Truncated great icosahedron

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Truncated great icosahedron
Great truncated icosahedron.png
Type Uniform star polyhedron
Elements F = 32, E = 90
V = 60 (χ = 2)
Faces by sides12{5/2}+20{6}
Wythoff symbol 2 5/2| 3
2 5/3| 3
Symmetry group Ih, [5,3], *532
Index references U 55, C 71, W 95
Dual polyhedron Great stellapentakis dodecahedron
Vertex figure Great truncated icosahedron vertfig.png
6.6.5/2
Bowers acronymTiggy
3D model of a truncated great icosahedron Truncated great icosahedron.stl
3D model of a truncated great icosahedron

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.

Contents

Cartesian coordinates

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.

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
stellated
dodecahedron
Truncated great stellated dodecahedron Great
icosidodecahedron
Truncated
great
icosahedron
Great
icosahedron
Coxeter-Dynkin
diagram
CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.png
Picture Great stellated dodecahedron.png Icosahedron.png Great icosidodecahedron.png Great truncated icosahedron.png Great icosahedron.png

Great stellapentakis dodecahedron

Great stellapentakis dodecahedron
DU55 great stellapentakisdodecahedron.png
Type Star polyhedron
Face DU55 facets.png
Elements F = 60, E = 90
V = 32 (χ = 2)
Symmetry group Ih, [5,3], *532
Index references DU 55
dual polyhedron Truncated great icosahedron
3D model of a great stellapentakis dodecahedron Great stellapentakis dodecahedron.stl
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 also

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References

  1. Maeder, Roman. "55: great truncated icosahedron". MathConsult.
Animated truncation sequence from { /2, 3} to {3, /2} Great stellated dodecahedron truncations.gif
Animated truncation sequence from {2, 3} to {3, 2}