{k})=\frac{2^{k+1}-1}{2^{k}}$
$length(T_{\infty})=2$
[FGM2022]
Frettloeh, D. and Garber, A. and Manibo, N.
Substitution tilings with transcendental inflation factor
2022,
10.48550/ARXIV.2208.01327