Infinite Rotations / Dense Tile Orientations
In many cases, the prototiles of a substitution tiling occur only in finitely many orientations. Here we list the tilings where this is not the case: the tiles occur in infinitely many different orientations. Thus these don’t show flc up to translation, even though they may show flc up to isometries.
In any primitive substitution tiling with tiles in infinitely many orientations the orientations are dense on the circle, and more: the orientations are equidistributed on the circle, see [fre08] .
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
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
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 …
Self Similar Substitution Infinite Rotations
10-fold cyclotomic aperiodic tiling with dense tile orientations (CAST DTO).
Finite Local Complexity Polytopal Tiles Self Similar Substitution Infinite Rotations
8-fold cyclotomic aperiodic tiling with dense tile orientations (CAST DTO).
Finite Local Complexity Polytopal Tiles Self Similar Substitution Infinite Rotations
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
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 …
Finite Local Complexity Polytopal Tiles Self Similar Substitution Infinite Rotations
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 …
With Decoration Finite Local Complexity Saduns Generalised Pinwheels Polytopal Tiles Self Similar Substitution Mld Class Pinwheel Infinite Rotations
References
[fre08]
Frettlöh, D.
Substitution tilings with statistical circular symmetry
European Journal of Combinatorics
2008,
29,
pp. 1881-1893,
arxiv 0704.2521