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A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art , especially in textiles , tiles , and wallpaper .
The 17 wallpaper groups, with finite fundamental domains, are given by International notation, orbifold notation, and Coxeter notation, classified by the 5 Bravais lattices in the plane: square, oblique (parallelogrammatic), hexagonal (equilateral triangular), rectangular (centered rhombic), and rhombic (centered rectangular).
A wallpaper group or plane symmetry group or plane crystallographic group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct groups.
A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern (an aperiodic set of prototiles). A tessellation of space, also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions.
Tilings and Patterns is such a book." [8] E. Schulte wrote the entry in zbMATH Open: "I hope that this review conveys my impression that Tilings and Patterns is an excellent book on one of the oldest mathematical disciplines. Most certainly this book will be the 'bible' for this kind of geometry." [9]
A pattern with 4-fold rotational symmetry has a square lattice of 4-fold rotocenters that is a factor √2 finer and diagonally oriented relative to the lattice of translational symmetry. With respect to reflection axes there are three possibilities: None. This is wallpaper group p4. In four directions. This is wallpaper group p4m.
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Broken down, 3 6; 3 6 (both of different transitivity class), or (3 6) 2, tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 3 4 .6, 4 more contiguous equilateral triangles and a single regular hexagon.