Stewart toroid

An example 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 Stewart (1964) 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) P has positive genus.

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

Quasi-convex Stewart toroids

See also: Quasi-convex

P* is a quasi-convex Stewart toroid.

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.

See also

• Johnson solid
• Toroidal polytope

External links

• Wikipedia Contributors. "Stewart toroid".

Bibliography

• Stewart, Bonnie (1964). Adventures Amoung the Toroids (2 ed.). ISBN 0686-119 36-3.