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Vertex-transitivity
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces. Technically, one says that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope '' acts transitively'' on its vertices, or that the vertices lie within a single '' symmetry orbit''. All vertices of a finite -dimensional isogonal figure exist on an -sphere. The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory. The pseudorhombicuboctahedronwhich is ''not'' isogonaldemonstrates that simply asserting that "all vertices look t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetry Group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object ''X'' is ''G'' = Sym(''X''). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure. Introduction We consider the "objects" possessing symmetry to be geometric figures, images, and patterns, such as a wallpaper pattern. For symmetry of physical objects, one may also take their physical composition as part of the pattern. (A pattern may be specified formally as a scalar field, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Star Polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex polygon, convex'' or ''star polygon, star''. In the limit (mathematics), limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a Line (geometry), straight line), if the edge length is fixed. General properties These properties apply to all regular polygons, whether convex or star polygon, star: *A regular ''n''-sided polygon has rotational symmetry of order ''n''. *All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. *Together with the property of equal-length sides, this implies that every regular polygon also h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew Apeirogon
In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all Collinearity, colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of a Cylinder (geometry), cylinder. Regular infinite skew polygons exist in the Petrie polygons of the affine and hyperbolic Coxeter groups. They are constructed a single operator as the composite of all the reflections of the Coxeter group. Regular zig-zag skew apeirogons in two dimensions A regular zig-zag skew apeirogon has (2*∞), D∞d Frieze group symmetry. Regular zig-zag skew apeirogons exist as Petrie polygons of the three regular tilings of the plane: , , and . These regular zig-zag skew apeirogons have internal angles of 90°, 120°, and 60° respectively, from the regular polygons within the tilings: Is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Apeirogon2d
Isogonal, a mathematical term meaning "having similar angles", may refer to: *Isogonal figure or polygon, polyhedron, polytope or tiling *Isogonal trajectory, in curve theory *Isogonal conjugate, in triangle geometry See also *Isogonic line A contour line (also isoline, isopleth, isoquant or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensi ..., in the study of Earth's magnetic field, a line of constant magnetic declination {{disambig Geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apeirogon
In geometry, an apeirogon () or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an infinite dihedral group of symmetries. Definitions Geometric apeirogon Given a point ''A''0 in a Euclidean space and a translation ''S'', define the point ''Ai'' to be the point obtained from ''i'' applications of the translation ''S'' to ''A''0, so ''Ai'' = ''Si''(''A''0). The set of vertices ''Ai'' with ''i'' any integer, together with edges connecting adjacent vertices, is a sequence of equal-length segments of a line, and is called the regular apeirogon as defined by H. S. M. Coxeter. A regular apeirogon can be defined as a partition of the Euclidean line ''E''1 into infinitely many equal-length segments. It generalizes the regular ''n''-gon, which may be defined as a partition of the circle ''S''1 into ''finitely'' many equal-length ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
