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If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane. The Conway criterion is a sufficient, but not necessary, set of rules for deciding whether a given shape tiles the plane periodically without reflections: some tiles fail the criterion, but still tile the plane. [ 19 ]
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.
In geometry, a tiling is a partition of the plane (or any other geometric setting) into closed sets (called tiles), without gaps or overlaps (other than the boundaries of the tiles). [1] A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself.
In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way. Such a shape is called an einstein, a word play on ein Stein, German for "one stone". [2]
A periodic tiling of the plane is the regular repetition of a "unit cell", in the manner of a wallpaper, without any gaps. Such tilings can be seen as a two-dimensional crystal, and because of the crystallographic restriction theorem , the unit cell is restricted to a rotational symmetry of 2-fold, 3-fold, 4-fold, and 6-fold.
1-uniform tilings include 3 regular tilings, and 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the number m of distinct vertex figures, which are also called m-Archimedean tilings.
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.
These shapes are known as daggers, fish-bladders, or mouchettes. [1] Starting in the late 13th century and at the beginning of the 14th century, tracery took on more fluid characteristics. A common shape used in curvilinear tracery was that of the ogee, which was too weak for structural application and was instead used as a decorative element.