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- Point_groups_in_four_dimensions abstract "In geometry, a point group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere.A polychoric group is one of five symmetry groups of the 4-dimensional regular polytopes. There are also three polyhedral prismatic groups, and an infinite set of duoprismatic groups. Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes. The dihedral angles between the mirrors determine order of dihedral symmetry. The Coxeter–Dynkin diagram is a graph where nodes represent mirror planes, and edges are called branches, and labeled by their dihedral angle order between the mirrors.Like the 3D polyhedral groups, the names of the 4D polychoric groups given are constructed by the Greek prefixes of the cell counts of the corresponding triangle-faced regular polytopes. Extended symmetries exist in uniform polychora with symmetric ring-patterns within the Coxeter diagram construct. Chiral symmetries exist in alternated uniform polychora. The groups are named in this article in Coxeter's Bracket notation (1985). Coxeter notation has a direct correspondence the Coxeter diagram like [3,3,3], [4,3,3], [31,1,1], [3,4,3], [5,3,3], and [p,2,q].For cross-referencing, also given here are quaternion based notations by Patrick du Val (1964) and John Conway (2003). Conway's notation allows the order of the group to be computed as a product of elements with chiral polyhedral group orders: (T=12, O=24, I=60). In Conway's notation, a (±) prefix implies central inversion, and a suffix (.2) implies mirror symmetry. Simiarly Du Val's notation has an asterisk (*) superscript for mirror symmetry.".
- Point_groups_in_four_dimensions thumbnail Polychoral_group_tree.png?width=300.
- Point_groups_in_four_dimensions wikiPageExternalLink productCd-0471010030.html.
- Point_groups_in_four_dimensions wikiPageID "39715492".
- Point_groups_in_four_dimensions wikiPageRevisionID "604164106".
- Point_groups_in_four_dimensions title "Uniform polychoron".
- Point_groups_in_four_dimensions urlname "UniformPolychoron".
- Point_groups_in_four_dimensions subject Category:Four-dimensional_geometry.
- Point_groups_in_four_dimensions subject Category:Polychora.
- Point_groups_in_four_dimensions comment "In geometry, a point group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere.A polychoric group is one of five symmetry groups of the 4-dimensional regular polytopes. There are also three polyhedral prismatic groups, and an infinite set of duoprismatic groups. Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes.".
- Point_groups_in_four_dimensions label "Point groups in four dimensions".
- Point_groups_in_four_dimensions sameAs m.0x29j5n.
- Point_groups_in_four_dimensions sameAs Q17099598.
- Point_groups_in_four_dimensions sameAs Q17099598.
- Point_groups_in_four_dimensions wasDerivedFrom Point_groups_in_four_dimensions?oldid=604164106.
- Point_groups_in_four_dimensions depiction Polychoral_group_tree.png.
- Point_groups_in_four_dimensions isPrimaryTopicOf Point_groups_in_four_dimensions.