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 . Numberphile.
"The Holey Monster (with 934 Faces)." Jul. 31, 2025. https://www.youtube.com/watch?v=BhSlqIe1CRQ . 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 More... Less...
(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 (
).
