Truncated great icosahedron

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.

Truncated great icosahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Tiggy
Coxeter diagram	o5/2x3x ()
Elements
Faces	12 pentagrams, 20 hexagons
Edges	30+60
Vertices	60
Vertex figure	Isosceles triangle, edge lengths (√5–1)/2, √3, √3
Measures (edge length 1)
Circumradius	$\sqrt{\frac{29-9\sqrt5}{8}} ≈ 1.05329$
Volume	$\frac{125-43\sqrt5}{4} ≈ 7.21227$
Dihedral angles	6–5/2: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81232^\circ$
	6–6: $\arccos\left(\frac{\sqrt5}{3}\right) ≈ 41.81031^\circ$
Central density	7
Number of external pieces	192
Level of complexity	13
Related polytopes
Army	Semi-uniform Srid, edge lengths 1 (pentagons) and $\frac{3-\sqrt5}{2}$ (triangles)
Regiment	Tiggy
Dual	Great stellapentakis dodecahedron
Conjugate	Truncated icosahedron
Convex core	Icosahedron
Abstract & topological properties
Flag count	360
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

Vertex coordinates

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

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

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 ()

External links

• Bowers, Jonathan. "Polyhedron Category 2: Truncates" (#16).

• Klitzing, Richard. "tiggy".

• Wikipedia Contributors. "Truncated great icosahedron".
• McCooey, David. "Truncated Great Icosahedron"