Toroids with Regular Faces

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This Demonstration shows five toroids: three with regular faces (equilateral triangles, squares, pentagons) and then two more, with faces compounded of multiple equilateral triangles or squares.

Contributed by: Izidor Hafner (April 2016)
Open content licensed under CC BY-NC-SA


According to the MathWorld entry for Toroidal Polyhedron, the smallest known single-hole toroidal polyhedron made up of only equilateral triangles was found by Conway [1] and consists of 36 triangles. However, some adjacent triangles are coplanar.


[1] J. H. Conway. "RE: Polyhedra of Positive Genus." (Sep 23, 1997)

[2] R. Webb. "Stella: Polyhedron Navigator." (Apr 15, 2016)

[3] S. Werbeck, "Multiple Unit Constructions of Face-Connected Regular Dodecahedra: A Comparative Study of Simple and Complex Forms," Symmetry: Culture and Science, 11(1–4), 2000 pp. 393–403.

[4] L. Szilassi, "On Three Classes of Regular Toroids," Symmetry: Culture and Science, 11(1–4), 2000 pp. 317–335.


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