Toroids with Regular Faces

triangle

square

pentagon

triangle 2

square 2

photo/graphics

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.

Details

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.

References

[1] J. H. Conway. "RE: Polyhedra of Positive Genus." (Sep 23, 1997) groups.google.com/group/geometry.research/msg/70178885eb20c524.

[2] R. Webb. "Stella: Polyhedron Navigator." (Apr 15, 2016) www.software3d.com/Stella.php.

[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.

External Links

Toroidal Polyhedron (Wolfram MathWorld)

Five Toroidal Polyhedra and Their Nets

Drilled Truncated Octahedron

The Parametrized Szilassi Polyhedron

Twenty Dodecahedra and a Golden Rhombohedron

Tetrahedral Loops (

Permanent Citation

Izidor Hafner

"Toroids with Regular Faces" from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/ToroidsWithRegularFaces/
Published: April 18, 2016