# Isotoxal figure

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

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.

The term isotoxal is derived from the Greek τοξον meaning arc.

## Isotoxal polygons

An isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons.

In general, an isotoxal 2n-gon will have Dn (*nn) dihedral symmetry. A rhombus is an isotoxal polygon with D2 (*22) symmetry.

All regular polygons (equilateral triangle, square, etc.) are isotoxal, having double the minimum symmetry order: a regular n-gon has Dn (*nn) dihedral symmetry. A regular 2n-gon is an isotoxal polygon and can be marked with alternately colored vertices, removing the line of reflection through the mid-edges.

Example isotoxal polygons
D2 (*22) D3 (*33) D4 (*44) D5 (*55)
Rhombus Equilateral triangle Concave hexagon Self-intersecting hexagon Convex octagon Regular pentagon Self-intersecting (regular) pentagram Self-intersecting decagram

## Isotoxal polyhedra and tilings

An isotoxal polyhedron or tiling must be either isogonal (vertex-transitive) or isohedral (face-transitive) or both.

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.

Examples
Quasiregular
polyhedron
Quasiregular dual
polyhedron
Quasiregular
star polyhedron
Quasiregular dual
star polyhedron
Quasiregular
tiling
Quasiregular dual
tiling

A cuboctahedron is isogonal and isotoxal polyhedron

A rhombic dodecahedron is an isohedral and isotoxal polyhedron

A great icosidodecahedron is isogonal and isotoxal star polyhedron

A great rhombic triacontahedron is an isohedral and isotoxal star polyhedron

The trihexagonal tiling is an isogonal and isotoxal tiling

The rhombille tiling is an isohedral and isotoxal tiling with p6m (*632) symmetry.

Not every polyhedron or 2-dimensional 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.

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

There are nine convex isotoxal polyhedra formed from the Platonic solids, 8 formed by the Kepler–Poinsot polyhedra, and six more as quasiregular (3 | p q) star polyhedra and their duals.

There are 5 polygonal tilings of the Euclidean plane that are isotoxal, 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.

## References

• Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 371 Transitivity
• Grünbaum, Branko; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (6.4 Isotoxal tilings, 309-321)
• Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246: 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446
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