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A Penrose tiling with rhombi exhibiting fivefold symmetry. A Penrose tiling is an example of an aperiodic tiling.Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches.
Glass was used in mosaics as early as 2500 BC, but it was not until the 3rd century BC that innovative artisans in Greece, Persia, and India created glass tiles.. Whereas clay tile is dated as early as 8,000 BC, there were significant barriers to the development of glass tile, including the high temperatures required to melt glass and the complexities of annealing glass curves.
A glass tile among clay tiles. Glass tiles, also referred to as skylight tiles, are used as accessories alongside clay roof tiles. These were first developed in the 1890s and designed to allow light into spaces roofed with interlocking tiles, such as warehouses and factories. [39]
A tiling is usually understood to be a covering with no overlaps, and so the Gummelt tile is not considered an aperiodic prototile. An aperiodic tile set in the Euclidean plane that consists of just one tile–the Socolar–Taylor tile –was proposed in early 2010 by Joshua Socolar and Joan Taylor. [ 7 ]
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A tiling T is a set of prototile placements whose regions have pairwise disjoint interiors. We say that the tiling T is a tiling of W where W is the union of the regions of the placements in T. A tile substitution is often loosely defined in the literature. A precise definition is as follows. [3]