Toroidal Polyhedron
A toroidal polyhedron is a polyhedron with genus g>=1 (i.e., one having one or more holes). Examples of toroidal polyhedra include the Császár polyhedron and Szilassi polyhedron, both of which have genus 1 (i.e., the topology of a torus).
The only known toroidal polyhedron with no polyhedron diagonals is the Császár polyhedron. If another exists, it must have 12 or more polyhedron vertices and genus g>=6 (Gardner 1975). The smallest known single-hole toroidal polyhedron made up of only equilateral triangles was found by Conway (1997) and consists of 36 triangles. Borisov shows pictures of an assembled version. This construction has 6 diamonds (two attached triangles in the same plane) and 3 triamonds (three attached triangles in the same plane), and so basically consists of 3 octahedra and 9 tetrahedra (3×6+9×2=36).
Stewart (1984) discusses and illustrates many new toroidal polyhedron constructions in a hand-printed and illustrated book.
See also
Császár Polyhedron, Szilassi Polyhedron, ToroidExplore with Wolfram|Alpha
References
Borisov, N. "tomr polyhedron." http://gallery.nikita.ca/tomrhedron/.Borisov, N. "Toroidal Polyhedron Movie." http://gallery.nikita.ca/albums/tomrhedron/mvi_0240.avi.Conway, J. H. "RE: Polyhedra of Positive Genus." 23 Sep 1997. https://groups.google.com/g/geometry.research/c/4gvG8ZieFuE/m/JMUg64WIF3AJ.Gardner, M. "Mathematical Games: On the Remarkable Császár Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102-107, May 1975.Gardner, M. Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, p. 141, 1988.Hart, G. "Toroidal Polyhedra." https://www.georgehart.com/virtual-polyhedra/toroidal.html.Stewart, B. M. Adventures Among the Toroids, a Study of Quasi-Convex, Aplanar, Tunneled Orientable Polyhedra of Positive Genus Having Regular Faces with Disjoint Interiors, 2nd rev. ed. Okemos, MI: B. M. Stewart, 1984.Webb, R. "Miscellaneous Polyhedra: Stewart Toroids." https://www.software3d.com/Misc.php#Stewart.Referenced on Wolfram|Alpha
Toroidal PolyhedronCite this as:
Weisstein, Eric W. "Toroidal Polyhedron." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ToroidalPolyhedron.html