Without Decoration

Preview Ammann A3
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

Preview Ammann Chair
Ammann Chair

One of the tilings discovered by R. Ammann in 1977, published in [GS87] . The other ones (published there) are Ammann A3, Ammann A4, and Ammann A5 (better known as Ammann Beenker). The inflation factor of this substitution is quite small. It is the square root of the golden ratio, approx 1.272. These tilings are the dual tilings of the golden triangle tilings. The matching rules for the Ammann chair tilings can be expressed by using Ammann bars.

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

Preview Coloured Golden Triangle
Coloured Golden Triangle

In order to generate the golden triangle tilings by matching rules, L. Danzer and G. van Ophuysen found this substitution for coloured prototiles. The list of its vertex stars serves as matching rules. For more details, see golden triangle and the references there.

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

Preview Danzer's 7-fold
Danzer's 7-fold

A substitution tiling with three triangles as prototiles, based on 7-fold symmetry. The four different edge lengths occurring are $\sin(\frac{\pi}{7})$, $\sin(\frac{2\pi}{7})$, $\sin(\frac{3\pi}{7})$, $\sin(\frac{2\pi}{7}) + \sin(\frac{3\pi}{7})$, The inflation factor is $1+{\sin(\frac{2\pi}{7})}/{\sin(\frac{\pi}{7})}$ , which is not a PV number. There are simple matching rules for the tiling. In fact, the list of all vertex stars occurring in the substitution tiling serves as one. This is stated in [ND96], but never really published, up to my knowledge.

Without Decoration Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Danzer's 7-fold original
Danzer's 7-fold original

A tiling based on 7-fold (resp. 14-fold) symmetry [ND96]. The inflation factor is $1+{\sin(\frac{2\pi}{7})}/{\sin(\frac{\pi}{7})}$. The three different edge lengths are proportional to $\sin(\frac{\pi}{7})$, $\sin(\frac{2\pi}{7})$, $\sin(\frac{3\pi}{7})$. On a first glance, there seems to exist a centre of perfect 14-fold symmetry: a 14-tipped star in the upper right corner. But in fact it is only 2-fold symmetric. The symmetry is broken by the right- and left-handedness of the tiles. On rings around the 14-tipped star, this manifests in tiles pointing clockwise or counterclockwise, thus breaking the symmetry.

Without Decoration Finite Rotations Polytopal Tiles Self Similar Substitution

Preview Penrose Kite Dart
Penrose Kite Dart

A classic, using a kite (blue) and a dart (orange) as prototiles. See Penrose Rhomb for more details.

Without Decoration Finite Rotations Polytopal Windowed Tiling Polytopal Tiles Mld Class Penrose

Preview Penrose Rhomb
Penrose Rhomb

Certainly the most popular substitution tilings. Discovered in 1973 and 1974 by R. Penrose in - at least - three versions (Rhomb, Penrose kite-dart and Penrose Pentagon boat star), all of them forcing nonperiodic tilings by matching rules. It turns out that the three versions are strongly related: All three generate the same mld-class. These tiles, their matching rules and the corresponding substitution was studied thoroughly in [GS87] . A lot of information can be found there.

Without Decoration Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Rhomb Tiles Mld Class Penrose

Preview Robinson Triangle
Robinson Triangle

A variation of the Penrose rhomb tilings, suggested by R. M. Robinson. The rhombs are cut into triangles, thus making the substitution volume hierarchic. Thus, this one is obviously mld with the other Penrose tilings. For more details, see Penrose rhomb tilings. Each triangle comes either left- or right-handed, which is indicated by the different colours. This distinction is important since the triangles itself are mirror symmetric, but their first substitutions are not.

Without Decoration Finite Rotations Polytopal Windowed Tiling Polytopal Tiles Self Similar Substitution Mld Class Penrose