Regular Hexagonal 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:

The toroids on this page are regular genus-1 toroids in the class {6,3}.

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Szilassi Polyhedron (version 1)

Regular Hexagonal Toroid with 8 faces (version 1)

Regular Hexagonal Toroid with 8 faces (D2 symmetry)

Hexagonal Toroid with 8 overarching faces

Regular Hexagonal Toroid with 9 faces (type A) (cubic form)

Regular Hexagonal Toroid with 9 faces (type A) (triangular form)

Regular Hexagonal Toroid with 9 faces (type B)

Regular Hexagonal Toroid with 10 faces (version 2)

Regular Hexagonal Toroid with 11 faces

Regular Hexagonal Toroid with 12 faces (type A) (trapezohedral form)

Regular Hexagonal Toroid with 12 faces (type B)

Regular Hexagonal Toroid with 12 faces (type C) (simple form)

Regular Hexagonal Toroid with 12 faces (type C) (knot form)

Regular Hexagonal Toroid with 12 faces (type D) (version 2)

Regular Hexagonal Toroid with 13 faces (type A)

Regular Hexagonal Toroid with 13 faces (type B) (version 2)

Regular Hexagonal Toroid with 14 faces (type A) (version 3)

Regular Hexagonal Toroid with 14 faces (type A) (knot form)

Regular Hexagonal Toroid with 14 faces (type B)

Regular Hexagonal Toroid with 14 faces (type B) (D2 symmetry) (version 1)

Regular Hexagonal Toroid with 15 faces (type A) (version 1)

Regular Hexagonal Toroid with 15 faces (type A) (knot form 2)

Regular Hexagonal Toroid with 15 faces (type D) (version 6)

Regular Hexagonal Toroid with 24 faces