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 …

Socolar found the basis for this tiling already in 1987, but recently added a substitution tiling. An interesting feature is that there exists a context-independent Ammann bar decoration of the tiles, similar to the one in the original Socolar 12-fold but with different relative phases, and hence a fairly simple set of matching rules.

10-fold cyclotomic aperiodic tiling with dense tile orientations (CAST DTO).

In order to generalize Danzer’s 7-fold tiling to n-fold symmetry, where n>5 is odd, L. Danzer and D. Frettlöh introduced trapezoidal tiles, each one the union of two triangles with edge lengths of the form $\sin(k \frac{\pi}{n})$. It needs some further effort, including the introduction of three …