Szilassi polyhedron
Szilassi polyhedron  

Rank  3 
Type  Regular toroid 
Elements  
Faces  2+2+2 irregular hexagons, 1 bilaterallysymmetric hexagon 
Edges  1+1+1+2+2+2+2+2+2+2+2+2 
Vertices  2+2+2+2+2+2+2 
Vertex figure  2+2+2+2+2+2+2 scalene triangles 
Related polytopes  
Dual  Császár polyhedron 
Abstract & topological properties  
Flag count  84 
Euler characteristic  0 
Orientable  Yes 
Genus  1 
Skeleton  Heawood graph 
Properties  
Symmetry  K_{2}+×I, order 2 
Flag orbits  42 
Convex  No 
History  
Discovered by  Lajos Szilassi 
First discovered  1977 
The Szilassi polyhedron (Hungarian: [ˈsilɒʃːi], approximate English pronunciation SEElashi) 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.
The Szilassi polyhedron is a realization of the Heawood map.
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.
Vertex coordinates[edit  edit source]
This polytope is missing vertex coordinates.April 2024) ( 
Gallery[edit  edit source]

The fundamental domain of the Szilassi polyhedron. Edges and vertices with the same label are identified.
External links[edit  edit source]
 Wikipedia contributors. "Szilassi polyhedron".
 Weisstein, Eric W. "Szilassi Polyhedron" at MathWorld.
 McCooey, David. "Szilassi Polyhedron"
 McNeil, Jim. "The Szilassi and Czászár Polyhedra"
 Szilassi, Lajos. "On some regular toroids"
 Ace, Tom. "the Szilassi polyhedron"