# 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*.

## 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.

pentagon |
pentagram |

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 concave quadrilateral (green). Its vertex arrangement is the set {A, B, C, D}. Its convex hull is the triangle ABC (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*.

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

Lattice points |
Triangular tiling |
rhombic tiling |
Zig-zag rhombic tiling |
Rhombille tiling |

## Edge arrangement

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*:

icosahedron (20 triangles) |
great dodecahedron (12 intersecting pentagons) |

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

## 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 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:

Grand stellated 120-cell (120 small stellated dodecahedra) |
Great stellated 120-cell (120 great stellated dodecahedra) |

## Classes of similar polytopes

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)