# Toroidal icosahedron faceting

Toroidal icosahedron faceting
Rank3
TypeToroid, orbiform
Notation
Bowers style acronymToif
Elements
Faces6 triangles, 6 pentagons
Edges6+6+12
Vertices6+6
Vertex figures6 nonconvex pentagons, edge lengths 1, (1+5)/2, (1+5)/2, 1, (1+5)/2
6 isosceles triangles, edge lengths 1, (1+5)/2, (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{8}}}\approx 0.95106}$
Volume${\displaystyle {\frac {2+{\sqrt {5}}}{2}}\approx 2.11803}$
Dihedral angles3-5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
5-5: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
Central density0
Related polytopes
ArmyIke
ConjugateToroidal great icosahedron faceting
Convex hullIcosahedron
Convex coreGyroelongated triangular bipyramid
Abstract & topological properties
Flag count96
Euler characteristic0
OrientableYes
Genus1
Properties
Symmetry(G2×A1)/2, order 12
Flag orbits8
ConvexNo

The toroidal icosahedron faceting or toif, also called the crazy tube, is an orbiform toroid. As its name suggests, it is a faceting of the icosahedron. Its faces are 6 pentagons and 6 triangles.

It has the fewest faces and vertices of any known regular faced toroid, with 12 of each. However is not a Stewart toroid as it self-intersects. The record for the fewest faces of a Stewart toroid is 21, with the tunnelled elongated triangular cupola and the record for the fewest vertices is 15, with the tortuous tunnel.

## Vertex coordinates

Its vertex coordinates are the same as those of the icosahedron.

## Related polytopes

Both the toroidal icosahedron faceting and the trigonic great dodecahedron faceting have 6 pentagonal and 6 triangular faces and they share a convex hull (the icosahedron). The two can be superimposed so that their triangular faces are exactly incident in which case their pentagonal faces are complementary, forming the full set of twelve pentagons of the great dodecahedron.

The toroidal icosahedron faceting appears as the cells of the small toroidal trigonal swirlprism, a noble scaliform polychoron.