Isotoxal figure

Isotoxal polygons

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 $4 n$ -gons are centrally symmetric, thus are also zonogons.

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

An isotoxal $2 n$ -gon with outer internal angle $α$ can be denoted by ${n_{α}} .$ The inner internal angle $(β)$ may be less or greater than $180$ $^{o},$ making convex or concave polygons respectively.

A star $2 n$ -gon can also be isotoxal, denoted by ${(n / q)_{α}},$ with $q \leq n - 1$ 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 $q < n / 2.$ If $D = gcd (n, q) \geq 2,$ then ${(n / q)_{α}} = {(D m / D p)_{α}}$ is "reduced" to a compound $D {(m / p)_{α}}$ of $D$ rotated copies of ${(m / p)_{α}} .$

Caution:

The vertices of

{(n / q)_{α}}

are not always placed like those of

{n_{α}},

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.

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Isotoxal polygons

Isotoxal polyhedra and tilings

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