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The Robinson triangles arising in P2 tilings (by bisecting kites and darts) are called A-tiles, while those arising in the P3 tilings (by bisecting rhombs) are called B-tiles. [31] The smaller A-tile, denoted A S , is an obtuse Robinson triangle, while the larger A-tile, A L , is acute ; in contrast, a smaller B-tile, denoted B S , is an acute ...
Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman [5] who subsequently won the Nobel prize in 2011. [6] However, the specific local structure of these materials is still poorly understood. Several methods for constructing aperiodic tilings are known.
He has received several prizes and awards, including the 1988 Wolf Prize in Physics, which he shared with Stephen Hawking for the Penrose–Hawking singularity theorems, [6] and the 2020 Nobel Prize in Physics "for the discovery that black hole formation is a robust prediction of the general theory of relativity".
Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex". Penrose P3 tiles: 2: E 2: 1978 [9] [10] Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex". Binary tiles: 2: E 2: 1988 [11] [12] Although similar in shape to the P3 tiles, the tilings are not MLD from each other.
All three of these tilings are isogonal and monohedral. [26] A Pythagorean tiling is not an edge‑to‑edge tiling. A semi-regular (or Archimedean) tessellation uses more than one type of regular polygon in an isogonal arrangement. There are eight semi-regular tilings (or nine if the mirror-image pair of tilings counts as two). [27]
Alternatively, an undecorated tile with no matching rules may be constructed, but the tile is not connected. The construction can be extended to a three-dimensional, connected tile with no matching rules, but this tile allows tilings that are periodic in one direction, and so it is only weakly aperiodic. Moreover, the tile is not simply connected.
Some substitution tilings are periodic, defined as having translational symmetry. Every substitution tiling (up to mild conditions) can be "enforced by matching rules"—that is, there exist a set of marked tiles that can only form exactly the substitution tilings generated by the system. The tilings by these marked tiles are necessarily aperiodic.
It is not difficult to design a set of tiles that admits non-periodic tilings as well as periodic tilings. (For example, randomly arranged tilings using a 2×2 square and 2×1 rectangle are typically non-periodic.) However, an aperiodic set of tiles can only produce non-periodic tilings.