In geometry , a VERTEX FIGURE, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. CONTENTS * 1 Definitions – theme and variations * 1.1 As a flat slice * 1.2 As a spherical polygon * 1.3 As the set of connected vertices * 1.4 Abstract definition * 2 General properties * 3 Dorman Luke construction * 4 Regular polytopes * 5 An example vertex figure of a honeycomb * 6 Edge figure * 7 See also * 8 References * 8.1 Notes * 8.2 Bibliography * 9 External links DEFINITIONS – THEME AND VARIATIONS Take some vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points. 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
AS A FLAT SLICE Make a slice through the corner of the polyhedron, cutting through
all the edges connected to the vertex. The cut surface is the vertex
figure. This is perhaps the most common approach, and the most easily
understood. Different authors make the slice in different places.
Wenninger (2003) cuts each edge a unit distance from the vertex, as
does
Coxeter
For irregular polyhedra, these approaches may produce a vertex figure that does not lie in a plane. A more general approach, valid for arbitrary convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices, but is otherwise arbitrary. This construction determines the combinatorial structure of the vertex figure, similar to a set of connected vertices (see below), but not its precise geometry; it may be generalized to convex polytopes in any dimension. AS A SPHERICAL POLYGON Cromwell (1999) makes a spherical cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere. AS THE SET OF CONNECTED VERTICES Many combinatorial and computational approaches (e.g. Skilling, 1975) treat a vertex figure as the ordered (or partially ordered) set of points of all the neighboring (connected via an edge) vertices to the given vertex. ABSTRACT DEFINITION In the theory of abstract polytopes , the vertex figure at a given vertex V comprises all the elements which are incident on the vertex; edges, faces, etc. More formally it is the (n−1)section Fn/V, where Fn is the greatest face. This set of elements is elsewhere known as a vertex star. GENERAL PROPERTIES A vertex figure for an npolytope is an (n−1)polytope. For example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a 4polytope is a polyhedron. By considering the connectivity of these neighboring vertices an (n−1)polytope, the vertex figure, can be constructed for each vertex of a polytope: * Each vertex of the vertex figure coincides with a vertex of the original polytope. * Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two alternate vertices from an original face. * Each face of the vertex figure exists on or inside a cell of the original npolytope (for n > 3). * ... and so on to higher order elements in higher order polytopes. Vertex figures are the most useful for uniform polytopes because one vertex figure can encode the entire polytope. For polyhedra, the vertex figure can be represented by a vertex configuration notation, by listing the faces in sequence around a vertex. For example 3.4.4.4 is a vertex with one triangle and three squares, and it represents the rhombicuboctahedron . If the polytope is vertextransitive , the vertex figure will exist in a hyperplane surface of the nspace. In general the vertex figure need not be planar. For nonconvex polyhedra, the vertex figure may also be nonconvex. Uniform polytopes, for instance, can have star polygons for faces or for vertex figures. DORMAN LUKE CONSTRUCTION For a uniform polyhedron, the face of the dual polyhedron may be found from the original polyhedron's vertex figure using the "Dorman Luke " construction. REGULAR POLYTOPES If a polytope is regular, it can be represented by a Schläfli symbol and both the cell and the vertex figure can be trivially extracted from this notation. In general a regular polytope with
Schläfli symbol
* For a regular polyhedron {p,q}, the vertex figure is {q}, a qgon. * Example, the vertex figure for a cube {4,3}, is the triangle {3}. * For a regular 4polytope or spacefilling tessellation {p,q,r}, the vertex figure is {q,r}. * Example, the vertex figure for a hypercube {4,3,3}, the vertex figure is a regular tetrahedron {3,3}. * Also the vertex figure for a cubic honeycomb {4,3,4}, the vertex figure is a regular octahedron {3,4}. Since the dual polytope of a regular polytope is also regular and
represented by the
Schläfli symbol
AN EXAMPLE VERTEX FIGURE OF A HONEYCOMB truncated cubic honeycomb (partial). The vertex figure of a truncated cubic honeycomb is a nonuniform square pyramid . One octahedron and four truncated cubes meet at each vertex form a spacefilling tessellation . Vertex figure: A nonuniform square pyramid Schlegel diagram Perspective Created as a square base from an octahedron (3.3.3.3) And four isosceles triangle sides from truncated cubes (3.8.8) EDGE FIGURE The truncated cubic honeycomb has two edge types, one with four truncated cubes, and the others with one octahedron, and two truncated cubes. These can be seen as two types of edge figures. These are seen as the vertices of the vertex figure. Related to the vertex figure, an edge figure is the vertex figure of a vertex figure. Edge figures are useful for expressing relations between the elements within regular and uniform polytopes. An edge figure will be a (n−2)polytope, representing the arrangement of facets around a given edge. Regular and singleringed coxeter diagram uniform polytopes will have a single edge type. In general, a uniform polytope can have as many edge types as active mirrors in the construction, since each active mirror produces one edge in the fundamental domain. Regular polytopes (and honeycombs) have a single edge figure which is also regular. For a regular polytope {p,q,r,s,...,z}, the edge figure is {r,s,...,z}. In four dimensions, the edge figure of a 4polytope or 3honeycomb is a polygon representing the arrangement of a set of facets around an edge. For example, the edge figure for a regular cubic honeycomb {4,3,4} is a square , and for a regular 4polytope {p,q,r} is the polygon {r}. Less trivially, the truncated cubic honeycomb t0,1{4,3,4}, has a square pyramid vertex figure, with truncated cube and octahedron cells. Here there are two types of edge figures. One is a square edge figure at the apex of the pyramid. This represents the four truncated cubes around an edge. The other four edge figures are isosceles triangles on the base vertices of the pyramid. These represent the arrangement of two truncated cubes and one octahedron around the other edges. SEE ALSO * Vertex configuration * List of regular polytopes REFERENCES NOTES * ^ Coxeter, H. et al. (1954). * ^ Skilling, J. (1975). * ^ Klitzing: Vertex figures, etc. BIBLIOGRAPHY * H. S. M.
Coxeter
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