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Regular Complex Polygon
In geometry, a regular complex polygon is a generalization of a regular polygon in real coordinate space, real space to an analogous structure in a Complex number, complex Hilbert space, where each real dimension is accompanied by an imaginary number, imaginary one. A regular polygon exists in 2 real dimensions, \mathbb^2, while a complex polygon exists in two complex dimensions, \mathbb^2, which can be given real representations in 4 dimensions, \mathbb^4, which then must be projected down to 2 or 3 real dimensions to be visualized. A ''complex polygon'' is generalized as a complex polytope in \mathbb^n. A complex polygon 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. The ''regular complex polygons'' have been completely characterized, and can be described using a symbolic notation developed by Harold Scott MacDonald Coxeter, Coxeter. A ''regular comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter–Dynkin Diagram
In geometry, a Harold Scott MacDonald Coxeter, Coxeter–Eugene Dynkin, Dynkin diagram (or Coxeter diagram, Coxeter graph) is a Graph (discrete mathematics), graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group. A class of closely related objects is the Dynkin diagrams, which differ from Coxeter diagrams in two respects: firstly, branches labeled "" or greater are Directed graph, directed, while Coxeter diagrams are Undirected graph, undirected; secondly, Dynkin diagrams must satisfy an additional (Crystallographic restriction theorem, crystallographic) restriction, namely that the only allowed branch labels are and Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras. Description A Coxeter group is a group that admits a presentation: \langle r_0,r_1,\dots,r_n \mid (r_i r_j)^ = 1 \rangle where the are integers that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-3 Duoprism
In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. Descriptions The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons. In the case of 3-3 duoprism is the simplest among them, and it can be constructed using Cartesian product of two triangles. The resulting duoprism has 9 vertices, 18 edges, and 15 faces—which include 9 squares and 6 triangles. Its cell has 6 triangular prism. It has Coxeter diagram , and symmetry , order 72. The hypervolume of a uniform 3-3 duoprism with edge length a is V_4 = a^4. This is the square of the area of an equilateral triangle, A = a^2. The 3-3 duoprism can be represented as a graph with the same number of vertices and edges. Like the Berlekamp–van Lint–Seidel graph and the unknown solution to Conway's 99-graph problem, every edge is part of a unique triangle and every non-adjacent pair of vertices is the diagonal of a unique squar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duopyramid
In geometry of 4 dimensions or higher, a double pyramid, duopyramid, or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two. The term fusil is used by Norman Johnson as a rhombic-shape. The term ''duopyramid'' was used by George Olshevsky, as the dual of a duoprism. Polygonal forms The lowest dimensional forms are 4 dimensional and connect two polygons. A ''p''-''q'' duopyramid or ''p''-''q'' fusil, represented by a composite Schläfli symbol + , and Coxeter-Dynkin diagram . The regular 16-cell can be seen as a 4-4 duopyramid or 4-4 fusil, , symmetry , order 128. A ''p-q duopyramid'' or ''p-q'' fusil has Coxeter group symmetry 'p'',2,''q'' order 4pq. When ''p'' and ''q'' are identical, the symmetry in Coxeter notation is doubled as or ''p'',2+,2''q'' order 8''p''2. Edges exist on all pairs of vertices between the ''p''-gon and ''q''-gon. The 1-skeleton of a ''p''-''q'' duopyramid represents edges of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duoprism
In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, where and are dimensions of 2 (polygon) or higher. The lowest-dimensional duoprisms exist in 4-dimensional space as 4-polytopes being the Cartesian product of two polygons in 2-dimensional Euclidean space. More precisely, it is the Set (mathematics), set of points: :P_1 \times P_2 = \ where and are the sets of the points contained in the respective polygons. Such a duoprism is Convex polytope, convex if both bases are convex, and is bounded by prism (geometry), prismatic cells. Nomenclature Four-dimensional duoprisms are considered to be prismatic 4-polytopes. A duoprism constructed from two regular polygons of the same edge length is a uniform duoprism. A duoprism made of ''n''-polygons and ''m''-polygons is named by prefixing 'duopr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Reflection Group
In mathematics, a complex reflection group is a Group (mathematics), finite group acting on a finite-dimensional vector space, finite-dimensional complex numbers, complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise in the study of the invariant theory of polynomial rings. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedron, regular polyhedra). Definition A (complex) reflection ''r'' (sometimes also called ''pseudo reflection'' or ''unitary reflection'') of a finite-dimensional complex vector space ''V'' is an element r \in GL(V) of finite order that fixes a complex hyperplane pointwise, that is, the ''fixed-space'' \operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternation (geometry)
In geometry, an alternation or ''partial truncation'', is an operation on a polygon, polyhedron, tessellation, tiling, or higher dimensional polytope that removes alternate vertices.Coxeter, Regular polytopes, pp. 154–156 8.6 Partial truncation, or alternation Coxeter labels an ''alternation'' by a prefixed ''h'', standing for ''hemi'' or ''half''. Because alternation reduces all polygon faces to half as many sides, it can only be applied to polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge. More generally any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be ''alternated''. For example, the alternation of a vertex figure with ''2a.2b.2c'' is ''a.3.b.3.c.3'' where the three is the number of elements in this vertex figure. A special case is square faces whose order divides in half into degenerate digons. So for example, the cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Number
In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which have the same order. This order is known as the Coxeter number. They are named after British-Canadian geometer H.S.M. Coxeter, who introduced the groups in 1934 as abstractions of reflection groups. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number of an irreducible root system. *The Coxeter number is the order of any Coxeter element;. *The Coxeter number is where is the rank, and is the number of reflections. In the crystallographic case, is half the number of roots; and is the dimension of the corresponding semisimple Lie algebra. *If th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Reflection
In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise in the study of the invariant theory of polynomial rings. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedra). Definition A (complex) reflection ''r'' (sometimes also called ''pseudo reflection'' or ''unitary reflection'') of a finite-dimensional complex vector space ''V'' is an element r \in GL(V) of finite order that fixes a complex hyperplane pointwise, that is, the ''fixed-space'' \operatorname(r) := \operatorname(r-\operatorname_V) has codimension 1. A (finite) complex reflecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shephard Group
In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise in the study of the invariant theory of polynomial rings. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedra). Definition A (complex) reflection ''r'' (sometimes also called ''pseudo reflection'' or ''unitary reflection'') of a finite-dimensional complex vector space ''V'' is an element r \in GL(V) of finite order that fixes a complex hyperplane pointwise, that is, the ''fixed-space'' \operatorname(r) := \operatorname(r-\operatorname_V) has codimension 1. A (finite) complex reflecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure. More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tessellation, tilings or, by extension, to Honeycomb (geometry), space-filling tessellation with polytope Cell (geometry), cells and other higher-dimensional polytopes. As a flat slice Make a slice through the corner of the polyhedron, cutting through all the edges conn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
