Quasi-convex polytope

This quasi-convex polyhedron based on the truncated cube retains the truncated cube as its convex hull, despite being modified in ways that make it nonconvex.

A polyhedron is quasi-convex if all of the edges of its convex hull are also edges of P . Quasi-convexity is itself a broad term, but it is most interesting together with the constraints of regular faces, non-self-intersection, and no coplanar adjacent faces. It is suspected that the set of quasi-convex polyhedra under these conditions is finite (apart from polyhedra whose convex hulls are prisms or antiprisms).

Although quasi-convex polyhedra are not necessarily toroidal, quasi-convexity is primarily examined with respect to Stewart toroids. The concept of quasi-convexity was originally suggested by Norman Johnson as an attempt to restrict regular faced toroidal polyhedra to a finite set; without quasi-convexity, blending a Stewart toroid with any appropriate acrohedron produces another Stewart toroid, causing an explosion in possibilities that renders the set much less interesting. The property first appears as the property "(Q)" in Adventures Among the Toroids, written by Bonnie Stewart.

Related notions

Weak quasi-convexity

A polyhedron, P , is weakly quasi-convex if all of the edges of its convex hull are entirely coincident on the 1-skeleton of P . This allows for edges in the convex edges which are coextant with multiple edges in P . Every quasi-convex polyhedron is weakly quasi-convex, but not every weakly quasi-convex polyhedron is quasi-convex. This property is called (Q') in Adventures Amoung the Toroids.

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  A polyhedron made of eight cubes that is (R) and (Q'), but not (Q) or (A). Some of the edges of its convex hull are made of 3 of its original edges.

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  A polyhedron that is (R)(A) and (Q') but not (Q). Some of the edges of its convex hull are made of 3 of its original edges.

Regular-faced quasi-convexity

The Webb toroid is (R)(A)(Q)(T), but not (Q"). Its convex hull, a truncated chamfered dodecahedron, is a near-miss Johnson solid.

A polyhedron, P , is regular-faced quasi-convex if it is quasi-convex and every face in the convex hull is regular. This property is called (Q") in Adventures Amoung the Toroids. Stewart conjectured that the number of polyhedra which were (R)(A)(Q)(T), but not (Q") is small and restricted his search only to polyhedra with the (Q") property. Others since have investigated polyhedra which are (R)(A)(Q)(T) but not (Q").

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  A polyhedron that is (R)(A) and has a regular-faced convex hull, but is not (Q).

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  A polyhedron from Adventures that is (R)(A) and (Q") (as well as (Q')), but not (Q), plus a view of its tunnel.

External links

• Doskey, Alex. "B.M. Stewart's - Adventures Among the Toroids - Index".

Bibliography

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