Truncated great icosahedron

Truncated great icosahedron
Great truncated icosahedron.png
Type Uniform star polyhedron
Elements F = 32, E = 90
V = 60 (χ = 2)
Faces by sides 12{5/2}+20{6}
Wythoff symbol 2 5/2 | 3
2 5/3 | 3
Symmetry group Ih, [5,3], *532
Index references U55, C71, W95
Dual polyhedron Great stellapentakis dodecahedron
Vertex figure Great truncated icosahedron vertfig.png
Bowers acronym Tiggy

In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices.[1] It is given a Schläfli symbol t{3,​52} or t0,1{3,​52} as a truncated great icosahedron.

3D model of a truncated great icosahedron

Cartesian coordinatesEdit

Cartesian coordinates for the vertices of a truncated great icosahedron centered at the origin are all the even permutations of

(±1, 0, ±3/τ)
(±2, ±1/τ, ±1/τ3)
(±(1+1/τ2), ±1, ±2/τ)

where τ = (1+√5)/2 is the golden ratio (sometimes written φ). Using 1/τ2 = 1 − 1/τ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 10−9/τ. The edges have length 2.

Related polyhedraEdit

This polyhedron is the truncation of the great icosahedron:

The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.

Name Great
Truncated great stellated dodecahedron Great

Great stellapentakis dodecahedronEdit

Great stellapentakis dodecahedron
Type Star polyhedron
Elements F = 60, E = 90
V = 32 (χ = 2)
Symmetry group Ih, [5,3], *532
Index references DU55
dual polyhedron Truncated great icosahedron
3D model of a great stellapentakis dodecahedron

The great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.

See alsoEdit


  1. ^ Maeder, Roman. "55: great truncated icosahedron". MathConsult.

External linksEdit

Animated truncation sequence from {​52, 3} to {3, ​52}