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The Info List - Vertex Figure



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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 (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 tilings , or space-filling tessellation with polytope cells .

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 (1948). For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint. Other authors make the cut through the vertex at the other end of each edge.

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 _n_-polytope is an (_n_−1)-polytope. For example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a 4-polytope 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 _n_-polytope (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 vertex-transitive , the vertex figure will exist in a hyperplane surface of the _n_-space. 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 {_a_,_b_,_c_,...,_y_,_z_} has cells as {_a_,_b_,_c_,...,_y_}, and _vertex figures_ as {_b_,_c_,...,_y_,_z_}.

* For a regular polyhedron {_p_,_q_}, the vertex figure is {_q_}, a _q_-gon.

* Example, the vertex figure for a cube {4,3}, is the triangle {3}.

* For a regular