# Isotoxal figure

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 D_{n} (*nn) dihedral symmetry. A rhombus is an isotoxal polygon with D_{2} (*22) symmetry.

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

D_{2} (*22) |
D_{3} (*33) |
D_{4} (*44) |
D_{5} (*55) | |||||
---|---|---|---|---|---|---|---|---|

Rhombus | Equilateral triangle | Concave hexagon | Self-intersecting hexagon | Convex octagon | Regular pentagon | Self-intersecting (regular) pentagram | Self-intersecting decagram | |

## Isotoxal polyhedra and tilings

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.

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 at least 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.

## See also

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

## 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