Truncated great icosahedron

Short description: Polyhedron with 32 faces

Truncated great icosahedron
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	6.6.5/2
Bowers acronym	Tiggy

File:Truncated great icosahedron.stl

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. It is given a Schläfli symbol t{3,​⁵⁄₂} or t_0,1{3,​⁵⁄₂} as a truncated great icosahedron.

Cartesian coordinates

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

[math]\displaystyle{ \begin{array}{crccc} \Bigl(& \pm\,1,& 0,& \pm\,\frac{3}{\varphi} &\Bigr) \\ \Bigl(& \pm\,2,& \pm\,\frac{1}{\varphi},& \pm\,\frac{1}{\varphi^3} &\Bigr) \\ \Bigl(& \pm \bigl[1+\frac{1}{\varphi^2}\bigr],& \pm\,1,& \pm\,\frac{2}{\varphi} &\Bigr) \end{array} }[/math]

where [math]\displaystyle{ \varphi = \tfrac{1+\sqrt 5}{2} }[/math] is the golden ratio. Using [math]\displaystyle{ \tfrac{1}{\varphi^2} = 1 - \tfrac{1}{\varphi} }[/math] one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to [math]\displaystyle{ 10-\tfrac{9}{\varphi}. }[/math] The edges have length 2.

Related polyhedra

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
Picture

Great stellapentakis dodecahedron

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

File:Great stellapentakis dodecahedron.stl 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

• List of uniform polyhedra

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5 

External links

• Weisstein, Eric W.. "Truncated great icosahedron". http://mathworld.wolfram.com/GreatTruncatedIcosahedron.html. 
• Weisstein, Eric W.. "Great stellapentakis dodecahedron". http://mathworld.wolfram.com/GreatStellapentakisDodecahedron.html. 
• Uniform polyhedra and duals

Animated truncation sequence from {​⁵⁄₂, 3} to {3, ​⁵⁄₂}