Isotoxal figure
- This article is about geometry. For edge transitivity in graph theory, see edge-transitive graph.
In geometry, a polytope (a polygon, 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.
The term isotoxal is derived from the Greek τοξον meaning arc.
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[edit] Isotoxal polygons
An isotoxal polygon is an equilateral polygon, but not all equilateral polygons are isotoxal.
In general, an isotoxal 2n-gon will have Dn (*n) dihedral symmetry. A rhombus is a isotoxal polygon with D2 (*2) symmetry.
All regular polygons (equilateral triangle, square, etc.) are isotoxal, having double the minimum symmetry order: a regular n-gon has Dn (*n) dihedral symmetry. A square is a isotoxal polygon with D4 (*4) symmetry.
| Dihedral symmetry | D2 (*2) | D3 (*3) | D4 (*4) | D5 (*5) | |||||
|---|---|---|---|---|---|---|---|---|---|
| Name | Rhombus | Equilateral triangle | Concave hexagon | Self-intersecting hexagon | Square | Convex octagon | Regular pentagon | Self-intersecting (regular) pentagram | Self-intersecting decagram |
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[edit] Isotoxal polyhedra and tilings
 The rhombille tiling is an isotoxal tiling with p6m (*632) symmetry. |
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.
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.
[edit] See also
[edit] References
- Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 9-521-55432-2, p. 371 Transitivity
- Grünbaum, Branko; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-716-71193-1. (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, ISSN 0080-4614, JSTOR 91532, MR0062446
