Stefan Pautze

Discovered Tilings

Preview Generalized Godreche-Lancon-Billard Binary
Generalized Godreche-Lancon-Billard Binary

This tiling is a generalization of the Godreche-Lancon-Billard Binary first derived by T. Hibma and later worked out in detail by S. Pautze. All interior angles are integer multiples of $\frac{\pi}{n}$. For $n=5$ it is identical to the Godreche-Lancon-Billard Binary tiling with 2 prototiles. For odd $n$ it has $\frac{n-1}{2}$ prototiles. For even $n$ it has $n+1$ prototiles. The inflation multiplier is $\sqrt{2 + 2 \cos(\frac{\pi}{n})}$. The example shown below is the tiling for $n=9$.

Finite Rotations Polytopal Tiles Parallelogramm Tiles Rhomb Tiles Finite Local Complexity

Preview FASS-Curve of the Pentagon Substitution Tiling
FASS-Curve of the Pentagon Substitution Tiling

The FASS-curve of the pentagon bases on an aperiodic substitution tiling with four substitution rules and appropriate decorations. The substitution tiling was derived from the Robinson Triangle Tiling. Its inflation factor is the golden mean $\frac{\sqrt{5}}{2} + \frac{1}{2} = 1.618033988\ldots$.

Polytopal Tiles Self Similar Substitution With Decoration FASS_curve

Preview Cyclotomic Aperiodic Substitution Tiling with Dense Tile Orientations, 10-fold
Cyclotomic Aperiodic Substitution Tiling with Dense Tile Orientations, 10-fold

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

Finite Local Complexity Polytopal Tiles Self Similar Substitution

Preview Cyclotomic Aperiodic Substitution Tiling with Dense Tile Orientations, 8-fold
Cyclotomic Aperiodic Substitution Tiling with Dense Tile Orientations, 8-fold

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

Finite Local Complexity Polytopal Tiles Self Similar Substitution

Preview Girih inspired 14-fold Tiling
Girih inspired 14-fold Tiling

A tiling resembling Islamic Girih patterns but using 14-fold symmetry rather than 8- or 10- or 12-fold. Its inflation factor is $1 + \cos(\frac{\pi}{14}) \csc(\frac{\pi}{7}) + 2 \cos(\frac{3 \pi}{14}) \csc(\frac{\pi}{7}) = 6.850855...$ which is a unit but not a PV number. It uses 11 prototiles altogether, 10 of them showing $D_2$-symmetry and one showing $D_{14}$-symmetry. This shows that all resulting tilings have local patches with 14-fold symmetry, and that the hull contains tilings with global 14-fold symmetry.

Finite Local Complexity Finite Rotations Polytopal Tiles Self Similar Substitution