# Regular toroid

A regular toroid is an orientable equivelar polyhedron of genus 1.[1] Despite the name, regular toroids are not necessarily abstractly regular in the standard sense. However, an orientable genus 1 polyhedron that is abstractly regular is a guaranteed regular toroid.

## Classes

For polyhedra of genus 1, the Euler characteristic must be 0. Thus

${\displaystyle V-E+F=0}$.

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

${\displaystyle {\frac {2E}{s}}+{\frac {2E}{d}}-E=0}$

Since ${\displaystyle E>0}$, this leads to the equation

${\displaystyle 2s+2d=sd}$

with the restrictions that ${\displaystyle s\geq 3}$ and ${\displaystyle d\geq 3}$, this leads to 3 solutions:

• ${\displaystyle s=3}$, ${\displaystyle d=6}$
• ${\displaystyle s=4}$, ${\displaystyle d=4}$
• ${\displaystyle s=6}$, ${\displaystyle d=3}$

These three solutions form three classes of regular toroids, called ${\displaystyle T_{1}}$, ${\displaystyle T_{2}}$, and ${\displaystyle T_{3}}$ by Lajos Szilassi.[1]

### T1

Regular toroids in the class ${\displaystyle T_{1}}$ have six triangular faces meeting at every vertex. They are topologically equivalent to looped portions of the triangular tiling.

### T2

Regular toroids in the class ${\displaystyle T_{2}}$ 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×d  faces where both n  and d  are at least 3. The vertex coordinates of these n -d  toroids can be given by ${\displaystyle \left((\sin({\tfrac {j\pi }{n}})+r)\sin({\tfrac {k\pi }{d}}),\,(\sin({\tfrac {j\pi }{n}})+r)\cos({\tfrac {k\pi }{d}}),\,\cos({\tfrac {j\pi }{n}})\right)}$ where ${\displaystyle 0\leq j and ${\displaystyle 0\leq k, 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 ${\displaystyle T_{2}}$ regular toroids, that do not rule out a realization. For example, it is unknown whether a ${\displaystyle T_{2}}$ regular toroid can be constructed with 10 or 11 faces.[1] It is known that no ${\displaystyle T_{2}}$ regular toroid has fewer than 9 faces / vertices.[1]

### T3

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

## References

1. Szilassi, Lajos (1986). "Regular toroids" (PDF). Structural Topology. 13: 69–80.