Tilings Encyclopedia | Generalized Robinson Triangles

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 $\frac {\cos(\frac{(n-1)\pi}{2n})} {\cos(\frac{\pi}{n})}$ . 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.

The cases $n\in\left\{ 17,21\right\}$ are Say-awen 17-fold and Say-awen 21-fold.

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

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

Say-awen 17-fold

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

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,