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 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 Constructions 3.1 From the adjacent vertices
3.2
4 An example vertex figure of a honeycomb 5 Edge figure 6 See also 7 References 7.1 Notes 7.2 Bibliography 8 External links Definitions[edit] "Whole-edge" vertex figure of the cube Spherical vertex figure of the cube Point-set vertex figure of the cube Take some corner or 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
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.
The vertex figure of the great icosahedron is a regular pentagram or star polygon 5/2 . 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
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
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
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 space-filling 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[edit] 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.[3] 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 single-ringed
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
List of regular polytopes References[edit] Notes[edit] ^ Coxeter, H. et al. (1954). ^ Skilling, J. (1975). ^ Klitzing: Vertex figures, etc. Bibliography[edit] H. S. M. Coxeter, Regular Polytopes, Hbk (1948), ppbk (1973).
H.S.M.
External links[edit] Wikimedia Commons has media related to Vertex figures. Weisstein, Eric W. "Vertex figure". MathWorld. Olshevsky, George. "Vertex figure". Glossary for Hyperspace. Archived from the original on 4 February 2007. Vertex Figures Consistent V |