Toroidal polytope

A toroidal polytope or toroid is a polytope whose surface has positive genus such as a torus. They are distinct from the more-studied spherical polytopes, whose surfaces are topologically equivalent to spheres. 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 "excavating" one polyhedron from another (like the "diminishing" operation used in Johnson solids but without the restriction of convexity), Bonnie Stewart found that a properly aligned polyhedral "tunnel" could be excavated from a larger polyhedron that shared a face-to-face height with the tunnel, leaving a hole in the polyhedron and making it toroidal. He referred to a genus-altering excavation of a polyhedron as a "tunneling."

In an attempt to find an noteworthy subclass of toroidal polyhedra, Stewart considered polyhedra with regular faces, where no two faces that shared an edge were coplanar, and where no self-intersection occurred, loosely referring to these as Stewart toroids. Stewart eventually found that the number of tunnelings of convex regular-faced polyhedra (that is, of the uniform polyhedra and Johnson solids) was finite, provided that no unnecessary excavations and augmentations were made. He called this subclass the quasi-convex Stewart toroids because their convex hulls remained unaltered by the tunneling process.

By connecting polyhedra face-to-face via the blending operation (like the "augmenting" operation used in Johnson solids but without the restriction of convexity), a Stewart toroid may be created where a chain of connected polyhedra loops back on itself. The result will likely not be quasi-convex, but it may display a high degree of symmetry and can even have a regular or uniform convex hull.

Quadrilateral toroids
There exists an infinite family of toroidal polyhedra that are topologically equivalent to looped portions of the square tiling. They can be inscribed in a torus.

The vertex coordinates of an n-d toroid can be given by $$\left((\sin(\tfrac{j\pi}n)+r)\sin(\tfrac{k\pi}d),\,(\sin(\tfrac{j\pi}n)+r)\cos(\tfrac{k\pi}d),\,\cos(\tfrac{j\pi}n)\right)$$ where $$0 ≤ j < n-1$$ and $$0 ≤ k < d-1$$, and r is the major radius of the torus.

They can be uniform and noble (but also skew) when embedded in 4D space, as the honeycomb product of a n-gon and a d-gon, or equivalently as a subset of the faces of a n-d duoprism. If n=d, they are regular.

Uniform polytopes
Some uniform polytopes are nonconvex and have facets 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 octahemioctahedron 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 butterflies for faces and vertex figures and the vertex layouts of various 3D prisms and antiprisms. They are noble and self-dual. Like the self-intersecting uniform polytopes, their Euler characteristic makes them topologically equivalent to certain tori.