Face configuration

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In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face.

There is no single standard accepted representation, but one common notation prefixes the description with a V and separates the vertices by a period (.) or a comma (,).

The rhombic dodecahedron alternates three faces per vertex and four faces per vertex
V3.4.3.4

For example, V3.4.3.4 represents the rhombic dodecahedron which is face-transitive: every face is a rhombus, and alternating vertices of the rhombus contain 3 or 4 faces each.

Another form of this notation, used in Tilings and Patterns, has brackets around the symbol, for instance [3.4.3.4].

Face-transitive polyhedra are generally the polyhedral duals of the vertex-transitive polyhedra, which are described by a parallel vertex configuration notation. That notation omits the V prefix and represents sequentially the number of edges of the faces around a vertex. For example, 3.4.3.4 is the cuboctahedron with alternating triangular and square faces around each vertex. Polyhedra have the same representation in face configuration notation (with the addition of the V) that their duals have in vertex configuration notation. The rhombic dodecahedron (V3.4.3.4) and the cubocahedron (3.4.3.4) above are dual polyhedra.

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