Parametrized Toroid with 42 Faces

​
net
toroid
fragment
dual
r
2
a
0.16
b
0.3
c
0.4
d
0.28
This Demonstration shows a parametrized toroid with 42 faces. Looking down along the axis of symmetry, the larger and smaller concave heptagons have circumscribing circles of radius
r
and 1, respectively. The vertical faces are double trapezoids with bases of length
2a
,
2b
(inner) and
2c
,
2d
(outer). This polyhedral toroid is colored so that it realizes the Heawood map.

Details

When constructing this toroid, care must be taken to make sure that the six points defining a face region are in the same plane[2, p. 325]. This toroid belongs to the class
T
3
, which means that it is a regular toroid with hexagonal faces, with three edges meeting at each vertex[2, p. 318]. The dual has triangular faces, with six edges meeting at each vertex, thus it belongs to the class
T
1
.

References

[1] L. Szilassi, "Regular Toroids," Structural Topology, Université de Montréal, 13, 1986 pp. 69–80. www-iri.upc.es/people/ros/StructuralTopology/ST13/st13-06-a3-ocr.pdf.
[2] L. Szilassi, "On Three Classes of Regular Toroids," Symmetry: Culture and Science, 11(1–4), 2000 pp. 317–335.
[3] "Shelf of Lajos Szilassi." www.kabai.hu/elte-mathematical-museum.
[4] B. M. Stewart, Adventures among the Toroids, Okemos, Michigan: B. M. Stewart, rev. 2nd ed., 1980 p. 199.

External Links

Toroid (Wolfram MathWorld)
Drilled Truncated Octahedron
Seven-Coloring of a Torus
The Parametrized Szilassi Polyhedron
Map Coloring on a Torus
Colored Szilassi Polyhedron
Toroidal Polyhedra

Permanent Citation

Lajos Szilassi, Izidor Hafner, Sándor Kabai
​
​"Parametrized Toroid with 42 Faces"​
​http://demonstrations.wolfram.com/ParametrizedToroidWith42Faces/​
​Wolfram Demonstrations Project​
​Published: May 1, 2016