Truncated great icosahedron

Truncated great icosahedron
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymTiggy
Coxeter diagramo5/2x3x ()
Elements
Faces12 pentagrams, 20 hexagons
Edges30+60
Vertices60
Vertex figureIsosceles triangle, edge lengths (5–1)/2, 3, 3
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{29-9\sqrt5}{8}} ≈ 1.05329}$
Volume${\displaystyle \frac{125-43\sqrt5}{4} ≈ 7.21227}$
Dihedral angles6–5/2: ${\displaystyle \arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232^\circ}$
6–6: ${\displaystyle \arccos\left(\frac{\sqrt5}{3}\right) ≈ 41.81031^\circ}$
Central density7
Number of external pieces192
Level of complexity13
Related polytopes
ArmySemi-uniform Srid, edge lengths 1 (pentagons) and ${\displaystyle \frac{3-\sqrt5}{2}}$ (triangles)
RegimentTiggy
DualGreat stellapentakis dodecahedron
ConjugateTruncated icosahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count360
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The truncated great icosahedron, or tiggy, also called the great truncated icosahedron, is a uniform polyhedron. It consists of 12 pentagrams and 20 hexagons. Each vertex joins one pentagram and two hexagons. As the name suggests, it can be obtained by the truncation of the great icosahedron.

Vertex coordinates

A truncated great icosahedron of edge length 1 has vertex coordinates given by all even permutations and all changes of sign of:

• ${\displaystyle \left(0,\,±\frac12,\,±3\frac{\sqrt5-1}{4}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{5-\sqrt5}{4},\,±\frac{\sqrt5-1}{2}\right),}$
• ${\displaystyle \left(±\frac{\sqrt5-1}{4},\,±1,\,±\frac{\sqrt5-2}{2}\right).}$

Related polyhedra

o3o5/2o truncations
Name OBSA Schläfli symbol CD diagram Picture
Great icosahedron gike {3,5/2} x3o5/2o ()
Truncated great icosahedron tiggy t{3,5/2} x3x5/2o ()
Great icosidodecahedron gid r{3,5/2} o3x5/2o ()
Truncated great stellated dodecahedron (degenerate, ike+2gad) t{5/2,3} o3x5/2x ()
Great stellated dodecahedron gissid {5/2,3} o3o5/2x ()
Small complex rhombicosidodecahedron (degenerate, sidtid+rhom) sicdatrid rr{3,5/2} x3o5/2x ()
Truncated great icosidodecahedron (degenerate, ri+12(10/2)) tr{3,5/2} x3x5/2x ()
Great snub icosidodecahedron gosid sr{3,5/2} s3s5/2s ()