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A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that ...
Figure 1. Part of a periodic tiling with two prototiles. Covering a flat surface ("the plane") with some pattern of geometric shapes ("tiles"), with no overlaps or gaps, is called a tiling. The most familiar tilings, such as covering a floor with squares meeting edge-to-edge, are examples of periodic tilings. If a square tiling is shifted by ...
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.
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are a well-known example of aperiodic tilings. [1] [2]
Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits of edges, as e-isotoxal.
A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic. [3] The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are ...