Stewart toroid

Stewart toroids are polyhedra of positive genus with regular faces, where no two faces that share an edge are coplanar, and where faces intersect only at edges. Many such polyhedra can be constructed by "excavating" one polyhedron from another (like the "diminishing" operation used in Johnson solids but without the restriction of convexity). Repeated excavations form "tunnels" passing through the faces of polyhedra and alter their genus.

Stewart's criteria
In several relevant properties are given single letter abbreviations and denoted in parentheses. Stewart toroids are polyhedra satisfying (R)(A)(T):


 * (R) Each face of $P$ is regular.
 * (A) Faces of $P$ which share an edge are not coplanar.
 * (T) See.

Other requirements dyadicity, connectivity, and non-self-intersection are implicit in the definition of polyhedra used.

(T)
Stewart defines the (T) requirement as follows:

A polyhedron is said to be tunnelled, or to have the property (T), if there exists a set of polyhedra $$A_1,A_2,\dots,A_t$$ such that $$P_{t+1} = \mathcal{P}$$ and every $$A_i$$ is either a tunnel or a rod.

Quasi-convex Stewart toroids
Consideration of Stewart toroids may further be restricted to just those which are quasi-convex:


 * (Q) Every edge in the convex hull of $P$ is in $P$

While the set of Stewart toroids, (R)(A)(T), is infinite, it is believed that the set of quasi-convex Stewart toroids, (R)(A)(Q)(T), is finite.

Many Quasi-convex Stewart toroids have regular faced convex hulls, however some do not. For example, the Webb toroid and the excavated expanded cuboctahedron are both quasi-convex Stewart toroids, but their convex hulls have irregular faces.

Knotted Stewart toroids
Genus 1 Stewart toroids can be created so they form knots under ambient isotopy. They are made by outer-blending many copies of a polyhedron together into a loop. A non-trivial knotted toroid cannot be quasi-convex. Because there are infinitely many knots, and many ways to represent each one as a Stewart toroid, knotted toroids do not receive much study.