# 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.

A group of polytopes that shares a vertex arrangement is called an *army*.

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*.

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

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

For example, the self-intersecting *great dodecahedron* shares its edge arrangement with the convex *icosahedron*:

A group polytopes that share both a *vertex arrangement* and an *edge arrangement* are called a *regiment*.

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 arrangements.

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:

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).