Isotoxal figure

In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal (from Greek τόξον  'arc') 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 ${\displaystyle 4n}$-gons are centrally symmetric, thus are also zonogons.

In general, a (non-regular) isotoxal ${\displaystyle 2n}$-gon has ${\displaystyle \mathrm {D} _{n},(^{*}nn)}$ dihedral symmetry. For example, a (non-square) rhombus is an isotoxal "${\displaystyle 2}$×${\displaystyle 2}$-gon" (quadrilateral) with ${\displaystyle \mathrm {D} _{2},(^{*}22)}$ symmetry. All regular ${\displaystyle {\color {royalblue}n}}$-gons (also with odd ${\displaystyle n}$) are isotoxal, having double the minimum symmetry order: a regular ${\displaystyle n}$-gon has ${\displaystyle \mathrm {D} _{n},(^{*}nn)}$ dihedral symmetry.

An isotoxal ${\displaystyle {\mathbf {2}}n}$-gon with outer internal angle ${\displaystyle \alpha }$ can be denoted by ${\displaystyle \{n_{\alpha }\}.}$ The inner internal angle ${\displaystyle (\beta )}$ may be less or greater than ${\displaystyle 180}$${\displaystyle {\color {royalblue}^{\mathsf {o}}},}$ making convex or concave polygons respectively.

A star ${\displaystyle {\color {royalblue}{\mathbf {2}}n}}$-gon can also be isotoxal, denoted by ${\displaystyle \{(n/q)_{\alpha }\},}$ with ${\displaystyle q\leq n-1}$ and with the greatest common divisor ${\displaystyle \gcd(n,q)=1,}$ where ${\displaystyle q}$ is the turning number or density. Concave inner vertices can be defined for ${\displaystyle q<n/2.}$ If ${\displaystyle D=\gcd(n,q)\geq 2,}$ then ${\displaystyle \{(n/q)_{\alpha }\}=\{(Dm/Dp)_{\alpha }\}}$ is "reduced" to a compound ${\displaystyle D\{(m/p)_{\alpha }\}}$ of ${\displaystyle D}$ rotated copies of ${\displaystyle \{(m/p)_{\alpha }\}.}$

Caution:

A set of "uniform" tilings, actually isogonal tilings using isotoxal polygons as less symmetric faces than regular ones, can be defined.

Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).