List Of Planar Symmetry Groups
This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane: *2 families of rosette groups – 2D point groups *7 frieze groups – 2D line groups *17 wallpaper groups – 2D space groups. Rosette groups There are two families of discrete two-dimensional point groups, and they are specified with parameter ''n'', which is the order of the group of the rotations in the group. Frieze groups The 7 frieze groups, the two-dimensional line groups, with a direction of periodicity are given with five notational names. The Schönflies notation is given as infinite limits of 7 dihedral groups. The yellow regions represent the infinite fundamental domain in each. Wallpaper groups The 17 wallpaper groups, with finite fundamental domains, are given by International notation, or ...
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Discrete Space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology. Definitions Given a set X: A metric space (E,d) is said to be '' uniformly discrete'' if there exists a ' r > 0 such that, for any x,y \in E, one has either x = y or d(x,y) > r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set \left\. Properties The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with one ...
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Cyclic Symmetry 3
Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social sciences ** Business cycle, the downward and upward movement of gross domestic product (GDP) around its ostensible, long-term growth trend Arts, entertainment, and media Films * ''Cycle'' (2008 film), a Malayalam film * ''Cycle'' (2017 film), a Marathi film Literature * ''Cycle'' (magazine), an American motorcycling enthusiast magazine * Literary cycle, a group of stories focused on common figures Music Musical terminology * Cycle (music), a set of musical pieces that belong together **Cyclic form, a technique of construction involving multiple sections or movements **Interval cycle, a collection of pitch classes generated from a sequence of the same interval class **Song cycle, individually complete songs designed to be performe ...
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IUC Notation
The International Union of Crystallography (IUCr) is an organisation devoted to the international promotion and coordination of the science of crystallography. The IUCr is a member of the International Council for Science (ICSU). Objectives The objectives of the IUCr are to promote international cooperation in crystallography and to contribute to all aspects of crystallography, to promote international publication of crystallographic research, to facilitate standardization of methods, units, nomenclatures and symbols, and to form a focus for the relations of crystallography to other sciences. The IUCr fulfils these objectives by publishing in print and electronically primary scientific journals through the ''Acta Crystallographica'' journal series, as well as ''Journal of Applied Crystallography'', ''Journal of Synchrotron Radiation'', '' IUCrJ'', the series of reference volumes ''International Tables for Crystallography'', distributing the quarterly ''IUCr Newsletter'', maint ...
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Schönflies Notation
The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule, the notation is often sufficient and commonly used for spectroscopy. However, in crystallography, there is additional translational symmetry, and point groups are not enough to describe the full symmetry of crystals, so the full space group is usually used instead. The naming of full space groups usually follows another common convention, the Hermann–Mauguin notation, also known as the international notation. Although Schoenflies notation without superscripts is a pure point group notation, optionally, superscripts can be added to further specify individual space groups. However, for space groups, the connection to the underlying symmetry elements is much more clear in Hermann–Mauguin notation, so the latter not ...
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Line Group
A line group is a mathematical way of describing symmetries associated with moving along a line. These symmetries include repeating along that line, making that line a one-dimensional lattice. However, line groups may have more than one dimension, and they may involve those dimensions in its isometries or symmetry transformations. One constructs a line group by taking a point group in the full dimensions of the space, and then adding translations or offsets along the line to each of the point group's elements, in the fashion of constructing a space group. These offsets include the repeats, and a fraction of the repeat, one fraction for each element. For convenience, the fractions are scaled to the size of the repeat; they are thus within the line's unit cell segment. One-dimensional There are 2 one-dimensional line groups. They are the infinite limits of the discrete two-dimensional point groups C''n'' and D''n'': Two-dimensional There are 7 frieze groups, which involve r ...
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Dihedral Symmetry Domains 6
Dihedral or polyhedral may refer to: * Dihedral angle, the angle between two mathematical planes * Dihedral (aeronautics), the upward angle of a fixed-wing aircraft's wings where they meet at the fuselage, dihedral ''effect'' of an aircraft, longitudinal dihedral angle of a fixed-wing aircraft * Dihedral group, the group of symmetries of the ''n''-sided polygon in abstract algebra ** Also Dihedral symmetry in three dimensions * Dihedral kite, also known as a bowed kite * Dihedral doors, also known as butterfly doors * Dihedral prime, also known as a dihedral calculator prime * In rock climbing, an inside corner of rock See also * Anhedral (other) * Euhedral, a crystal structure * Polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...

Dihedral Symmetry Domains 5
Dihedral or polyhedral may refer to: * Dihedral angle, the angle between two mathematical planes * Dihedral (aeronautics), the upward angle of a fixed-wing aircraft's wings where they meet at the fuselage, dihedral ''effect'' of an aircraft, longitudinal dihedral angle of a fixed-wing aircraft * Dihedral group, the group of symmetries of the ''n''-sided polygon in abstract algebra ** Also Dihedral symmetry in three dimensions * Dihedral kite, also known as a bowed kite * Dihedral doors, also known as butterfly doors * Dihedral prime, also known as a dihedral calculator prime * In rock climbing, an inside corner of rock See also * Anhedral (other) * Euhedral, a crystal structure * Polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

Dihedral Symmetry Domains 4
Dihedral or polyhedral may refer to: * Dihedral angle, the angle between two mathematical planes * Dihedral (aeronautics), the upward angle of a fixed-wing aircraft's wings where they meet at the fuselage, dihedral ''effect'' of an aircraft, longitudinal dihedral angle of a fixed-wing aircraft * Dihedral group, the group of symmetries of the ''n''-sided polygon in abstract algebra ** Also Dihedral symmetry in three dimensions * Dihedral kite, also known as a bowed kite * Dihedral doors, also known as butterfly doors * Dihedral prime, also known as a dihedral calculator prime * In rock climbing, an inside corner of rock See also * Anhedral (other) * Euhedral, a crystal structure * Polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

Dihedral Symmetry Domains 3
Dihedral or polyhedral may refer to: * Dihedral angle, the angle between two mathematical planes * Dihedral (aeronautics), the upward angle of a fixed-wing aircraft's wings where they meet at the fuselage, dihedral ''effect'' of an aircraft, longitudinal dihedral angle of a fixed-wing aircraft * Dihedral group, the group of symmetries of the ''n''-sided polygon in abstract algebra ** Also Dihedral symmetry in three dimensions * Dihedral kite, also known as a bowed kite * Dihedral doors, also known as butterfly doors * Dihedral prime, also known as a dihedral calculator prime * In rock climbing, an inside corner of rock See also * Anhedral (other) * Euhedral, a crystal structure * Polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

Dihedral Symmetry Domains 2
Dihedral or polyhedral may refer to: * Dihedral angle, the angle between two mathematical planes * Dihedral (aeronautics), the upward angle of a fixed-wing aircraft's wings where they meet at the fuselage, dihedral ''effect'' of an aircraft, longitudinal dihedral angle of a fixed-wing aircraft * Dihedral group, the group of symmetries of the ''n''-sided polygon in abstract algebra ** Also Dihedral symmetry in three dimensions * Dihedral kite, also known as a bowed kite * Dihedral doors, also known as butterfly doors * Dihedral prime, also known as a dihedral calculator prime * In rock climbing, an inside corner of rock See also * Anhedral (other) * Euhedral, a crystal structure * Polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

Dihedral Symmetry Domains 1
Dihedral or polyhedral may refer to: * Dihedral angle, the angle between two mathematical planes * Dihedral (aeronautics), the upward angle of a fixed-wing aircraft's wings where they meet at the fuselage, dihedral ''effect'' of an aircraft, longitudinal dihedral angle of a fixed-wing aircraft * Dihedral group, the group of symmetries of the ''n''-sided polygon in abstract algebra ** Also Dihedral symmetry in three dimensions * Dihedral kite, also known as a bowed kite * Dihedral doors, also known as butterfly doors * Dihedral prime, also known as a dihedral calculator prime * In rock climbing, an inside corner of rock See also * Anhedral (other) * Euhedral, a crystal structure * Polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

Dihedral Symmetry
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the -gon, a group of order . In abstract algebra, refers to this same dihedral group. This article uses the geometric convention, . Definition Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. Usually, we take n \ge 3 here. The associated rotations and reflections make up the dihedral group \mathrm_n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides and n/2 axes of symmetry connecting oppo ...
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