Tilings Encyclopedia |

A substitution rule that gives rise to a non-periodic tiling $\mathcal{T}{17}$ with $17$-fold dihedral symmetry. The substitution factor is $\mu{17}=1/(2\sin(\pi/34))$. The tiling $\mathcal{T}{17}$ is obtained by assigning orientations to relevant tiles in the 1-order supertiles to ensure the existence of a patch invariant under $17$-fold dihedral symmetry in a supertile of the substitution. This patch of $34$ triangles equivalent to $T{17,15}$ whose $\pi/17$ vertices meet at a point serves as the seed for $\mathcal{T}_{17}$.

Finite Rotations Polytopal Tiles Self Similar Substitution Generalized Robinson Triangles

Say-awen 21-fold

The substitution rule is similar to Say-awen 17-fold. It is invariant under $21-$fold dihedral symmetry and has infinite local complexity. The substitution factor is $\mu{21}=1/(2\sin(\pi/42))$, which is non-Pisot with minimal polynomial $p{21}(x)=x^6-8x^5+8x^4+6x^3-6x^2-x+1$.

Finite Rotations Polytopal Tiles Self Similar Substitution Generalized Robinson Triangles

5-fold Shuriken Tiling

A substitution rule that gives rise to an aperiodic tiling with dense tile orientations and 5-fold rotational symmetry. The inflation factor is $\sqrt{6+\sqrt{5}}$. The tiling has finite local complexity with respect to rigid motions.

Finite Local Complexity Polytopal Tiles Self Similar Substitution Infinite Rotations

Discovered Tilings