Edge-transitive

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

In geometry, a form is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged. An isotoxal polyhedron has the same dihedral angle for all edges.

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

[edit] See also

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

[edit] References

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