# Császár polyhedron

Császár polyhedron
An animation of the Császár polyhedron being rotated and unfolded
Type Toroidal polyhedron
Faces 14 triangles
Edges 21
Vertices 7
χ 0 (Genus 1)
Vertex configuration 3.3.3.3.3.3
Symmetry group C1, [ ]+, (11)
Dual polyhedron Szilassi polyhedron
Properties Nonconvex

In geometry, the Császár polyhedron (Hungarian pronunciation: [ˈtʃaːsaːr]) is a nonconvex polyhedron, topologically a toroid, with 14 triangular faces.

This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graph $K_7$ onto a topological torus. Of the 35 possible triangles from vertices of the polyhedron, only 14 are faces. If the seven vertices are numbered 1 through 7, the torus can be cut open to form a sheet topologically equivalent to this:

    5———4———7———2
/ \ / \ / \ / \
6———1———3———5———4
/ \ / \ / \ /
4———7———2———6
\ /
4


This pattern can be used to tile the plane. In the animated figure above right, the faces are the following (vertex 1 being at the top):

• Light blue:

(1, 2, 3) (1, 3, 4) (1, 4, 5) (1, 5, 6) (1, 6, 7) (1, 7, 2)

• Red

(2, 3, 6) (6, 3, 5)

• Yellow

(3, 5, 7) (7, 5, 2)

• Green

(6, 2, 4) (4, 2, 5)

• Dark blue

(4, 6, 7) (4, 7, 3)

In this numbering, the layout of the vertices at the end of the video clip, going clockwise from vertex 1, is 1, 2, 5, 4, 3, 7, 6, 5, 2, 7, 3, 4, 5, 6, 7.

There is some freedom in the positioning of the vertices, but some arrangements will lead to faces intersecting one another and no hole being formed.

All vertices are topologically equivalent, as can be seen from the tessellation of the plane that uses the above diagram.

The tetrahedron and the Császár polyhedron are the only two known polyhedra (having a manifold boundary) without any diagonals, although there are other known polyhedra such as the Schönhardt polyhedron for which there are no interior diagonals (that is, all diagonals are outside the polyhedron) as well as non-manifold surfaces with no diagonals (Szabó 1984, 2009). If a polyhedron with v vertices is embedded onto a surface with h holes, in such a way that every pair of vertices is connected by an edge, it follows by some manipulation of the Euler characteristic that

$h = \frac{(v-3)(v-4)}{12}.$

This equation is satisfied for the tetrahedron with h = 0 and v = 4, and for the Császár polyhedron with h = 1 and v = 7. The next possible solution, h = 6 and v = 12, would correspond to a polyhedron with 44 faces and 66 edges, but it is not realizable as a polyhedron; it is not known whether such a polyhedron exists with a higher genus (Ziegler 2008). More generally, this equation can be satisfied only when v is congruent to 0, 3, 4, or 7 modulo 12 (Lutz 2001).

The Császár polyhedron is named after Hungarian topologist Ákos Császár, who discovered it in 1949. The dual to the Császár polyhedron, the Szilassi polyhedron, was discovered later, in 1977, by Lajos Szilassi; it has 14 vertices, 21 edges, and seven hexagonal faces, each sharing an edge with every other face. Like the Császár polyhedron, the Szilassi polyhedron has the topology of a torus.

## References

• Császár, A. (1949), "A polyhedron without diagonals", Acta Sci. Math. Szeged 13: 140–142.
• Gardner, Martin (1988), Time Travel and Other Mathematical Bewilderments, W. H. Freeman and Company, pp. 139–152, ISBN 0-7167-1924-X
• Gardner, Martin (1992), Fractal Music, Hypercards and More: Mathematical Recreations from Scientific American, W. H. Freeman and Company, pp. 118–120, ISBN 0-7167-2188-0
• Lutz, Frank H. (2001), "Császár's Torus", Electronic Geometry Models: 2001.02.069.
• Szabó, Sándor (1984), "Polyhedra without diagonals", Periodica Mathematica Hungarica 15 (1): 41–49, doi:10.1007/BF02109370.
• Szabó, Sándor (2009), "Polyhedra without diagonals II", Periodica Mathematica Hungarica 58 (2): 181–187, doi:10.1007/s10998-009-10181-x.
• Ziegler, Günter M. (2008), "Polyhedral Surfaces of High Genus", in Bobenko, A. I.; Schröder, P.; Sullivan, J. M.; Ziegler, G. M., Discrete Differential Geometry, Oberwolfach Seminars 38, Springer-Verlag, pp. 191–213, doi:10.1007/978-3-7643-8621-4_10, ISBN 978-3-7643-8620-7, math.MG/0412093.