A toroidal polyhedron is a polyhedron with genus 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 ).
An eight-face locally regular hexagonal toroid has 16 polyhedron vertices , 24 polyhedron edges , eight faces ,
and equivelar map type equivelar
octahedron of genus 3 is a higher-genus eight-face example of type polyhedron vertices
and 36 polyhedron edges (Mizhaev 2026).
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 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 (
Stewart (1984) discusses and illustrates many new toroidal polyhedron constructions in a hand-printed and illustrated book.
See also Császár Polyhedron ,
Equivelar Map ,
Equivelar
Octahedron of Genus 3 ,
Szilassi Polyhedron ,
Toroid
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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. Hart, G. "Toroidal Polyhedra." https://www.georgehart.com/virtual-polyhedra/toroidal.html . McCooey,
D. I. "Regular Hexagonal Toroid with 8 Faces (Version 1)." https://dmccooey.com/polyhedra/RegularHexagonalToroid8_1.html . Mizhaev,
R. "Integer Realization of an Equivelar Octahedron of Genus 3." 15 Sep
2026. https://arxiv.org/abs/2609.17700 . 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. Szilassi, L. "Locally
Regular Toroids with Hexagonal Faces." Symmetry: Culture and Science 20 ,
269-296, 2009. https://journal-scs.symmetry.hu/content-pages/volume-20-numbers-1-4-pages-001-448-2009/ . Webb,
R. "Miscellaneous Polyhedra: Stewart Toroids." https://www.software3d.com/Misc.php#Stewart . Referenced
on Wolfram|Alpha Toroidal Polyhedron
Cite this as:
Weisstein, Eric W. "Toroidal Polyhedron."
From MathWorld https://mathworld.wolfram.com/ToroidalPolyhedron.html
Subject classifications
(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).
(Szilassi 2009; McCooey). The equivelar
octahedron of genus 3 is a higher-genus eight-face example of type 
with 24 polyhedron vertices
and 36 polyhedron edges (Mizhaev 2026).
(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 (
).
