Regular skew polyhedron

In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces. Infinite regular skew polyhedra that span 3-space or higher are called regular skew apeirohedra. According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to regular skew polyhedra. Coxeter offered a modified Schläfli symbol {l,mn} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes. The regular skew polyhedra, represented by {l,mn}, follow this equation: A first set {l,mn}, repeats the five convex Platonic solids, and one nonconvex Kepler–Poinsot solid: {| class=wikitable ! {l,m|n} !Faces !Edges !Vertices !p !Polyhedron !Symmetryorder |- BGCOLOR="#e0f0e0" align=center | {3,3| 3} = {3,3} || 4||6||4 || 0|| Tetrahedron||12 |- BGCOLOR="#f0e0e0" align=center | {3,4| 4} = {3,4} ||8||12||6 || 0|| Octahedron||24 |- BGCOLOR="#e0e0f0" align=center | {4,3| 4} = {4,3} ||6||12||8 || 0|| Cube||24 |- BGCOLOR="#f0e0e0" align=center | {3,5| 5} = {3,5} ||20||30||12 || 0||Icosahedron||60 |- BGCOLOR="#e0e0f0" align=center | {5,3| 5} = {5,3} ||12||30||20 || 0|| Dodecahedron||60 |- BGCOLOR="#e0f0e0" align=center | {5,5| 3} = {5,5/2} ||12||30||12 || 4|| Great dodecahedron||60 |} Coxeter also enumerated the a larger set of finite regular polyhedra in his paper "regular skew polyhedra in three and four dimensions, and their topological analogues". Just like the infinite skew polyhedra represent manifold surfaces between the cells of the convex uniform honeycombs, the finite forms all represent manifold surfaces within the cells of the uniform 4-polytopes. Polyhedra of the form {2p, 2q | r} are related to Coxeter group symmetry of [(p,r,q,r)], which reduces to the linear [r,p,r] when q is 2.

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Regular skew polyhedron

Coxeter notation

In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram.

Complex polytope

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Precise definitions exist only for the regular complex polytopes, which are configurations.

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