2-component Rauzy Fractal

A one dimensional substitution rule with a two component Rauzy Fractal. For a second example and more details see infinite component Rauzy fractal.

One Dimensional Euclidean Windowed Tiling Self Similar Substitution

2-component Rauzy Fractal (dual)

In the cut and project scheme for the 2-component Rauzy Fractal, just interchange the roles of direct space and internal space: The Rauzy fractal and its decomposition define a plane substitution for fractal tiles. Here we replace the fractal tiles with appropriate parallelograms. This plane tiling now has one-dimensional internal space, and its window is just an interval.

Border Forcing

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

A->AB, B->C, C->A

A classic simple substitution rule with Rauzy Fractal:

One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles

A->AB, B->C, C->A (dual)

The dual tiling of the 1D tiling a->ab, b->c, c->a, resp. the version with polygonal tiles.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogramm Tiles

Ammann A3

In 1977 Robert Ammann discovered a number of sets of aperiodic prototiles, i.e., prototiles with matching rules forcing nonperiodic tilings. These were published as late as 1987 in [GS87] , where they were named Ammann A2 (our Ammann Chair), Ammann A3, Ammann A4 and Ammann A5 (better known as Ammann Beenker tiling). The substitution of this one uses the golden ratio as inflation factor. It is certainly true that this is a cut and project tiling, but to our knowledge, noone bothered to compute the window of it up to now.

Without Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Self Similar Substitution