Webb toroid

The Webb toroid is a quasi-convex Stewart toroid. It can be obtained by outer-blending twelve tunnelled pentagonal rotundae, sixty pentagonal antiprisms, sixty pentagonal prisms, and twenty tridiminished icosahedra together. Although it has pentagons and decagons as pseudo-faces, its faces are all triangles or squares.

Its convex hull is an equilateral truncated chamfered dodecahedron, with 12 decagons, 30 equilateral dodecagons, and 80 triangles (divided into two sets of 20 and 60).

The toroid was first described by Robert Webb, creator of the proprietary polytope software Stella.

Related polyhedra
A similar figure to the Webb toroid can be made with ordinary pentagonal rotundae instead of tunnelled ones. It would have genus 29. Similar toroids can be made with a mixture of ordinary and tunnelled pentagonal rotundae, to achieve quasi-convex Stewart toroids in the genera 30-40, although these have lower symmetry than either the Webb toroid or the version with 12 ordinary pentagonal rotundae.

Its pentagonal prisms can also be removed (while bringing the other components together), and the toroid would maintain its quasi-convexity and genus. The convex hull would remain equilateral, but would no longer be a near-miss Johnson solid. Instead, it would be the Minkowski sum of a rectified chamfered dodecahedron and a regular dodecahedron. It would maintain the 12 decagons and 80 triangles, while the dodecagons would "contract" into rectangular-symmetric octagons (which would appear somewhat squashed). If the perpendicular edges of these octagons were further "contracted" into points, it would produce a rhombus with one diagonal double the length of the other.

The Webb toroid is also used in the construction of a quasi-convex Stewart toroid of even greater genus than the holey monster. The new toroid is an outer-blend of a Webb toroid, a holey monster, and a triangular cupola that connects the two larger toroids. (The holey monster most readily admits outer blends on its hexagonal faces, and the Webb toroid has no hexagons but plenty of triangles and squares.) The empty space in the middle of the Webb toroid is large enough to contain the holey monster, allowing the blend to share the convex hull of the Webb toroid and thus be quasi-convex.