Vertex arrangement

: See "vertex figure for the local description of faces around a vertex of a polyhedron or tiling."

In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.

For example a "square vertex arrangement" is understood to mean four points in a plane, equal distance and angles from a center point.

Two polytopes share the same "vertex arrangement" if they share the same 0-skeleton.

Vertex arrangement

The same "vertex arrangement" can be connected by edges in different ways. For example the "pentagon" and "pentagram" have the same "vertex arrangement", while the second connects alternate vertices.

Edge arrangement

Polyhedra can also have the same "edge arrangement" which means they have similar vertex and edge arrangements, but may differ in their faces.

For example the self-intersecting "great dodecahedron" shares it edge arrangement with the convex "icosahedron".

Face arrangement

4-polytopes can also have the same "face arrangement" which means they have similar vertex, edge, and face arrangements, but may differ in their cells.

For example, of the ten nonconvex regular Schläfli-Hess polychora, there are only 7 unique face arragements.

For example the grand stellated 120-cell and great stellated 120-cell, both with pentagrammic faces, appear visually indistinguishable without a representation of their cells:

Classes for similar polytopes

George Olshevsky advocates calling classes of polytopes with similar element arrangements as an army. More generally he defines "n-regiments" for polytopes that share elements up to dimension "n". So a regiment ("1-regiment") shares the same "edge and vertex arrangement". He called a set of polytopes with the same "2-regiment" as a company.

See also

*n-skeleton - a set of elements of dimension "n" and lower in a higher polytope.
*Vertex figure - A local arrangement of faces in a polyhedron (or arrangement of cells in a polychoron) around a single vertex.

External links

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