Vertex arrangement

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 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.

Infinite tilings can also share common vertex arrangements.

For example, this triangular lattice of points can be connected to form either isosceles triangles or rhombic faces.

Edge arrangement

Polyhedra can also share an edge arrangement while differing in their faces.