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Concept
Isotoxal figure
Summary
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.
An isotoxal polygon is an even-sided i.e. equilateral polygon, but not all equilateral polygons are isotoxal. The duals of isotoxal polygons are isogonal polygons. Isotoxal -gons are centrally symmetric, so are also zonogons.
In general, an isotoxal -gon has dihedral symmetry. For example, a rhombus is an isotoxal "×-gon" (quadrilateral) with symmetry. All regular polygons (equilateral triangle, square, etc.) are isotoxal, having double the minimum symmetry order: a regular -gon has dihedral symmetry.
An isotoxal -gon with outer internal angle can be labeled as The inner internal angle may be greater or less than degrees, making convex or concave polygons.
Star polygons can also be isotoxal, labeled as with and with the greatest common divisor where is the turning number or density. Concave inner vertices can be defined for If then is "reduced" to a compound of rotated copies of
Caution: The vertices of are not always placed like those of whereas the vertices of the regular are placed like those of the regular
A set of "uniform tilings", actually isogonal tilings using isotoxal polygons as less symmetric faces than regular ones can be defined.
List of isotoxal polyhedra and tilings
Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).
Quasiregular polyhedra, like the cuboctahedron and the icosidodecahedron, are isogonal and isotoxal, but not isohedral. Their duals, including the rhombic dodecahedron and the rhombic triacontahedron, are isohedral and isotoxal, but not isogonal.
Not every polyhedron or 2-dimensional tessellation constructed from regular polygons is isotoxal.
Official source
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