A hobby mathematician finds the long-awaited Einstein Tile

The well tried and tested method of proof using Robinson tiles can be illustrated with six different tiles. The lines drawn on the tiles are similar to the colored edges in Wang’s squares: you just place Robinson tiles next to each other in such a way that lines of the same color continue seamlessly. Following this rule results in a recognizable pattern: the yellow lines form squares with a larger square corner starting in the center, and so on. So you can set a hierarchical structure for tiles: tiles that get bigger and bigger and intersect. So tessellation cannot be periodic. If this is the case, sections of the form will appear elsewhere in exactly the same way. But if you do part of the tiling and move it around, you inevitably destroy the hierarchical structure.

Kaplan, Goodman Strauss, and Myers were able to show something similar to the Einstein hat-shaped tiles proposed by Smith. However, they did not look at the hat itself, but at four different combinations, some of which consisted of several tiles: a hexagonal structure with four hat tiles, a pentagon with two pieces, a parallelogram with two pieces and one tile, which the researchers represent with a triangle. By defining the appropriate rules on how the four cluster polygons combine, the entire plane can be covered. At the same time, cluster polygons form a pattern that, they proved, never repeats itself.

The combination of the four groups together results in one of the four polygons (hexagon, pentagon, parallelogram, or triangle). Myers, Kaplan, and Goodman-Strauss were also able to show that this larger structure is unique: it can only arise from a single arrangement of smaller block polygons. Now you can merge the larger polygons together again to form larger structures: these also result in either a six, five, four, or a triangle – these also consist of a clear arrangement of the smaller polygons. This can be repeated indefinitely, giving a hierarchical structure, as in the case of Robinson’s tiling. An exception is periodicity: larger and larger polygons can be generated, which consist of a clear arrangement of smaller groups. If you simply move sections of it to another location, the superstructure may lose its uniqueness. As it turns out, this proof can be made not only with the “hat” that Smith provided, but also with the various shapes of the tiles.