In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling 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.

Isotoxal polygons

An isotoxal polygon is an even-sided i.e. equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons-gons are centrally symmetric, thus are also zonogons.

dihedral symmetry. For example, a (non-square) rhombus symmetry. All {\color{royalblue}n}-gons dihedral symmetry.

{\color{royalblue}^\mathsf{o}}, making convex or concave polygons respectively.

A {\color{royalblue}\bold{2}n}-gon and with the greatest common divisor is the turning number or density.

Caution:

A set of "uniform" tilings, actually isogonal tilings using isotoxal polygons as less symmetric faces than regular ones, can be defined.

Isotoxal polyhedra and tilings

Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).

Quasiregular polyhedra, like the cuboctahedron and the icosidodecahedron, are isogonal and isotoxal, but not isohedral. Their duals, including the rhombic dodecahedron and the rhombic triacontahedron, are isohedral and isotoxal, but not isogonal.Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal. For instance, the truncated icosahedron (the familiar soccerball) is not isotoxal, as it has two edge types: 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.

An isotoxal polyhedron has the same dihedral angle for all edges.

The dual of a convex polyhedron is also a convex polyhedron.

The dual of a non-convex polyhedron is also a non-convex polyhedron. (By contraposition.)

The dual of an isotoxal polyhedron is also an isotoxal polyhedron. (See the Dual polyhedron article.)

There are nine convex isotoxal polyhedra: the five (regular) Platonic solids, the two (quasiregular) common cores of dual Platonic solids, and their two duals.

There are thirty-eight non-convex isotoxal polyhedra:

• the four regular Kepler-Poinsot polyhedra

• their two rectifications and their duals

• the three ditrigonary uniforms and their duals

• nine out of the ten uniform hemipolyhedra (all except the great dirhombicosidodecahedron)

• four isotoxal polyhedra constructed as the duals of dyadic interpretations of degenerate polyhedral compounds

• eleven isotoxal polyhedra which are neither isohedral nor isogonal

There are at least five isotoxal polyhedral compounds: the five regular polyhedral compounds; their five duals are also the five regular polyhedral compounds (or one chiral twin).

There are at least five isotoxal polygonal tilings of the Euclidean plane, and infinitely many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the regular hyperbolic tilings {p,q}, and non-right (p q r) groups.

See also

• Table of polyhedron dihedral angles

• Vertex-transitive

• Face-transitive

• Cell-transitive

References

• Peter R. Cromwell, Polyhedra, Cambridge University Press, 1997, , Transitivity, p. 371

• (6.4 Isotoxal tilings, pp. 309–321)