Vertex-uniform
From Wikipedia, the free encyclopedia
In geometry, a polyhedron (or tiling) is vertex-uniform if all its vertices are the same, that is, if each vertex is surrounded by the same faces, in the same order. Technically, vertex-uniform means that for any two vertices there exists a symmetry of the polyhedron mapping the first isometrically onto the second.
A polyhedron is:
- Regular if it is vertex-uniform, edge-uniform and face-uniform; this implies that every face is a regular polygon or a regular star polygon.
- Quasi-regular if it is edge-uniform and face-uniform but not vertex-uniform, and every face is a regular polygon.
- Semi-regular if it is vertex-uniform but neither edge-uniform nor face-uniform, and every face is a regular polygon.
- Uniform if it is vertex-uniform and every face is a regular polygon, i.e. it is regular, quasi-regular, or semi-regular.
A vertex-uniform polyhedron has a single vertex figure and can be represented by a vertex configuration notation sequencing the faces around each vertex.
These definitions can be extended to higher dimensional polytopes.
