Polyhedron with 92 faces
3D model of a 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] Schläfli symbol sr{5 ⁄2 Coxeter-Dynkin diagram
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 Magnus Wenninger , the polyhedron is misnamed great inverted snub icosidodecahedron
Cartesian coordinates [ edit ] Cartesian coordinates for the vertices of a great retrosnub icosidodecahedron are all the even permutations of
( ± 2 α , ± 2 , ± 2 β ) , ( ± [ α − β φ − 1 φ ] , ± [ α φ + β − φ ] , ± [ − α φ − β φ − 1 ] ) , ( ± [ α φ − β φ + 1 ] , ± [ − α − β φ + 1 φ ] , ± [ − α φ + β + φ ] ) , ( ± [ α φ − β φ − 1 ] , ± [ α + β φ + 1 φ ] , ± [ − α φ + β − φ ] ) , ( ± [ α − β φ + 1 φ ] , ± [ − α φ − β − φ ] , ± [ − α φ − β φ + 1 ] ) , {\displaystyle {\begin{array}{crrrc}{\Bigl (}&\pm \,2\alpha ,&\pm \,2,&\pm \,2\beta &{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha -\beta \varphi -{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta -\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi -{\frac {\beta }{\varphi }}-1{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha \varphi -{\frac {\beta }{\varphi }}+1{\bigr ]},&\pm {\bigl [}-\alpha -\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta +\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha \varphi -{\frac {\beta }{\varphi }}-1{\bigr ]},&\pm {\bigl [}\alpha +\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta -\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha -\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}-\beta -\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi -{\frac {\beta }{\varphi }}+1{\bigr ]}&{\Bigr )},\\\end{array}}} with an even number of plus signs, where
α = ξ − 1 ξ , β = − ξ φ + 1 φ 2 − 1 ξ φ , {\displaystyle {\begin{aligned}\alpha &=\xi -{\frac {1}{\xi }},\\[4pt]\beta &=-{\frac {\xi }{\varphi }}+{\frac {1}{\varphi ^{2}}}-{\frac {1}{\xi \varphi }},\end{aligned}}} where φ = 1 + 5 2 {\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}} golden ratio and ξ is the negative real root of:
ξ 3 − 2 ξ = − 1 φ ⟹ ξ ≈ − 1.5488772. {\displaystyle \xi ^{3}-2\xi =-{\frac {1}{\varphi }}\quad \implies \quad \xi \approx -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
R = 1 2 2 − x 1 − x = 0.64502 … {\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-x}{1-x}}}=0.64502\dots } where
x = − 0.505561 {\displaystyle x=-0.505561} is the second largest real root of the polynomial
[2] x 3 + 2 x 2 = φ − 2 = ( 1 + 5 2 ) − 2 = ( 1 − 5 2 ) 2 . {\displaystyle x^{3}+2x^{2}=\varphi ^{-2}=\left({\tfrac {1+{\sqrt {5}}}{2}}\right)^{-2}=\left({\tfrac {1-{\sqrt {5}}}{2}}\right)^{2}.} The four positive real roots of the sextic in R 2
4096 R 12 − 27648 R 10 + 47104 R 8 − 35776 R 6 + 13872 R 4 − 2696 R 2 + 209 = 0 {\displaystyle 4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0} are, in order, the circumradii of the
great retrosnub icosidodecahedron (U
74 ), great snub icosidodecahedron (U
57 ),
great inverted snub icosidodecahedron (U
69 ) and
snub dodecahedron (U
29 ).
Related polyhedra [ edit ] Great pentagonal hexecontahedron [ edit ] 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 [ edit ] Denote the golden ratio by ϕ {\displaystyle \phi } ξ ≈ − 0.199 510 322 83 {\displaystyle \xi \approx -0.199\,510\,322\,83} P = 8 x 3 − 8 x 2 + ϕ − 2 {\displaystyle P=8x^{3}-8x^{2}+\phi ^{-2}} arccos ( ξ ) ≈ 101.508 325 512 64 ∘ {\displaystyle \arccos(\xi )\approx 101.508\,325\,512\,64^{\circ }} arccos ( − ϕ − 1 + ϕ − 2 ξ ) ≈ 133.966 697 949 42 ∘ {\displaystyle \arccos(-\phi ^{-1}+\phi ^{-2}\xi )\approx 133.966\,697\,949\,42^{\circ }} l {\displaystyle l}
l = 2 − 4 ξ 2 1 − 2 ξ ≈ 1.315 765 089 00 {\displaystyle l={\frac {2-4\xi ^{2}}{1-2\xi }}\approx 1.315\,765\,089\,00} The dihedral angle equals arccos ( ξ / ( ξ + 1 ) ) ≈ 104.432 268 611 86 ∘ {\displaystyle \arccos(\xi /(\xi +1))\approx 104.432\,268\,611\,86^{\circ }} P {\displaystyle P} great inverted pentagonal hexecontahedron and the great pentagrammic hexecontahedron .
See also [ edit ] References [ edit ] External links [ edit ]
Kepler-Poinsot (nonconvex Uniform truncations Nonconvex uniform hemipolyhedra Duals of nonconvex Duals of nonconvex
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![{\displaystyle {\begin{array}{crrrc}{\Bigl (}&\pm \,2\alpha ,&\pm \,2,&\pm \,2\beta &{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha -\beta \varphi -{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta -\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi -{\frac {\beta }{\varphi }}-1{\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha \varphi -{\frac {\beta }{\varphi }}+1{\bigr ]},&\pm {\bigl [}-\alpha -\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta +\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha \varphi -{\frac {\beta }{\varphi }}-1{\bigr ]},&\pm {\bigl [}\alpha +\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta -\varphi {\bigr ]}&{\Bigr )},\\{\Bigl (}&\pm {\bigl [}\alpha -\beta \varphi +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}-{\frac {\alpha }{\varphi }}-\beta -\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi -{\frac {\beta }{\varphi }}+1{\bigr ]}&{\Bigr )},\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f7e1b1ee5692b0155448d8f632b2ffab5088df1)
![{\displaystyle {\begin{aligned}\alpha &=\xi -{\frac {1}{\xi }},\\[4pt]\beta &=-{\frac {\xi }{\varphi }}+{\frac {1}{\varphi ^{2}}}-{\frac {1}{\xi \varphi }},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e6a3e86c3eab5801ad1f161516d1c4927862671)
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