Great truncated icosidodecahedron

Short description: Polyhedron with 62 faces

Great truncated icosidodecahedron
Type	Uniform star polyhedron
Elements	F = 62, E = 180 V = 120 (χ = 2)
Faces by sides	30{4}+20{6}+12{10/3}
Wythoff symbol	2 3 5/3 |
Symmetry group	Ih, [5,3], *532
Index references	U68, C87, W108
Dual polyhedron	Great disdyakis triacontahedron
Vertex figure	4.6.10/3
Bowers acronym	Gaquatid

File:Great truncated icosidodecahedron.stl In geometry, the great truncated icosidodecahedron (or great quasitruncated icosidodecahedron or stellatruncated icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U₆₈. It has 62 faces (30 squares, 20 hexagons, and 12 decagrams), 180 edges, and 120 vertices. It is given a Schläfli symbol t_0,1,2{​⁵⁄₃,3}, and Coxeter-Dynkin diagram, .

Cartesian coordinates

Cartesian coordinates for the vertices of a great truncated icosidodecahedron centered at the origin are all the even permutations of $$\begin{array}{ccclc}(& \pm \phi ,& \pm \phi ,& \pm [3-\frac{1}{\phi }]& ),\\ (& \pm 2\phi ,& \pm \frac{1}{\phi },& \pm \frac{1}{{\phi }^3}& ),\\ (& \pm \phi ,& \pm \frac{1}{{\phi }^2},& \pm [1+\frac{3}{\phi }]& ),\\ (& \pm \sqrt{5},& \pm 2,& \pm \frac{\sqrt{5}}{\phi }& ),\\ (& \pm \frac{1}{\phi },& \pm 3,& \pm \frac{2}{\phi }& ),\end{array}$$

where $\phi =\frac{1+\sqrt{5}}{2}$ is the golden ratio.

Related polyhedra

Great disdyakis triacontahedron

Great disdyakis triacontahedron
Type	Star polyhedron
Face
Elements	F = 120, E = 180 V = 62 (χ = 2)
Symmetry group	Ih, [5,3], *532
Index references	DU68
dual polyhedron	Great truncated icosidodecahedron

File:Great disdyakis triacontahedron.stl The great disdyakis triacontahedron (or trisdyakis icosahedron) is a nonconvex isohedral polyhedron. It is the dual of the great truncated icosidodecahedron. Its faces are triangles.

Proportions

The triangles have one angle of $\arccos (\frac{1}{6}+\frac{1}{15}\sqrt{5})\approx 71.59463622088^{\circ }$, one of $\arccos (\frac{3}{4}+\frac{1}{10}\sqrt{5})\approx 13.19299904074^{\circ }$ and one of $\arccos (\frac{3}{8}-\frac{5}{24}\sqrt{5})\approx 95.21236473838^{\circ }.$ The dihedral angle equals $\arccos \left(\frac{-179+24\sqrt{5}}{241}\right)\approx 121.33625080739^{\circ }.$ Part of each triangle lies within the solid, hence is invisible in solid models.

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  p. 96

External links

• Weisstein, Eric W.. "Great truncated icosidodecahedron". http://mathworld.wolfram.com/GreatTruncatedIcosidodecahedron.html. 
• Weisstein, Eric W.. "Great disdyakis triacontahedron". http://mathworld.wolfram.com/GreatDisdyakisTriacontahedron.html.