Toroidal blend of 20 triangular hebesphenorotundae
|Toroidal blend of 20 triangular hebesphenorotundae|
|Faces||20+60+60+120 triangles, 60 pentagons, 20 hexagons|
|Vertex figures||60 rectangles, edge lengths 1 and (1+√)/2|
|60 isosceles trapezoids, edge lengths 1, 1, 1, (1+√)/2|
|Measures (edge length 1)|
|Dihedral angles||3–3 at J92 join off of digonal-symmetry axis:|
|3–3 at J92 join on digonal-symmetry axis:|
|3–3 near J92 "tips":|
|3–5 near J92 "tips":|
|Convex hull||Semi-uniform Tid, edge lengths 1 (triangles) and (1+√)/2 (between dipentagons)|
|Abstract & topological properties|
|Symmetry||H3, order 120|
The toroidal blend of 20 triangular hebesphenorotundae is a Stewart toroid. It can be obtained by outer-blending twenty triangular hebesphenorotundae together at their square faces, leaving no squares behind. It has these squares, as well as pentagons and star pentambi, as pseudo-faces.
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:
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 outer-blended 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 outer-blends in outward directions that contribute to almost-complete space-fillings, like a less-strict version of an aperiodic tiling. The gaps in these partial space-fillings can take the form of polyhedra with irregular but equilateral faces such as rhombi.
Plus 60 more bilunabirotundae and 60 more tridiminished icosahedra. This takes care of the non-dyadicity. To fill in the gaps at the trigonal- and digonal-symmetry 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.
[edit | edit source]
- Klitzing, Richard. "oxFx3xfox5fovo&#zxt".