Toroidal blend of 20 triangular hebesphenorotundae
Toroidal blend of 20 triangular hebesphenorotundae  

Rank  3 
Type  Stewart toroid 
Notation  
Coxeter diagram  oxFx3xfox5fovo&#zxt 
Elements  
Faces  20+60+60+120 triangles, 60 pentagons, 20 hexagons 
Edges  30+30+60+60+60+120+120+120 
Vertices  60+60+60+60 
Vertex figures  60 rectangles, edge lengths 1 and (1+√5)/2 
60 isosceles trapezoids, edge lengths 1, 1, 1, (1+√5)/2  
60 (3.3.5)^{2}  
60 (3^{4}.6^{2})  
Measures (edge length 1)  
Volume  
Dihedral angles  3–3 at J_{92} join off of digonalsymmetry axis: 
3–3 at J_{92} join on digonalsymmetry axis:  
6–6:  
3–5 rotundaic:  
3–3 near J_{92} "tips":  
3–6:  
3–5 near J_{92} "tips":  
Central density  0 
Related polytopes  
Convex hull  Semiuniform Tid, edge lengths 1 (triangles) and (1+√5)/2 (between dipentagons) 
Abstract & topological properties  
Flag count  2400 
Euler characteristic  –20 
Orientable  Yes 
Genus  11 
Properties  
Symmetry  H_{3}, order 120 
Convex  No 
Nature  Wild 
The toroidal blend of 20 triangular hebesphenorotundae is a Stewart toroid. It can be obtained by outerblending twenty triangular hebesphenorotundae together at their square faces, leaving no squares behind. It has these squares, as well as pentagons and star pentambi, as pseudofaces.
Vertex coordinates[edit  edit source]
The vertices of a toroidal blend of 20 triangular hebesphenorotundae, centered at the origin and with unit edge length, are given by all even permutations of:
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Relations[edit  edit source]
Twice the 4–6 dihedral angle of a triangular hebesphenorotunda, plus the 6–6 dihedral angle of a truncated icosahedron, is 360°. This can be taken as an explanation of why the toroid forms: it is possible to blend the toroid with a truncated icosahedron placed in the center. Doing this will remove all tunnels (making the genus 0) and hexagonal faces.
The triangular hebesphenorotunda's relation to the small rhombicosidodecahedron can also provide some insight on the formation of the toroid. The two polyhedra have certain arrangements of faces in common.

The three "lunes" (arrangements of a square with triangles joining opposite edges of it) of a triangular hebesphenorotunda, and the congruent lunes in the small rhombicosidodecahedron.

Twenty small rhombicosidodecahedra can be outerblended together in the same way as this toroid's parts.
If one inserts 30 cubes in between the triangular hebesphenorotundae, the resulting toroid will admit many outerblends in outward directions that contribute to almostcomplete spacefillings, like a lessstrict version of an aperiodic tiling. The gaps in these partial spacefillings can take the form of polyhedra with irregular but equilateral faces such as rhombi.

Plus 30 bilunabirotundae. This is related to a potential blend of the original nonenlarged toroid and 30 icosahedra. Also, note how the enlarged toroid can be blended with a great rhombicosidodecahedron placed in the center.

Plus 60 dodecahedra and 20 tridiminished icosahedra. Some coincident edges of the dodecahedra violate the diamond property here by being part of 4 faces.

Plus 60 more bilunabirotundae and 60 more tridiminished icosahedra. This takes care of the nondyadicity. To fill in the gaps at the trigonal and digonalsymmetry axes, one could use a polyhedron containing 36°rhombus faces, and a polyhedron containing faces that are the inner blend of a pentagon and a golden rhombus, respectively.

The enlarged toroid will also admit 12 pentagonal rotundae. This specific blend puts 4 faces at some edges unless the aforementioned great rhombicosidodecahedron is also blended in.
External links[edit  edit source]
 Klitzing, Richard. "oxFx3xfox5fovo&#zxt".