Regular toroid

A regular toroid is a polyhedron of genus 1 in which every vertex has the same degree and every face has the same number of vertices. While every abstractly regular toroid is a regular toroid, the reverse is not true.

Classes
For polyhedra of genus 1, the Euler characteristic must be 0. Thus
 * $$V-E+F=0$$.

For a regular toroid, if each face has $s$ sides, then the number of faces can be rewritten as $$F=\frac{2E}{s}$$. Likewise if the degree of each vertex is $d$ then the number of vertices can be rewritten as $$V=\frac{2E}{d}$$. Thus the Euler characteristic can be written as:
 * $$\frac{2E}{s}+\frac{2E}{b}-E=0$$

Since $$E>0$$, this leads to the equation
 * $$2s+2d=sd$$

with the restrictions that $$s\geq 3$$ and $$d\geq 3$$, this leads to 3 solutions:
 * $$s=3$$, $$d=6$$
 * $$s=4$$, $$d=4$$
 * $$s=6$$, $$d=3$$

These three solutions form three classes of regular toroids, called $$T_1$$, $$T_2$$, and $$T_3$$ by Lajos Szilassi.

$T_{1}$


Regular toroids in the class $$T_1$$ have six triangular faces meeting at every vertex. They are topologically equivalent to looped portions of the triangular tiling.

$T_{2}$
Regular toroids in the class $$T_2$$ have four square faces meeting at every vertex. They have an equal number of faces and vertices. They are topologically equivalent to looped portions of the square tiling.

Constructions are known for regular toroids with $n&times;d$ faces where both $n$ and $d$ are at least 3. The vertex coordinates of these $n$-$d$ toroids can be given by $$\left((\sin(\tfrac{j\pi}n)+r)\sin(\tfrac{k\pi}d),\,(\sin(\tfrac{j\pi}n)+r)\cos(\tfrac{k\pi}d),\,\cos(\tfrac{j\pi}n)\right)$$ where $$0 \leq j < n-1$$ and $$0 \leq k < d-1$$, and $r$ is the major radius of the torus. The resulting toroid has trapezoidal faces. However there are other numbers of faces for which there exist abstract $$T_2$$ regular toroids, that do not rule out a realization. For example, it is unknown whether a $$T_2$$ regular toroid can be constructed with 10 or 11 faces. It is known that no $$T_2$$ regular toroid has fewer than 9 faces / vertices.

$T_{3}$
Regular toroids in the class $$T_3$$ have 3 hexagonal faces meeting at every vertex. They are topologically equivalent to looped portions of the hexagonal tiling.