Vertex arrangement
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- See vertex figure for the local description of faces around a vertex of a polyhedron or tiling.
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.
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[edit] Vertex arrangement
The same vertex arrangement 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.
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| ABCD is concave quadrilateral (green). It's vertex arrangement is set {A, B, C, D}. It's convex hull is triangle ABC (blue). Vertex arrangement of the convex hull is set {A, B, C}, which is not the same as vertex arrangement of the quadrilateral. So here, convex hull is not the way for describe the vertex arrangement. |
Infinte tilings can also share common vertex arrangements.
For example, this triangular lattice of point can be connected either as a set of isosceles triangles or rhombic faces.
| Two tilings with same vertex arrangement. | |
 Triangular tiling |
 rhombic tiling |
[edit] Edge arrangement
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) |
[edit] 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 arragements.
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) |
[edit] Classes for similar polytopes
George Olshevsky advocates calling classes of polytopes with similar element arrangements as an army. 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.
[edit] 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.
[edit] External links
- Olshevsky, George, Army at Glossary for Hyperspace.
- Olshevsky, George, Regiment at Glossary for Hyperspace.
- Olshevsky, George, Company at Glossary for Hyperspace.
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