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Ungovernability of Sustainable Global Democracy ?

Towards engaging appropriately with time


Toroidal polyhedra  A polyhedral torus can be constructed to approximate a torus surface, from a net of quadrilateral faces,

(Toroidal Solids with Regular Polygon Faces)

Bonnie M. Stewart. Adventures Among the Toroids. 1970 Adventures Among the Toroids, a Study of Quasi-Convex, Aplanar, Tunneled Orientable Polyhedra of Positive Genus Having Regular Faces with Disjoint Interiors, 2nd rev. ed. Okemos, MI: B. M. Stewart, 1984.

Once a toroidal polyhedron has been found, we can derive an infinite number of others by tacking polyhedra onto free faces. Most of these are trivial.

To eliminate trivial cases, Stewart concentrated on polyhedra that were aplanar, that is, adjacent pairs of faces were not coplanar. He also concentrated on polyhedra that were quasi-convex, that is, were derived by tunneling into a convex polyhedron.

Since all the Stewart toroids have regular faces, they have a close connection with the Johnson Solids, the convex solids with regular polygon faces. Stewart's notation was a modification of Johnson's, of the form xxx/yyy, where xxx is the notation for the enclosing polyhedron and yyy is the notation for the portion excavated.

Steven Dutch. Stewart Toroids Based on Truncated Cube, 1999

Excavation was the term used for subtracting one smaller polyhedron from a larger one, to create a hole. The term Drilling was used when an excavation cut right through the model. See my paper Stella: Polyhedron Navigator for more details.

Various Stewart toroids are built into Small Stella and Great Stella. The latter also supports augmentation/excavation/drilling so that you can create your own new toroids.