Ingeometry, avertex arrangement is a set ofpoints in space described by their relative positions. They can be described by their use inpolytopes.
For example, asquare vertex arrangement is understood to mean four points in a plane, equal distance and angles from a center point.
Two polytopes share the samevertex arrangement if they share the same0-skeleton.
A group of polytopes that shares a vertex arrangement is called anarmy.
The same set of vertices can be connected by edges in different ways. For example, thepentagon andpentagram have the samevertex arrangement, while the second connects alternate vertices.
 pentagon |  pentagram |
Avertex arrangement is often described by theconvex hull polytope which contains it. For example, the regularpentagram can be said to have a (regular)pentagonal vertex arrangement.
 | ABCD is aconcavequadrilateral (green). Itsvertex arrangement is the set {A, B, C, D}. Its convex hull is thetriangleABC (blue). Thevertex 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 commonvertex arrangements.
For example, thistriangular lattice of points can be connected to form eitherisosceles triangles orrhombic faces.
 Lattice points |  Triangular tiling |  rhombic tiling |  Zig-zag rhombic tiling |  Rhombille tiling |
Polyhedra can also share anedge arrangement while differing in their faces.
For example, the self-intersectinggreat dodecahedron shares its edge arrangement with the convexicosahedron:
 icosahedron (20 triangles) |  great dodecahedron (12 intersecting pentagons) |
A group polytopes that share both avertex arrangement and anedge arrangement are called aregiment.
4-polytopes can also have the sameface arrangement which means they have similar vertex, edge, and face arrangements, but may differ in their cells.
For example, of the ten nonconvex regularSchläfli-Hess polychora, there are only 7 unique face arrangements.
For example, thegrand stellated 120-cell andgreat stellated 120-cell, both withpentagrammic faces, appear visually indistinguishable without a representation of theircells:
 Grand stellated 120-cell (120small stellated dodecahedra) |  Great stellated 120-cell (120great stellated dodecahedra) |
George Olshevsky advocates the termregiment for a set of polytopes that share an edge arrangement, and more generallyn-regiment for a set of polytopes that share elements up to dimensionn. Synonyms for special cases includecompany for a 2-regiment (sharing faces) andarmy for a 0-regiment (sharing vertices).