Vertex_figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.

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.

But compare the detailed descriptions across standard reference works, and you will soon find there is little agreement on a formal definition.

Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion.
Cromwell (1999) treats it as a spherical polygon marked on a sphere centered on the vertex.
Skilling (1975) and most abstract theorists use the definition discussed below.

General properties

One kind of vertex figure represents the arrangement of a connected set of points of all the neighboring vertices, in a polytope to a given vertex. This applies equally well to infinite tilings, or space-filling tessellation with polytope cells.

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 polychoron is a polyhedron figure.

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 imply 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 3 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.

Also nonconvex polyhedra, the vertex figure may also be nonconvex. Uniform polytopes can have either star polygon faces and vertex figures for instance.

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 polychoron 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 construction.

an example vertex figure of a honeycomb

The vertex figure of a truncated cubic honeycomb is a nonuniform square pyramid. One octahedron and four truncated cube meet at each vertex for 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

References

H.S.M. Coxeter, Regular Polytopes, Hbk (1948), ppbk (1973).
H.S.M. Coxeter (et al), Uniform Polyhedra, Phil. Trans. 246 A (1954) pp. 401-450.
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).