# Toroidal great icosahedron faceting

Toroidal great icosahedron faceting
Rank3
TypeToroid, orbiform
Notation
Bowers style acronymTogif
Elements
Faces6 triangles, 6 pentagrams
Edges6+6+12
Vertices6+6
Vertex figures6 nonconvex pentagons, edge lengths 1, (5–1)/2, (5–1)/2, 1, (5–1)/2
6 isosceles triangles, edge lengths 1, (5–1)/2, (5–1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{8}}}\approx 0.58779}$
Volume${\displaystyle {\frac {{\sqrt {5}}-2}{2}}\approx 0.11803}$
Dihedral angles5/2–5/2: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
3-5/2: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
Central density0
Related polytopes
ArmyIke
ConjugateToroidal icosahedron faceting
Convex hullIcosahedron
Abstract & topological properties
Flag count96
Euler characteristic0
OrientableYes
Genus1
Properties
Symmetry(G2×A1)/2, order 12
ConvexNo

The toroidal great icosahedron faceting or togif, is an orbiform toroid and an edge-faceting of the small stellated dodecahedron. Its faces are 6 pentagrams and 6 triangles. It has the same face counts as the trigonic small stellated dodecahedron faceting but uses the other set of 6 pentagrams compared to that polyhedron.

## Vertex coordinates

Its vertex coordinates are the same as those of the small stellated dodecahedron.