Toroidal polyhedron

Toroidal polyhedra are defined as collections of polygons that meet at their edges and vertices, forming a manifold as they do.

[2] In this area, it is important to distinguish embedded toroidal polyhedra, whose faces are flat polygons in three-dimensional Euclidean space that do not cross themselves or each other, from abstract polyhedra, topological surfaces without any specified geometric realization.

In all of these cases the toroidal nature of a polyhedron can be verified by its orientability and by its Euler characteristic being non-positive.

[6] It and the tetrahedron are the only known polyhedra in which every possible line segment connecting two vertices forms an edge of the polyhedron.

[7] Its dual, the Szilassi polyhedron, has seven hexagonal faces that are all adjacent to each other,[8] hence providing the existence half of the theorem that the maximum number of colors needed for a map on a (genus one) torus is seven.

[14] A polyhedron that is formed by a system of crossing polygons corresponds to an abstract topological manifold formed by its polygons and their system of shared edges and vertices, and the genus of the polyhedron may be determined from this abstract manifold.

A polyhedral torus can be constructed to approximate a torus surface, from a net of quadrilateral faces, like this 6x4 example.
Interactive Csaszar polyhedron model – in the SVG image, move the mouse left and right to rotate it. [ 4 ]
Interactive Szilassi polyhedron model – in the SVG image, move the mouse to rotate it. [ 5 ]
Conway's toroidal deltahedron
Conway's toroidal deltahedron
Pentagonal stephanoid. This stephanoid has pentagonal dihedral symmetry and has the same vertices as the uniform pentagonal prism .