Great snub icosidodecahedron

In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U₅₇. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices.[1] It can be represented by a Schläfli symbol sr{​⁵⁄₂,3}, and Coxeter-Dynkin diagram .

Great snub icosidodecahedron
Type	Uniform star polyhedron
Elements	F = 92, E = 150 V = 60 (χ = 2)
Faces by sides	(20+60){3}+12{5/2}
Wythoff symbol	| 2 5/2 3
Symmetry group	I, [5,3]^+, 532
Index references	U57, C88, W113
Dual polyhedron	Great pentagonal hexecontahedron
Vertex figure	3^4.5/2
Bowers acronym	Gosid

3D model of a great snub icosidodecahedron

This polyhedron is the snub member of a family that includes the great icosahedron, the great stellated dodecahedron and the great icosidodecahedron.

In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great inverted snub icosidodecahedron, and vice versa.

Cartesian coordinates

Cartesian coordinates for the vertices of a great snub icosidodecahedron are all the even permutations of

(±2α, ±2, ±2β),

(±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),

(±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),

(±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and

(±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),

with an even number of plus signs, where

α = ξ−1/ξ

and

β = −ξ/τ+1/τ²−1/(ξτ),

where τ = (1+√5)/2 is the golden mean and ξ is the negative real root of ξ³−2ξ=−1/τ, or approximately −1.5488772. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.

The circumradius for unit edge length is

${\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-x}{1-x}}}=0.64502\dots }$

where ${\displaystyle x}$ is the appropriate root of ${\displaystyle x^{3}+2x^{2}={\Big (}{\tfrac {1\pm {\sqrt {5}}}{2}}{\Big )}^{2}}$. The four positive real roots of the sextic in ${\displaystyle R^{2},}$

${\displaystyle 4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0}$

are the circumradii of the snub dodecahedron (U₂₉), great snub icosidodecahedron (U₅₇), great inverted snub icosidodecahedron (U₆₉), and great retrosnub icosidodecahedron (U₇₄).

Related polyhedra

Great pentagonal hexecontahedron

Great pentagonal hexecontahedron
Type	Star polyhedron
Face
Elements	F = 60, E = 150 V = 92 (χ = 2)
Symmetry group	I, [5,3]^+, 532
Index references	DU57
dual polyhedron	Great snub icosidodecahedron

3D model of a great pentagonal hexecontahedron

The great pentagonal hexecontahedron (or great petaloid ditriacontahedron) is a nonconvex isohedral polyhedron and dual to the uniform great snub icosidodecahedron. It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices.

Proportions

Denote the golden ratio by ${\displaystyle \phi }$. Let ${\displaystyle \xi \approx -0.199\,510\,322\,83}$ be the negative zero of the polynomial ${\displaystyle P=8x^{3}-8x^{2}+\phi ^{-2}}$. Then each pentagonal face has four equal angles of ${\displaystyle \arccos(\xi )\approx 101.508\,325\,512\,64^{\circ }}$ and one angle of ${\displaystyle \arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 133.966\,697\,949\,42^{\circ }}$. Each face has three long and two short edges. The ratio ${\displaystyle l}$ between the lengths of the long and the short edges is given by

${\displaystyle l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 1.315\,765\,089\,00}$.

The dihedral angle equals ${\displaystyle \arccos(\xi /(\xi +1))\approx 104.432\,268\,611\,86^{\circ }}$. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial ${\displaystyle P}$ play a similar role in the description of the great inverted pentagonal hexecontahedron and the great pentagrammic hexecontahedron.

See also

• List of uniform polyhedra
• Great inverted snub icosidodecahedron
• Great retrosnub icosidodecahedron

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208

1. Maeder, Roman. "57: great snub icosidodecahedron". MathConsult.

External links

• Weisstein, Eric W. "Great pentagonal hexecontahedron". MathWorld.
• Weisstein, Eric W. "Great snub icosidodecahedron". MathWorld.
• Uniform polyhedra and duals