Aperiodic set of prototiles

The aperiodicity referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic.

As early as AD 325, Pappus of Alexandria knew that only 3 types of regular polygons (the square, equilateral triangle, and hexagon) can fit perfectly together in repeating tessellations on a Euclidean plane.

They are built from flat faces and straight edges and have sharp corner turns at the vertices.

The problem as stated was solved by Karl Reinhardt in 1928, but sets of aperiodic tiles have been considered as a natural extension.

[7] The specific question of aperiodic sets of tiles first arose in 1961, when logician Hao Wang tried to determine whether the Domino Problem is decidable — that is, whether there exists an algorithm for deciding if a given finite set of prototiles admits a tiling of the plane.

Hence, when in 1966 Robert Berger found an aperiodic set of prototiles this demonstrated that the tiling problem is in fact not decidable.

[8] (Thus Wang's procedures do not work on all tile sets, although that does not render them useless for practical purposes.)

The question of whether an aperiodic set exists with only a single prototile is known as the einstein problem.

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A periodic tiling with a fundamental unit (triangle) and a primitive cell (hexagon) highlighted. A tiling of the entire plane can be generated by fitting copies of these triangular patches together. To do this, the basic triangle must be rotated 60 degrees to fit edge-to-edge to a neighboring triangle. Thus a triangular tiling of fundamental units is generated that is mutually locally derivable from the tiling by the colored tiles. The other figure drawn onto the tiling, the white hexagon, represents a primitive cell of the tiling. Copies of the corresponding patch of coloured tiles can be translated to form an infinite tiling of the plane. It is not necessary to rotate this patch to achieve this.
The Penrose tiles are an aperiodic set of tiles, since they admit only non-periodic tilings of the plane (see next image).
All of the infinitely many tilings by the Penrose tiles are aperiodic . That is, the Penrose tiles are an aperiodic set of prototiles.
These Wang tiles yield only non-periodic tilings of the plane, and so are aperiodic.