Infinite Local Complexity

Here tilings are listed where the prototiles occur in infinitely many orientations, and the tilings are not flc .

Preview Cubic Pinwheel
Cubic Pinwheel

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

Preview Quartic pinwheel
Quartic pinwheel

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