Tilings Encyclopedia | Infinite Rotations / Dense Tile Orientations

A substitution rule that gives rise to an aperiodic tiling with dense tile orientations and 12-fold rotational symmetry. The inflation factor is $\sqrt{5+2\sqrt{3}}$. The tiling has finite local complexity with respect to rigid motions.

Finite Local Complexity Polytopal Tiles Self Similar Substitution Infinite Rotations

5-fold Shuriken Tiling

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 Infinite Rotations

Cesi's Substitution

The substitution system uses 4 letters. With: $x = \frac{\pi}{7}$, $c = \cos(x)$ and $s = \sin(x)$ They are: two squares of side lengths $1$ and $2-c-s$; a rectangle with sides $c+s$ and $2-c-s$: and a right triangle with legs $c$ and $s$. The substitution is indicated in the figure. Up to our knowledge, this was the first example of a substitution where the tiles occur in infinitely many orientations. Obviously, the substitution is not primitive.

Self Similar Substitution Infinite Rotations

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 Infinite Rotations

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 Infinite Rotations

Double Angle Plastic

Tiling submitted by Andrew Hudson. The scaling factor is the smallest PV number, the Plastic Number which is a root of the polynomial $x^3 - x - 1 = 0$.

Self Similar Substitution Plastic Number Infinite Rotations

Golden Pinwheel

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 Infinite Rotations

Pinwheel

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 Infinite Rotations

Infinite Rotations / Dense Tile Orientations

References