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.
A group of polytopes that shares a vertex arrangement is called an army.
Vertex arrangement
The same set of vertices 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.
A vertex arrangement is often described by the convex hull polytope which contains it. For example, the regular pentagram can be said to have a (regular) pentagonal vertex arrangement.
ABCD is a concavequadrilateral (green). Its vertex arrangement is the set {A, B, C, D}. Its convex hull is the triangleABC (blue). The vertex arrangement of the convex hull is the set {A, B, C}, which is not the same as that of the quadrilateral; so here, the convex hull is not a way to describe the vertex arrangement.
Infinite tilings can also share common vertex arrangements.
George Olshevsky advocates the term regiment for a set of polytopes that share an edge arrangement, and more generally n-regiment for a set of polytopes that share elements up to dimension n. Synonyms for special cases include company for a 2-regiment (sharing faces) and army for a 0-regiment (sharing vertices).
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
Olshevsky, George. "Army". Glossary for Hyperspace. Archived from the original on 4 February 2007. (Same vertex arrangement)
Olshevsky, George. "Regiment". Glossary for Hyperspace. Archived from the original on 4 February 2007. (Same vertex and edge arrangement)
Olshevsky, George. "Company". Glossary for Hyperspace. Archived from the original on 4 February 2007. (Same vertex, edge and face arrangement)
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