Vertex figure
|
The vertex figure of a triangular prism is an isosceles triangle. The triangle face makes the short edge, and the two square faces make the long edges. A short-hand notation for this vertex figure is 3.4.4 |
In
geometry, a
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.
By considering the connectivity of these neigboring vertices, a full imaginary
(n-1)-polytope 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-uniform, 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.
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 the dual of the
vertex figure is the cell of the dual polytope.
|
truncated cubic 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.
In higher order polytopes other lower order figures can be useful. For instance an
edge figure is a polygon representing the set of faces around an edge. For example the
edge figure for a regular
cubic honeycomb {4,3,4} is a
square, and for a regular
polychoron {p,q,r} is the polygon {r}.
*
Vertex configuration*
List of regular polytopes
