Generalized Robinson Triangles

It is possible to generalize the Robinson Triangle which is mld to the Penrose Rhomb tiling for odd $n \geq 3$.

The inflation factor is equal to the longest diagonal of a regular n-gon, in detail $\frac {\cos(\frac{(n-1)*\pi}{2*n})} {\cos(\frac{\pi}{n})}$. All tilings have $m=\frac{n-1}{2}$ prototiles in the shape of isosceles triangles with all interior angles are multiples of $\frac{\pi}{n}$. The substitution matrix was derived by D. H. Warrington in [War1988] :

$ m m-1 ... 3 2 1$ $m-1 m-1 ... 3 2 1$ $...$ $ 3 3 ... 3 2 1$ $ 2 2 ... 2 2 1$ $ 1 1 ... 1 1 1$

The case $n=3$ is a periodic tiling of equilateral triangles.

The case $n=5$ is the Robinson Triangle tiling.

The case $n=7$ is Danzer’s 7-fold variant.

The cases $n\in\left\{ 9,11,13,15\right\}$ were derived by T. Hibma, see [Hib2015a] for details.

Preview Danzer's 7-fold variant
Danzer's 7-fold variant

Substitution tiling with isosceles triangles as prototiles allow several variations: For each tile in the first order supertiles, one can choose whether it is a left-handed or a right-handed version. By playing around with these possibilities, one obtains this variant from Danzer’s 7-fold.

Finite Rotations Polytopal Tiles Self Similar Substitution Generalized Robinson Triangles

Preview Robinson Triangle
Robinson Triangle

A variation of the Penrose rhomb tilings, suggested by R. M. Robinson. The rhombs are cut into triangles, thus making the substitution volume hierarchic. Thus, this one is obviously mld with the other Penrose tilings. For more details, see Penrose rhomb tilings. Each triangle comes either left- or right-handed, which is indicated by the different colours. This distinction is important since the triangles itself are mirror symmetric, but their first substitutions are not.

Without Decoration Finite Rotations Polytopal Windowed Tiling Polytopal Tiles Self Similar Substitution Mld Class Penrose Generalized Robinson Triangles

References

[Hib2015a]
T. Hibma
Half Rhombs
2015?, https://www.aperiodictiling.org/wpaperiodictiling/index.php/fractional-rhomb-tilings/half-rhombs/

[War1988]
Warrington, D. H.
Two Dimensional Inflation Patterns and Corresponding Diffraction Patterns
Quasicrystalline materials: proceedings of the I.L.L./CODEST workshop, World Scientific: Singapore 1998, 243-254,