One of the rare examples of a tiling where the prototiles occur in infinitely many orientations. Apart from the pinwheel tiling and its generalizations [Sad98] there are only a few examples known which show infinite rotations. The inflation factor of this one is a complex algebraic PV number of degree four. As the scaling and the rotations for the tiles are all given by algebraic units, every vertex of the tiling lies within a finitely generated Z-module.

Infinite Rotations Infinite Local Complexity Polytopal Tiles Self Similar Substitution

A companion to infinite component Rauzy fractal. As mentioned for that rule, it was hoped that the result for two symbol substitution rules that the window is connected if and only if the rule is invertible. This substitution rules is not invertible and yet the Rauzy fractal is connected:

Euclidean Windowed Tiling One Dimensional Polytopal Tiles Self Simmilar Substitution

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

Finite Rotations Polytopal Tiles Parallelogram Tiles Rhomb Tiles Harrisss Rhomb

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

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

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

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

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

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

A pinwheel substitution rule with cubic scaling. As the scaling and the rotations for the tiles are all given by algebraic units, every vertex of the tiling lies within a finitely generated Z-module.

Infinite Rotations Infinite Local Complexity Polytopal Tiles Self Similar Substitution

The substitution rule a1->a1 b1, a2->b2 a2, b1->a2, b2->a1. The tilings generated become Fibonacci tilings under the projection a1,a2->a and b1,b2->b. Alternatively one can simply remove the colour labels on the tiles. The name comes from the projection structure of the tiling. The expansion predecessor of the tiling is itself a projection tiling with the window lying at the center of the window for the full tiling. For more information see [HL].

Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling One Dimensional Polytopal Tiles Parallelogram Tiles Self Similar Substitution Mld Class Fibonacci

An invertible substitution rule with a disconnected Rauzy Fractal. For two letter substitution rules the Rauzy fractal is connected if and only if the substitution is invertible. In fact as the window is one dimensional for these tilings it is an interval. It was hoped that the connectedness property extended to the higher dimensional case. Unfortunately, as this example shows, this is not the case. A second example, with just two components is 2-component Rauzy fractal.

One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles

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

This is the dual partner of Conch, which has more details. The scaaling factor of this rule is either of the (complex conjugate) expanding roots of $x^4 - x^3 + 1 = 0$.

Finite Rotations Euclidean Windowed Tiling Polytopal Tiles

A volume hierarchic version of Conch.

Finite Rotations Euclidean Windowed Tiling Self Similar Substitution

This tiling and Nautilus are dual tilings generated by non-PV morphisms. As such they are the first step in a generalisation of the work of G. Rauzy, P. Arnoux, S. Ito and others for PV substitution rules. The work that developed out of G. Rauzy’s seminal paper [Rau82] . The inflation factor for this substitution rule is either of the expanding roots of: $x^{4}-x+1 = 0$. Note that this it is related to R.

Finite Rotations Euclidean Windowed Tiling

A volume hierarchic version of Nautilus

Finite Rotations Euclidean Windowed Tiling Self Similar Substitution