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Tessellation in two dimensions, also called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules. These rules can be varied.
The original form of Penrose tiling used tiles of four different shapes, but this was later reduced to only two shapes: either two different rhombi, or two different quadrilaterals called kites and darts. The Penrose tilings are obtained by constraining the ways in which these shapes are allowed to fit together in a way that avoids periodic tiling.
Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials.
Dry stone, sometimes called drystack or, in Scotland, drystane, is a building method by which structures are constructed from stones without any mortar to bind them together. [1] A certain amount of binding is obtained through the use of carefully selected interlocking stones.
These shapes are said to tessellate (from the Latin tessella, 'tile') and such a tiling is called a tessellation. Geometric patterns of some Islamic polychrome decorative tilings are rather complicated (see Islamic geometric patterns and, in particular, Girih tiles ), even up to supposedly quaziperiodic ones, similar to Penrose tilings .
Such a shape is called an einstein, a word play on ein Stein, German for "one stone". [ 2 ] Several variants of the problem, depending on the particular definitions of nonperiodicity and the specifications of what sets may qualify as tiles and what types of matching rules are permitted, were solved beginning in the 1990s.
The most efficient way to pack different-sized circles together is not obvious. In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap.
The 12 pentominoes can form 18 different shapes, with 6 of them (the chiral pentominoes) being mirrored. A pentomino (or 5-omino) is a polyomino of order 5; that is, a polygon in the plane made of 5 equal-sized squares connected edge to edge.