Toroidal blend of 20 triangular hebesphenorotundae

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Toroidal blend of 20 triangular hebesphenorotundae
TypeStewart toroid
Coxeter diagramoxFx3xfox5fovo&#zxt
Faces20+60+60+120 triangles, 60 pentagons, 20 hexagons
Vertex figures60 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 (34.62)
Measures (edge length 1)
Dihedral angles3–3 at J92 join off of digonal-symmetry axis:
 3–3 at J92 join on digonal-symmetry axis:
 3–5 rotundaic:
 3–3 near J92 "tips":
 3–5 near J92 "tips":
Central density0
Related polytopes
Convex hullSemi-uniform Tid, edge lengths 1 (triangles) and (1+5)/2 (between dipentagons)
Abstract & topological properties
Flag count2400
Euler characteristic–20
SymmetryH3, 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.

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.

External links[edit | edit source]