In geometry, a vertex figure, broadly speaking, is the figure exposed
when a corner of a polyhedron or polytope is sliced off.
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
3.1 From the adjacent vertices
Dorman Luke construction
3.3 Regular polytopes
4 An example vertex figure of a honeycomb
5 Edge figure
6 See also
8 External links
"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
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, by extension, to space-filling tessellation with
polytope cells and other higher-dimensional polytopes.
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
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 an irregular polyhedron, 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
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. The
geometrical vertex figure and the vertex star may be understood as
distinct realizations of the same abstract section.
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.
Vertex figures are especially significant for uniforms and other
isogonal (vertex-transitive) polytopes because one vertex figure can
define the entire polytope.
For polyhedra with regular faces, the vertex figure can be represented
by a vertex configuration notation, by listing the faces in sequence
around a vertex. For example 184.108.40.206 is a vertex with one triangle and
three squares, and it defines the uniform rhombicuboctahedron.
If the polytope is isogonal, 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.
From the adjacent vertices
By considering the connectivity of these neighboring vertices, a
vertex figure can be constructed for each vertex of a polytope:
Each vertex of the vertex figure coincides with a vertex of the
Each edge of the vertex figure exists on or inside of a face of the
original polytope connecting two alternate vertices from an original
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.
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"
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
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
4-polytope or space-filling 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 indices reversed, it is easy to
see the dual of the vertex figure is the cell of the dual polytope.
For regular polyhedra, this is a special case of the Dorman Luke
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 space-filling tessellation.
Vertex figure: A nonuniform square pyramid
Created as a square base from an octahedron
And four isosceles triangle sides from truncated cubes
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 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
4-polytope or 3-honeycomb 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
4-polytope 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
List of regular polytopes
^ Coxeter, H. et al. (1954).
^ Skilling, J. (1975).
^ Klitzing: Vertex figures, etc.
H. S. M. Coxeter, Regular Polytopes, Hbk (1948), ppbk (1973).
Coxeter (et al.), Uniform Polyhedra, Phil. Trans. 246 A (1954)
P. Cromwell, Polyhedra, CUP pbk. (1999).
H.M. Cundy and A.P. Rollett, Mathematical Models, OUP (1961).
J. Skilling, The Complete Set of Uniform Polyhedra, Phil. Trans. 278 A
(1975) pp. 111–135.
M. Wenninger, Dual Models, CUP hbk (1983) ppbk (2003).
The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim
Goodman-Strass, ISBN 978-1-56881-220-5 (p289 Vertex figures)
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.