Aperiodic Tessellations

Constructing Penrose tiles with de Bruijn projections

In mathematics, a tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. The most familiar tilings (e.g., by squares or triangles) are periodic: a perfect copy of the tiling can be obtained by translating all of the tiles by a fixed distance in a given direction.

Fig. 1: A regular tessellation with hexagons

If a tiling has no periods, it is said to be non-periodic, and it's said to be aperiodic if every tiling of the plane with the tiles is non-periodic. Penrose tilings use two different quadrilateral prototiles which cannot tessellate periodically.

Fig. 2: An aperiodic tessellation with Penrose tiles

In 1980, N. G. de Bruijn showed that Penrose tilings can be seen as the projection of a two-dimensional plane in five-dimensional space. In the figure below, you can click and drag to translate the viewport along the plane.

Fig. 3: Interactive de Bruijn projection explorer

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