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A vertex figure of an n-polytope is an (n−1)-polytope. For example, a vertex figure of a polyhedron is a polygon, and the vertex figure for a 4-polytope is a polyhedron. In general a vertex figure need not be planar. For nonconvex polyhedra, the vertex figure may also be nonconvex.
A vertex of an angle is the endpoint where two lines or rays come together. In geometry, a vertex (pl.: vertices or vertexes) is a point where two or more curves, lines, or edges meet or intersect. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. [1] [2] [3]
k-uniform tilings with the same vertex figures can be further identified by their wallpaper group symmetry. 1-uniform tilings include 3 regular tilings, and 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings and 673 6-uniform tilings.
In geometry, an apex (pl.: apices) is the vertex which is in some sense the "highest" of the figure to which it belongs. The term is typically used to refer to the vertex opposite from some "base". The word is derived from the Latin for 'summit, peak, tip, top, extreme end'. The term apex may used in different contexts:
The relations can be made apparent by examining the vertex figures obtained by listing the faces adjacent to each vertex (remember that for uniform polyhedra all vertices are the same, that is vertex-transitive). For example, the cube has vertex figure 4.4.4, which is to say, three adjacent square faces.
For instance, in the octahedron graph, shown in the figure, each vertex has a neighbourhood isomorphic to a cycle of four vertices, so the octahedron is locally C 4. For example: Any complete graph K n is locally K n-1. The only graphs that are locally complete are disjoint unions of complete graphs. A Turán graph T(rs,r) is locally T((r-1)s,r ...
Faces on a vertex figure are considered to progress in one direction. Some uniform polyhedra have vertex figures with inversions where the faces progress retrograde. A vertex figure represents this in the star polygon notation of sides p/q such that p<2q, where p is the number of sides and q the number of turns around a circle. For example, "3/ ...
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G).
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