Tilings Encyclopedia | MLD Class Penrose

The tiling shares a mld-class with the Penrose Tilings, e.g. Penrose Rhomb, Penrose kite-dart and Penrose Pentagon boat star). The inflation factor is the square of the golden mean $(\frac{\sqrt{5}}{2} + \frac{1}{2})^{2} = \frac{\sqrt{5}}{2} + \frac{3}{2} = 2.618033988\ldots$. In contrast to the Penrose Tilings the interior angles of the prototiles are larger than $36^{\circ}$.

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

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

Penrose Pentagon Boat Star

One manifestation of the famous Penrose tilings. In fact, this is the first manifestation found by Penrose, the Penrose rhomb, the Penrose kite-dart and the Robinson triangle tilings are refinements of this one. (You may also click ‘Penrose’ below ‘MLD-class’ above to see the others.) Their properties are discussed on the page Penrose rhomb. For a more detailed discussion see [GS87] .

Finite Rotations Polytopal Tiles Mld Class Penrose

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 Matching Rules

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