Java Applets for Visualizing Regular Toroids


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) by the Euler-Poincaré formula:

V + F - E = 2 - 2 × G (1)

A toroid is said to be regular if every face has p vertices, and every vertex joins q faces. Regular toroids can thus be classified using the designation {p,q} for particular values of p ≥ 3 and q ≥ 3. 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 (also called equivelar).

In any polyhedron, each edge is adjacent to exactly two vertices and two faces. For a regular {p,q} toroid, this fact implies the following equations:

q × V = 2 × E = p × F (2)

For a genus-1 toroid, equation (1) gives the following:

V + F = E (3)

Combining equations (2) and (3) and using some algebra, we get the following relationship between p and q for a genus-1 toroid:

(p - 2) × (q - 2) = 4 (4)

For integers p ≥ 3 and q ≥ 3, equation (4) has only three solutions:

Using similar algebraic reasoning, we get the following relationship between p and q for a higher genus (G > 1) toroid:

(p - 2) × (q - 2) > 4 (5)

There are infinitely many solutions to this inequality, the simplest being {7,3}, {3,7}, {5,4}, and {4,5}.


The links below can be used to view some examples of regular toroids. Each toroid's page contains a Java applet for visualizing the toroid, followed by a summary of the toroid's vital statistics. The Java applet provides an opaque visual mode, a translucent visual mode, and a metrics mode. The metrics mode computes the toroid's vital statistics empirically.


{6,3} Genus-1 Toroids
{3,6} Genus-1 Toroids
{4,4} Genus-1 Toroids
Regular Higher Genus Toroids
Six-Pentagon Decomposition of the Real Projective Plane

Copyright © 2010 David I. McCooey. All Rights Reserved