Toroids1

The Szilassi and Czászár Polyhedra and beyond ...

The Szilassi polyhedron above is an example of a rare category of polyhedra termed 'face-neighbourly' [1]. These share the following characteristics:

- Each face shares an edge with every other face.
- All faces are planar
- Faces do not intersect.

If we consider Euler's formula: For a polyhedron with F faces, V vertices, E edges of genus g this is:

V - E +F = 2 - 2g

If we require every face with gonality G to share an edge with every other face, we need the number of faces to exceed the gonality by one, so F = G + 1.

The number of edges is F.G / 2

For all faces to share edges, the valence of every vertex must be 3. If we have four faces at a vertex, the opposing faces will not share an edge. So V = F.G / 3

Euler's formula can now ve restated solely in terms of F.
F.(F-1) / 3 - F.(F-1) / 2 + F = 2 - 2g

Simplifying and solving for g gives
g = (F² - 7F + 12) / 12 (see Footnote 1)

g has to be an integer so we require (F² - 7F + 12) ≡ 0 mod 12.

This means that F mod 12 = 3, 4, 7 or 0.

A list of (F,g) pairs would then start with (3,0), (4,0), (7,1), (12,6) etc.

(3,0) is a somewhat strange case as it requires 2-sided polygons, or 'digons'. Three coincident digons is technically a face neighbourly polyhedron, albeit highly degenerate.
(4,0) is a tetrahedron
(7,1) is the Szilassi toroid above
(12,6) is an as yet undiscovered toroid, but 'Code Parade' got very close - see below.
... higher order cases exist (15,11), (16, 13) etc.

Stewart in 1980 [2] showed that it is also possible to colour a number of Stewart Toroids to simulate a Szilassi toroid in that all faces of the same colour are contiguous and each of the seven colours is contiguous with the other six. The image above is of a K₃ / Q₃T₃ coloured in such a fashion.

Császár Polyhedra

This triangulated polyhedron is the Császár polyhedron (discovered in 1949) [3]. It is the dual of the Szilassi toroid. It consists of 7 vertices, 21 edges and 14 triangular faces. Along with the tetrahedron these are the only known polyhedra to have no diagonals. Every pair of verices is connected by an edge.

In 1986, Bokowski and Eggert showed there are four variations of the Császár polyhedron [4]. With the original being known as Cs₀, the variations as known as Cs_1,Cs_{2 and}Cs_4.(The variant Cs₃ was shown to be self intersecting). Duals to these variants are self-intersecting so there is only one valid Szilassi toroid.

Beyond Szilassi

For many years the Szilassi toroid was the only known toroidal face neighbourly polyhedron. The next possibility, the hypothetical 12-sided genus 6 toroid, is known as the 'Dual K₁₂ toroid'. It's name derives from the equivalent of the Császár polyhedron, which is a K₁₂ triangulated manifold with 12 vertices, 66 edges and 44 triangular faces and with every single vertex connected to all 11 others. In 1996 Bokowski, Altshuler and Schuchert catalogued the 59 distinct topological maps capable of embedding the K₁₂ graph onto a genus-6 surface [5]. Then in 2000 Bokowski and Guedes de Oliveira proved that none of these can be realised in 3D space with flat surfaces [6]. The K₁₂ equivalent of the Császár polyhedron therefore does not exist.

This has not stopped people searching for the Dual K₁₂ toroid though. In 2024 youtube user 'Code Parade' (Kevin H.) tried to generate precisely this toroid using a brute force numerical optimisation approach [7]. While he was not able to generate an entirely intersection free version, he did manage to generate a highly optimal near-miss toroid with only four intersections. He named this the 'Razorcross'. His code is available at the HackerPoet NeighborlyPolyhedra GitHub Repository

In 2009 Grünbaum and Szilassi considered the possibility of neighbouring faces sharing more than one edge [8]. They termed these 'overarching faces'. They did not though describe any new face-neighbourly polyhedra.

The calculation above now needs to be revisited to reflect these additional edges.

Case 1.

If we now allow each face to share two edges with one other face, we require F = G. Note that to share two edges the gonality G ≥ 6 so F ≥ 6.

Our formula becomes
g = (F² - 6F + 12) / 12

This means that F mod 12 = 6 or 0.

Our (F,g) pairs now become (6,1), (12,7) etc.

(6,1) was shown to be impossible by Grünbaum & Szilassi in 2009 [8]

(12,7) has not been found, nor proved to be impossible.

Case 2.

If we now allow each face to share two edges with two other faces, We now require F = G - 1. The limit for gonality is now G ≥ 7 so F ≥ 6.

Our formula becomes
g = (F² - 5F + 12) / 12

This means that F mod 12 = 5, 8, 9 or 0.

Our (F,g) pairs now become (5,1), (8,3), (9,4), (12,8) etc.

(5,1) is impossible as F < 6

(8,3) toroids have been discovered in 2026 firstly by Mizhaev and then a second example by Röst and Vigh (see below)

(9,4) has not been found, nor proved to be impossible.

(12,8) toroid was found by Mizhaev in 2026 (see below)

The next solution is (17,18).

Face neighbourly toroids with F=8 and genus = 3.

The first torid in this category was described by Mizhaev in 2020 [9], and then redefined with integer coordinates in 2026 [10]. It has S4 symmetry and 24 vertices, 36 edges and 8 9-gonal faces. This is the first face neighbourly polyhedron to be discovered since Szilassi in 1977.

A second toroid was then discovered by Röst and Vigh in 2026 [11]. It necessarily has the same vertex, edge and face counts as the Mizhaev toroid, but has D2 symmetry.

Face neighbourly toroids with F=12 and genus = 8.

This was discovered by Mizhaev in 2026 [12]. It has S4 symmetry and 52 vertices, 78 edges and 12 13-gonal faces. It is noticeable in that it has colinear vertices in sets of 8 on each face. [Footnote 2]

Resources

A zip file containing OFF files for all polyhedra on this page is available here.

References

[1] L. Szilassi, Regular toroids, Structural Topology 13 (1986), 69–80. Originally published in Hungarian: L. Szilassi, Egy poliéder, amelynek bármely két lapja szomszédos [A polyhedron in which any two faces are adjacent], A Juhász Gyula Tanárképző Főiskola Tudományos Közleményei (Szeged) (1977), 131–139.
[2] B. M. Stewart, Adventures Among the Toroids, rev. 2nd ed., B. M. Stewart, Okemos, MI, 1980
[3] Á. Császár, A polyhedron without diagonals, Acta Sci. Math. (Szeged) 13 (1949), 140–142[1][2].
[4] J. Bokowski and A. Eggert, Toutes les réalisations du tore de Moebius avec sept sommets [All realizations of Möbius' torus with 7 vertices], Structural Topology 17 (1991), 59–78; preprint, TH Darmstadt, 1986.
[5] A. Altshuler, J. Bokowski, and P. Schuchert, Neighborly 2-manifolds with 12 vertices, J. Combin. Theory Ser. A 75 (1996), no. 1, 148–162
[6] J. Bokowski and A. Guedes de Oliveira, On the generation of oriented matroids, Discrete Comput. Geom. 24 (2000), no. 2–3, 197–208
[7] CodeParade [Kevin H.], Can I Solve This Unsolved Math Problem?, online video, YouTube, Dec. 23, 2024, youtu.be/y53UNskR-zU
[8] B. Grünbaum and L. Szilassi, Geometric realizations of special toroidal complexes, Contributions to Discrete Mathematics 4 (2009), no. 1, 21–39
[9] R. Mizhaev, Equivelar octahedron of genus 3 in 3-space, OSF Preprints, 2020, \doi{10.31219/osf.io/hvtey}
[10] R. Mizhaev, Integer Realization of an Equivelar Octahedron of Genus 3, preprint, arXiv:2609.17700 [math.CO], 2026
[11] G. Röst and V. Vígh, A second eight-faced polyhedron in which every two faces share an edge, preprint, arXiv:2609.32998 [math.MG], 2026.
[12] R. Mizhaev, Equivelar dodecahedron of genus 8 in 3-space, preprint, ResearchGate, Jan. 2026.

Footnotes

Fn 1 The formula g = (F² - 7F + 12) / 12 is commonly displayes factored as g = (F - 3)(F - 4) / 12. That has not been done here due to the doubled pairs of edges studied by Mizhaev.
Fn 2 Mizhaev's paper unfortunately does contain some typos in the vertex and face data tables. The OFF file linked here has had these errors corrected.

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