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.
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.
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.
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