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.
Contents |
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 |
Polyhedra can also have the same edge arrangement which means they have similar vertex and edge arrangements, but may differ in their faces.
For example the self-intersecting great dodecahedron shares it edge arrangement with the convex icosahedron.
icosahedron (20 triangles) |
great dodecahedron (12 intersecting pentagons) |
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 dodecahedrons) |
Great stellated 120-cell (120 great stellated dodecahedrons) |
George Olshevsky advocates the term army for a class of polytopes that share an element arrangement. More generally he defines n-regiments for polytopes that share elements up to dimension n. So a regiment (1-regiment) shares the same edge and vertex arrangement. He called a set of polytopes with the same 2-regiment as a company.