# Szilassi polyhedron

Szilassi polyhedron Rank3
TypeRegular toroid
SpaceSpherical
Elements
Faces2+2+2 irregular hexagons, 1 bilaterally-symmetric hexagon
Edges1+1+1+2+2+2+2+2+2+2+2+2
Vertices2+2+2+2+2+2+2
Vertex figure2+2+2+2+2+2+2 scalene triangles
Related polytopes
DualCsászár polyhedron
Abstract & topological properties
Flag count84
Euler characteristic0
OrientableYes
Genus1
SkeletonHeawood graph
Properties
SymmetryK2+×I, order 2
ConvexNo
History
Discovered byLajos Szilassi
First discovered1977

The Szilassi polyhedron (Hungarian: [ˈsilɒʃːi], approximate English pronunciation SEE-la-shi) is a toroidal polyhedron. With seven irregular hexagonal faces, it has the least number of faces out of any toroid. It also has the unusual property that each of its faces is adjacent to all of the other faces. It is the dual of the Császár polyhedron.

It has 14 vertices and 21 edges. Three faces meet at each vertex. Among the faces, there are three pairs of identical faces and one unique face. The unique face is the only convex face among the seven.

It is an example of a regular toroid. It is not a regular polyhedron in the traditional sense, but as an abstract polyhedron it is weakly regular, meaning that it is abstractly vertex-transitive, edge-transitive, and face-transitive.

The only other known polyhedron with all of its faces adjacent to all of its other faces is the tetrahedron. It is theoretically possible to embed in 3D space a polyhedron with 12 faces, 44 vertices, and 66 edges with this property, but it has not been accomplished. It would have irregular hendecagonal faces and genus 6.