Inverted snub dodecadodecahedron

Polyhedron with 84 faces
Inverted snub dodecadodecahedron
Type Uniform star polyhedron
Elements F = 84, E = 150
V = 60 (χ = −6)
Faces by sides 60{3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol | 5/3 2 5
Symmetry group I, [5,3]+, 532
Index references U60, C76, W114
Dual polyhedron Medial inverted pentagonal hexecontahedron
Vertex figure
3.3.5.3.5/3
Bowers acronym Isdid
3D model of an inverted snub dodecadodecahedron

In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60.[1] It is given a Schläfli symbol sr{5/3,5}.

Cartesian coordinates

Cartesian coordinates for the vertices of an inverted snub dodecadodecahedron are all the even permutations of

( ± 2 α   , ± 2   , ± 2 β   ) , ( ± [ α + β φ + φ ] , ± [ − α φ + β + 1 φ ] , ± [ α φ + β φ − 1 ] ) , ( ± [ − α φ + β φ + 1 ] , ± [ − α + β φ − φ ] , ± [ α φ + β − 1 φ ] ) , ( ± [ − α φ + β φ − 1 ] , ± [ α − β φ − φ ] , ± [ α φ + β + 1 φ ] ) , ( ± [ α + β φ − φ ] , ± [ α φ − β + 1 φ ] , ± [ α φ + β φ + 1 ] ) , {\displaystyle {\begin{array}{crrrc}{\Bigl (}&\pm \,2\alpha \ ,&\pm \,2\ ,&\pm \,2\beta \ &{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}\alpha +{\frac {\beta }{\varphi }}+\varphi {\bigr ]},&\pm {\bigl [}-\alpha \varphi +\beta +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta \varphi -1{\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta \varphi +1{\bigr ]},&\pm {\bigl [}-\alpha +{\frac {\beta }{\varphi }}-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi +\beta -{\frac {1}{\varphi }}{\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}-{\frac {\alpha }{\varphi }}+\beta \varphi -1{\bigr ]},&\pm {\bigl [}\alpha -{\frac {\beta }{\varphi }}-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi +\beta +{\frac {1}{\varphi }}{\bigr ]}&{\Bigr )},\\[2pt]{\Bigl (}&\pm {\bigl [}\alpha +{\frac {\beta }{\varphi }}-\varphi {\bigr ]},&\pm {\bigl [}\alpha \varphi -\beta +{\frac {1}{\varphi }}{\bigr ]},&\pm {\bigl [}{\frac {\alpha }{\varphi }}+\beta \varphi +1{\bigr ]}&{\Bigr )},\end{array}}}

with an even number of plus signs, where

β =     α 2 φ + φ       α φ − 1 φ   , {\displaystyle \beta ={\frac {\ \ {\frac {\alpha ^{2}}{\varphi }}+\varphi \ \ }{\ \alpha \varphi -{\frac {1}{\varphi }}}}\ ,}
φ = 1 + 5 2 {\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}} is the golden ratio, and α is the negative real root of
φ α 4 − α 3 + 2 α 2 − α − 1 φ ⟹ α ≈ − 0.3352090. {\displaystyle \varphi \alpha ^{4}-\alpha ^{3}+2\alpha ^{2}-\alpha -{\frac {1}{\varphi }}\quad \implies \quad \alpha \approx -0.3352090.}
Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one. Taking α to be the positive root gives the snub dodecadodecahedron.

Related polyhedra

Medial inverted pentagonal hexecontahedron

Medial inverted pentagonal hexecontahedron
Type Star polyhedron
Face
Elements F = 60, E = 150
V = 84 (χ = −6)
Symmetry group I, [5,3]+, 532
Index references DU60
dual polyhedron Inverted snub dodecadodecahedron
3D model of a medial inverted pentagonal hexecontahedron

The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron. Its faces are irregular nonconvex pentagons, with one very acute angle.

Proportions

Denote the golden ratio by ϕ {\displaystyle \phi } , and let ξ ≈ − 0.236 993 843 45 {\displaystyle \xi \approx -0.236\,993\,843\,45} be the largest (least negative) real zero of the polynomial P = 8 x 4 − 12 x 3 + 5 x + 1 {\displaystyle P=8x^{4}-12x^{3}+5x+1} . Then each face has three equal angles of arccos ⁡ ( ξ ) ≈ 103.709 182 219 53 ∘ {\displaystyle \arccos(\xi )\approx 103.709\,182\,219\,53^{\circ }} , one of arccos ⁡ ( ϕ 2 ξ + ϕ ) ≈ 3.990 130 423 41 ∘ {\displaystyle \arccos(\phi ^{2}\xi +\phi )\approx 3.990\,130\,423\,41^{\circ }} and one of 360 ∘ − arccos ⁡ ( ϕ − 2 ξ − ϕ − 1 ) ≈ 224.882 322 917 99 ∘ {\displaystyle 360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 224.882\,322\,917\,99^{\circ }} . Each face has one medium length edge, two short and two long ones. If the medium length is 2 {\displaystyle 2} , then the short edges have length

1 − 1 − ξ ϕ 3 − ξ ≈ 0.474 126 460 54 , {\displaystyle 1-{\sqrt {\frac {1-\xi }{\phi ^{3}-\xi }}}\approx 0.474\,126\,460\,54,}
and the long edges have length
1 + 1 − ξ ϕ − 3 − ξ ≈ 37.551 879 448 54. {\displaystyle 1+{\sqrt {\frac {1-\xi }{\phi ^{-3}-\xi }}}\approx 37.551\,879\,448\,54.}
The dihedral angle equals arccos ⁡ ( ξ / ( ξ + 1 ) ) ≈ 108.095 719 352 34 ∘ {\displaystyle \arccos(\xi /(\xi +1))\approx 108.095\,719\,352\,34^{\circ }} . The other real zero of the polynomial P {\displaystyle P} plays a similar role for the medial pentagonal hexecontahedron.

See also

References

  1. ^ Roman, Maeder. "60: inverted snub dodecadodecahedron". MathConsult.

External links


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