- Isotoxal figure
:"This article is about geometry. For edge transitivity in graph theory, see
edge-transitive graph ."Ingeometry , apolytope (apolygon ,polyhedron or tiling, for example) 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 samedihedral angle for all edges.Not every
polyhedron ortessellation constructed fromregular polygons is isotoxal. For instance, thetruncated 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 areisohedral (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 regularPlatonic solid s
**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
*MathWorld | urlname=Edge-TransitiveGraph | title=Edge-transitive graph
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