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In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t {3,6} (as a truncated triangular tiling).
The Socolar–Taylor tile was proposed in 2010 as a solution to the einstein problem, but this tile is not a connected set. In 1996, Petra Gummelt constructed a decorated decagonal tile and showed that when two kinds of overlaps between pairs of tiles are allowed, the tiles can cover the plane, but only non-periodically. [6]
A patch of 25 monotiles, showing the triangular hierarchical structure. The Socolar–Taylor tile is a single non-connected tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane (due to the Sierpinski's triangle-like tiling that occurs), with rotations and reflections of the tile allowed. [1]
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The six coordinates of each pentagon can be grouped into two triples of coordinates, in which each triple gives the coordinates of a hexagon in an analogous three-dimensional coordinate system for each of the two overlaid hexagon tilings. [10] The number of tiles that are steps away from any given tile, for =,,, …, is given by the ...