This article is about geometry. For edge transitivity in graph theory, see edge-transitive graph.

Not every polyhedron or tessellation constructed from regular polygons is isotoxal. For instance, the truncated icosahedron (the familiar soccerball) has two types of edges: hexagon-hexagon and hexagon-pentagon, and it is not possible for a symmetry of the solid to move a hexagon-hexagon edge onto a hexagon-pentagon edge. However, regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive) and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral; their duals are isohedral and isotoxal, but not isogonal.

There are nine convex isotoxal polyhedra:

• The five regular Platonic solids
• Tetrahedron
• Cube
• Octahedron
• Dodecahedron
• Icosahedron
• The two quasiregular Archimedeans:
• cuboctahedron
• icosidodecahedron
• The two Catalans which are dual to the quasiregular Archimedeans:
• Rhombic dodecahedron
• Rhombic triacontahedron

See also

• Table of polyhedron dihedral angles
• Vertex-transitive
• Face-transitive
• Cell-transitive

References

• Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 9-521-55432-2, p.371 Transitivity