In geometry, a polytope (for example, a polygon or a polyhedron), or a tiling, is isotoxal or edgetransitive 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 2ngon 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 ngon has D_{n} (*nn) dihedral symmetry. A regular 2ngon is an isotoxal polygon and can be marked with alternately colored vertices, removing the line of reflection through the midedges.
D_{2} (*22)  D_{3} (*33)  D_{4} (*44)  D_{5} (*55)  

Rhombus  Equilateral triangle  Concave hexagon  Selfintersecting hexagon  Convex octagon  Regular pentagon  Selfintersecting (regular) pentagram  Selfintersecting decagram  
Regular polyhedra are isohedral (facetransitive), isogonal (vertextransitive) 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 2dimensional tessellation constructed from regular polygons is isotoxal. For instance, the truncated icosahedron (the familiar soccerball) has two types of edges: hexagonhexagon and hexagonpentagon, and it is not possible for a symmetry of the solid to move a hexagonhexagon edge onto a hexagonpentagon 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 nonright (p q r) groups.
In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertextransitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.
Technically, we say that for any two vertices there exists a symmetry of the polytope mapping the first isometrically onto the second. Other ways of saying this are that the group of automorphisms of the polytope is transitive on its vertices, or that the vertices lie within a single symmetry orbit.
All vertices of a finite ndimensional isogonal figure exist on an (n1)sphere.The term isogonal has long been used for polyhedra. Vertextransitive is a synonym borrowed from modern ideas such as symmetry groups and graph theory.
The pseudorhombicuboctahedron – which is not isogonal – demonstrates that simply asserting that "all vertices look the same" is not as restrictive as the definition used here, which involves the group of isometries preserving the polyhedron or tiling.
1–10 sides  

11–20 sides  
21–100 sides (selected) 

>100 sides  
Star polygons (5–12 sides)  
Others  
Classes 
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