Polyhedron with 84 faces
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] Schläfli symbol sr{5/3,5}.
Cartesian coordinates [ edit ] 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 [ edit ] Medial inverted pentagonal hexecontahedron [ edit ] 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 [ edit ] Denote the golden ratio by ϕ {\displaystyle \phi } ξ ≈ − 0.236 993 843 45 {\displaystyle \xi \approx -0.236\,993\,843\,45} P = 8 x 4 − 12 x 3 + 5 x + 1 {\displaystyle P=8x^{4}-12x^{3}+5x+1} arccos ( ξ ) ≈ 103.709 182 219 53 ∘ {\displaystyle \arccos(\xi )\approx 103.709\,182\,219\,53^{\circ }} arccos ( ϕ 2 ξ + ϕ ) ≈ 3.990 130 423 41 ∘ {\displaystyle \arccos(\phi ^{2}\xi +\phi )\approx 3.990\,130\,423\,41^{\circ }} 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 }} 2 {\displaystyle 2}
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 [ edit ] References [ edit ] External links [ edit ]
From Wikipedia the free encyclopedia
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c98f3d7fd373cecd5a28e79e9dc36acbefb52a6)
