Szilassi polyhedron

The Szilassi polyhedron ( Hungarian: [ˈsilɒʃːi], approximate English pronunciation SEE-la-shi) is a toroidal polyhedron with the property that each of its faces is adjacent to all of the other faces. It is the dual of the Császár polyhedron. As an abstract polytope, it is regular.

It has seven irregular hexagons as faces, and 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 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.

It can be seen as a 3D topological equivalent of the 7-2 gyrochoron, sharing its vertex counts and which also has cells adjacent to all other cells.