Truncated great icosahedron

Truncated great icosahedron
File: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}
Coxeter diagram	Template:CDD
Wythoff symbol	3 2 5/3 | 3
Symmetry group	Ih, [5,3], *532
Index references	U55, C71, W95
Dual polyhedron	Great stellapentakis dodecahedron
Vertex figure	File:Great truncated icosahedron vertfig.png 6.6.5/2
Bowers acronym	Tiggy

File: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. 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

$$\begin{array}{crccc}(& \pm 1,& 0,& \pm \frac{3}{\phi }& )\\ (& \pm 2,& \pm \frac{1}{\phi },& \pm \frac{1}{{\phi }^3}& )\\ (& \pm [1+\frac{1}{{\phi }^2}],& \pm 1,& \pm \frac{2}{\phi }& )\end{array}$$

where $\phi =\frac{1+\sqrt{5}}{2}$ is the golden ratio. Using $\frac{1}{{\phi }^2}=1-\frac{1}{\phi }$ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to $10-\frac{9}{\phi }.$ 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	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Picture	File:Great stellated dodecahedron.png	File:Icosahedron.png	File:Great icosidodecahedron.png	File:Great truncated icosahedron.png	File:Great icosahedron.png

Great stellapentakis dodecahedron

Great stellapentakis dodecahedron
File:DU55 great stellapentakisdodecahedron.png
Type	Star polyhedron
Face	File:DU55 facets.png
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

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

• List of uniform polyhedra

External links

• Uniform polyhedra and duals

File:Great stellated dodecahedron truncations.gif

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

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