# Webb toroid

Webb toroid
Rank3
TypeQuasi-convex Stewart toroid
Elements
Faces
Edges9×60​+14×120
Vertices7×60​+3×120
Measures (edge length 1)
Volume${\displaystyle {\frac {175}{2}}+{\frac {185{\sqrt {5}}}{6}}+15{\sqrt {25+10{\sqrt {5}}}}\approx 259.67407}$
Surface area${\displaystyle 360+250{\sqrt {3}}\approx 793.01270}$
Dihedral angles3-3 (outside of pero to pap): ${\displaystyle 180^{\circ }+\arctan \left(2\right)\approx 243.43495^{\circ }}$
3-3 (tunnelled pap): ${\displaystyle 360^{\circ }-\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 221.81031^{\circ }}$
3-4 (in pero): ${\displaystyle 360^{\circ }-\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 200.90516^{\circ }}$
3-4 (pip to diminished ikes): ${\displaystyle 90^{\circ }+\arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 190.81232^{\circ }}$
3-3 (in diminished ikes): ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
3-4 (pap to pero through 10): ${\displaystyle 180^{\circ }-\arccos \left({\sqrt {\frac {7+3{\sqrt {5}}}{30}}}\right)\approx 132.52979^{\circ }}$
3-4 (pap to pecu in tunnel): ${\displaystyle 180^{\circ }-\arcsin \left({\sqrt {\frac {3+{\sqrt {5}}}{6}}}\right)\approx 110.90516^{\circ }}$
4-4 (in pip): 108°
3-3 (pero to pecu through 10): ${\displaystyle \arcsin \left({\frac {2}{3}}\right)\approx 41.81031^{\circ }}$
3-3 (pero to tunnelled pap): ${\displaystyle \arcsin \left({\frac {2}{3}}\right)\approx 41.81031^{\circ }}$
Related polytopes
Convex hullTruncated chamfered dodecahedron
Abstract & topological properties
Flag count8880
Euler characteristic–80
OrientableYes
Genus41
Properties
SymmetryH3, order 120
Flag orbits74
ConvexNo
NatureTame
History
Discovered byRobert Webb
First discovered2019

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.

Unlike the majority of Stewart toroids studied, its convex hull has irregular faces. The convex hull can be described as "truncated chamfered dodecahedron", having as faces 12 regular decagons, 80 triangles, and 30 dodecagons that are irregular but equilateral. This is the largest hull of any known quasi-convex Stewart toroid.

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. In the resulting figure, the holey monster is off-center, so there is no obvious way to add more supports between the figures to increase the genus further.