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 4n-gons are
centrally symmetric, thus are also
zonogons. In general, a (non-regular) isotoxal 2n-gon has \mathrm{D}_n, (^*nn)
dihedral symmetry. For example, a (non-square)
rhombus is an isotoxal "2×2-gon" (quadrilateral) with \mathrm{D}_2, (^*22) symmetry. All
regular {\color{royalblue}n}-gons (also with odd n) are isotoxal, having double the minimum symmetry order: a regular n-gon has \mathrm{D}_n, (^*nn) dihedral symmetry. An isotoxal \bold{2}n-gon with outer internal angle \alpha can be denoted by \{n_\alpha\}. The inner internal angle (\beta) may be less or greater than 180
{\color{royalblue}^\mathsf{o}}, making convex or concave polygons respectively. A
star {\color{royalblue}\bold{2}n}-gon can also be isotoxal, denoted by \{(n/q)_\alpha\}, with q \le n - 1 and with the
greatest common divisor \gcd(n,q) = 1, where q is the
turning number or
density. Concave inner vertices can be defined for q If D = \gcd(n,q) \ge 2, then \{(n/q)_\alpha\} = \{(Dm/Dp)_\alpha\} is "reduced" to a compound D \{(m/p)_\alpha\} of D rotated copies of \{(m/p)_\alpha\}. Caution: : The vertices of \{(n/q)_\alpha\} are not always placed like those of \{n_\alpha\}, whereas the vertices of the regular \{n/q\} are placed like those of the regular \{n\}. A set of
"uniform" tilings, actually
isogonal tilings using isotoxal polygons as less symmetric faces than regular ones, can be defined. == Isotoxal polyhedra and tilings ==