Toroidal polytope

A toroidal polytope or toroid is a polytope whose surface is topologically equivalent to a torus. All toroidal polytopes are nonconvex.

Like all polytopes, toroids are bounded by distinct vertices, edges, and other higher-dimensional elements. This, however, differentiates them from the torus that their name comes from, since the torus is a topological object that can be infinitely deformed while the toroids strictly adhere to the layout determined by their elements.

Non-self-intersecting polyhedra
Toroidal polyhedra have been explored far more thoroughly than toroids in other dimensions. In 3 dimensions, an integer quantity called genus, written as g, can provide a more intuitive description of a toroid than the Euler characteristic does. The genus can be determined by the relation $$\chi=2-2g$$ and it directly corresponds to the number of “holes” in the toroid.

Császár and Szilassi polyhedra
The Szilassi polyhedron consists of 7 irregular hexagonal faces, each of which shares an edge with all of the other faces. It is one of only two known polyhedra where any two given faces share an edge, the other one being the tetrahedron, and has the fewest faces of any toroid in 3D. Since it has 14 vertices and 21 edges, its genus is 1, which corresponds with it only having one "hole."

The dual of the Szilassi polyhedron is the Császár polyhedron. Being the Szilassi's dual, it has the fewest vertices of any toroid in 3D.

Stewart toroids
By "subtracting" one polyhedron from another, akin to the "diminishing" operation found in the Johnson solids but without the restriction of convexity, Bonnie Stewart found that a polyhedral "tunnel" could be cut out of a larger polyhedron that shares a face-to-face height with the tunnel. Since the tunnel goes through the larger polyhedron and comes out at more than one face, the process yields a toroidal polyhedron.

Searching for a finite subclass of these toroids, Stewart restricted himself to polyhedra with regular faces, where no two faces that shared an edge were coplanar, and where no self-intersection occurred, calling these the Stewart toroids. Eventually, further restricting the search by only considering the simplest possible toroids (that is, without any modifications that didn't contribute to completing the tunnels) and mandating that the edges of convex hulls be contained within the set of edges of the entire toroid, Stewart proved that there were finitely many such polyhedra. He called this subclass the quasi-convex Stewart toroids.

By "adding" polyhedra together, akin to the "augmenting" operation found in Johnson solids but without the restriction of convexity, Stewart toroids can be made when a series of augmented polyhedra loops back on itself. The result will likely not be quasi-convex, but it may display the symmetries of a regular polyhedron or even have a regular or uniform convex hull.

Uniform polytopes
Some uniform polytopes are nonconvex and have faces that intersect one another, causing their Euler characteristic to deviate from the usually-observed value in that dimension. While these polytopes may not appear to be toroidal, on an abstract level each one is topologically equivalent to a certain torus. For instance, the great dodecahedron has Euler characteristic 0, making it equivalent to a genus-1 torus despite the fact that no hole is clearly visible in it.

Crown polyhedra
The crown polyhedra, also known as stephanoids, are polyhedra with self-crossing quadrilaterals for faces and the vertex layouts of various 3D prisms and antiprisms. They are self-dual and therefore noble. Like the self-intersecting uniform polytopes, their Euler characteristic makes them topologically equivalent to certain tori.