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

One of several substitution tilings found and submitted in February 2016 by L. Andritz with fivefold symmetry using tiles inspired by Islamic Girih patterns. Despite the similarities the tiling is different to the “Tie and Navette” tiling as discussed in [Lue1990] and [LL1994] .

Finite Rotations Model Set Polytopal Tiles Self Similar Substitution Finite Local Complexity

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

Finite Local Complexity Polytopal Tiles Self Similar Substitution

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

Finite Local Complexity Polytopal Tiles Self Similar Substitution

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

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

In [Lan88], energetic properties of certain decorations of Penrose Rhomb tilings were studied. A binary tiling was defined as a tiling by Penrose rhombs, where at each vertex all angles are either in {$\frac{\pi}{5}$, $3\frac{\pi}{5}$}, or in {$2\frac{\pi}{5}$, $4\frac{\pi}{5}$}. (‘Binary’ because the decorations were used to model binary alloys, i.e., alloys consisisting of two metallic elements). The authors did not mention the substitution rule explicitly, but it is obvious from the diagrams in this paper.

Finite Rotations Polytopal Tiles Parallelogramm Tiles Rhomb Tiles Finite Local Complexity

Using the prototiles of the golden triangle tiling, this substitution yields tilings where the tiles occur in infinitely many orientations. The inflation factor is $\tau + 1 = 2.618033988 \ldots $, the square of the golden mean. This is a PV number of algebraic degree 2. The expansion contains no rotational part. Nevertheless, the first substitution of the larger tile shows two small tiles, rotated against each other by an angle a incommensurate to pi (i.e., $\frac{a}{\pi}$ is irrational).

Finite Local Complexity Polytopal Tiles Self Similar Substitution

The Gosper Curve is a FASS-curve which can be derived by a substitution tiling with one substitution rule and appropriate decorations. The inflation factor $q$ is $sqrt(7)$.

Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve

The original Heighway Dragon Curve as described in [gar1967] , can be derived by a substitution tiling with one substitution rule and appropriate decoration. However, it is not a FASS-curve because it is not self avoiding. With the results in [pau2021] it is possible to derive a substitution tiling which generates a Heighway Dragon FASS-Curve without disturbing self similarity. In detail the decoration on the proto tile is shifted away from the corners in different ways.

Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve

The Hilbert Curve is one of the earliest FASS-curves. The original algorithm in [hil1891] bases on one substitution rule and an additional rule which describes how the substitutes have to be connected. As briefly mentioned in [pau2021] it is also possible to create the Hilbert Curve by a substitution tiling with two substitution rules and appropriate decorations. The inflation factor $q$ is 2 and the lines are shifted slightly away from the center of the sides to illustrate the matching rules.

Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve

This is a variation of the pinwheel substitution. The kite-domino tilings are mld to the pinwheel tilings. The two prototiles are made of two pinwheel triangles, glued together at their long edge. There are two ways to do so, one gives a kite (a quadrilateral with edge lengths 1,1,2,2) and a domino (a rectangle with edge lengths 1,2,1,2). Then the substitution rule is obtained by considering two steps of the pinwheel substitution as one step.

With Decoration Finite Local Complexity Polytopal Tiles Self Similar Substitution Mld Class Pinwheel

A tiling with 7-fold symmetry and a lot of locally 7-fold symmetric patches. There are only three tile shapes, but nine different prototiles. The inflation factor is a PV number: $2+2\cos\left(\frac{\pi}{7}\right)+2\cos\left(\frac{2\pi}{7}\right) = 5.04891733952231\ldots$ which is the largest root of $x^{3}-6x^{2}+5x-1$.

Polytopal Tiles Self Similar Substitution Finite Local Complexity Rhomb Tiles Finite Rotations

A substitution for three triangular prototiles, based on 7-fold symmetry. The lengths of the edges of the tiles are $\sin(\frac{\pi}{7})$, $\sin(\frac{2\pi}{7})$, $\sin(\frac{3\pi}{7})$. These tilings are essentially different from Danzer’s 7-fold examples, see for instance Danzer’s 7-fold.

Finite Rotations Polytopal Tiles Self Similar Substitution Finite Local Complexity

Finite Rotations Polytopal Tiles Self Similar Substitution Rhomb Tiles Finite Local Complexity

J. Millar discovered a set of tilings with patches of dihedral symmetry $D_2n$ and inflation multiplier $\sqrt{2 + 2 \cos(\frac{\pi}{n})}$, which is the same inflation multiplier as of the Generalized Godreche-Lancon-Billard Binary. All interior angles of all prototiles are integer multiples of $\frac{\pi}{n}$. All prototiles have sides with unit length. All tilings have a prototile in the shape of a rhomb with interior angle $\frac{\pi}{n}$. The longer diagonal also defines the inflation multiplier.

Finite Rotations Polytopal Tiles Finite Local Complexity

The Monnier Trapezium and Diamond tiling uses two prototiles, a trapezium and a rhomb. The inflation multiplier is $2$. By changing the chiralities of the prototiles within the first level supertiles several further variants can be derived.

Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic

Polytopal Tiles Self Similar Substitution Finite Local Complexity Finite Rotations

The Peano Curve is one the earliest known FASS-curves. The original algorithm in [pea1890] bases on one substitution rule and an additional rule which describes how the substitutes have to be connected. As briefly mentioned in [pau2021] it is also possible to create the Peano Curve by a substitution tiling with two substitution rules and appropriate decorations. The inflation factor $q$ is 3 and the lines are shifted slightly away from the center of the sides to illustrate the matching rules.

Finite Local Complexity Polytopal Tiles Self Similar Substitution With Decoration Limitperiodic FASS_curve

This substitution tiling is the example of substitution tilings with infinite rotations. Its statistical and dynamical properties were studied in several papers by C. Radin, see for instance [Rad92] , [Rad97] . In particular, it was shown that the orientations of triangles in the pinwheel tiling are equally distributed in the circle. Despite the occurrance of irrational edge lengths and incommensurate angles, all vertices of the pinwheel tiling have rational coordinates.

With Decoration Finite Local Complexity Saduns Generalised Pinwheels Polytopal Tiles Self Similar Substitution Mld Class Pinwheel

A simple example of an infinite series of substitutions with tilings of statistical circular symmetry. It is shown in [Frettloeh:STWCS not found], that all tilings in this series posses statistical circular symmetry. The substitution factors are $s2m$, where s is the largest root of $xm-xk-1$. Each pair of integers $(m,k)$, where $m>k, m>2, k>0$, encodes a such a Pythia substitution. The case $m=4, k=2$ yields the golden pinwheel substitution.

Finite Local Complexity Polytopal Tiles Self Similar Substitution

Schaad’s 7-fold is a variation of Madison’s 7-Fold, hence it shares many properties with it. It allows for tilings with global 7-fold symmetry and a lot of locally 7-fold symmetric patches. There are three tile shapes, but only seven instead of nine different prototiles. The inflation factor is a PV number: $2+2\cos\left(\frac{\pi}{7}\right)+2\cos\left(\frac{2\pi}{7}\right) = 5.04891733952231\ldots$ which is the largest root of $x^{3}-6x^{2}+5x-1$.

Polytopal Tiles Self Similar Substitution Finite Local Complexity Rhomb Tiles Finite Rotations

The source of the tiling can be found in [WSI95] Fig. 2 (iii) and Fig. 3. Its inflation factor is $2+\sqrt{3}$ and it has finite local complexity with respect to rigid motions. Unfortunately the corresponding substitution rules given in Fig. 2 (iii) of the paper are not unique. For some time the exact substitution rules which generate the square level-3-supertile in Fig. 3 of the paper remained unclear. Finally Alessandro Musesti and Maurizio Paolini submitted the correct set of substitution rules in January 2023 to us.

Finite Rotations Polytopal Tiles Self Similar Substitution Finite Local Complexity

This tiling was originally introduced in [WSI87] , however the description given there admits several substitution rules. This is the version given explicitly in [WSI95] . This is an example of a cut and project with a mixed internal space, a product of Euclidean and $p$-adic spaces, namely $\mathbb{R}^2 \times \mathbb{Q}_2$.

Finite Rotations Model Set Rhomb Tiles Polytopal Tiles Self Similar Substitution Finite Local Complexity