Regular Tetragonal Toroidal Solids |

A toroidal solid, or toroid, is an orientable polyhedron without self-intersections that has genus greater than zero, meaning that it contains one or more holes. An orientable polyhedron's genus (G) is related to the number of vertices (V), faces (F), and edges (E) as follows:

V + F − E = 2 − 2 * G

A toroid is said to be regular if every face has p vertices, and every vertex joins q faces. This notion of regularity is strictly topological, in the sense that it does not specify any geometric criteria, such as equal edge lengths or equal vertex angles. A regular toroid can be further classified as being either globally regular or locally regular by examining its set of mutually incident face-edge-vertex triples. If all such triples are topologically equivalent, then the toroid is globally regular. Otherwise, it is locally regular. The set of regular genus-1 toroids can be divided into three classes:

- {6,3} toroids where 3 hexagonal faces meet at every vertex
- {3,6} toroids where 6 triangular faces meet at every vertex
- {4,4} toroids where 4 tetragonal (quadrilateral) faces meet at every vertex

The toroids on this page are regular genus-1 toroids in the class {4,4}.