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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.

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Isotoxal polygonsEdit

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 tilingsEdit

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.

See alsoEdit

ReferencesEdit