In geometry, a polyhedron is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. Table of Geometry, from the 1728 Cyclopaedia. ... A polyhedron is a geometric object with flat faces and straight edges. ... In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. ... See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...

Isohedral polyhedra can be described by their face configuration. A form that is isohedral and has regular vertices is also edge-transitive (isotoxal) and is said to be a quasiregular dual: some theorists regard these figures as truly quasiregular because they share the same symmetries, but this is not generally accepted. In geometry, a face configuration is notational description of a face-uniform polyhedron. ... In geometry, a form is isotoxal or edge-transitive if its symmetries act transitively on its edges. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...

The terminology extends to n-dimensional polytopes, when the n - 1 dimensional facets are transitive. In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... A facet of an n-dimensional simplex is its (n-1)-dimensional face. ...

A polyhedron which is isohedral has a dual polyhedron that is vertex-transitive (isogonal). The Catalan solids, the bipyramids and the trapezohedra are all isohedral. They are the duals of the isogonal Archimedean solids, prisms and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex, edge, and face-transitive (isogonal, isotoxal, and isohedral). A polyhedron which is isohedral and isogonal but not isotoxal is said to be noble. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... In mathematics, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f : G → G such that f ( v1 ) = v2. ... Wikipedia does not yet have an article with this exact name. ... Regular octahedron triangular dipyramid J12 Pentagonal dipyramid J13 An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal pyramid and its mirror image base-to-base. ... An Archimedean solid or semiregular solid is a convex polyhedron with regular polygons as faces, such that at least two different types of regular polygons are used, and all vertices are identical (in the sense that the polygons are arranged in the same way about each vertex, and if someone... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. ...

References

• Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 9-521-55432-2, p.367 Transitivity

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