Snub dodecadodecahedron

Uniform star polyhedron with 84 faces
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 | 2 5/2 5
Symmetry group I, [5,3]+, 532
Index references U40, C49, W111
Dual polyhedron Medial pentagonal hexecontahedron
Vertex figure
3.3.5/2.3.5
Bowers acronym Siddid
3D model of a snub dodecadodecahedron

In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U40. It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices.[1] It is given a Schläfli symbol sr{5⁄2,5}, as a snub great dodecahedron.

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 positive real root of
φ α 4 − α 3 + 2 α 2 − α − 1 φ ⟹ α ≈ 0.7964421. {\displaystyle \varphi \alpha ^{4}-\alpha ^{3}+2\alpha ^{2}-\alpha -{\frac {1}{\varphi }}\quad \implies \quad \alpha \approx 0.7964421.}
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 negative root gives the inverted snub dodecadodecahedron.

Related polyhedra

Medial pentagonal hexecontahedron

Medial pentagonal hexecontahedron
Type Star polyhedron
Face
Elements F = 60, E = 150
V = 84 (χ = −6)
Symmetry group I, [5,3]+, 532
Index references DU40
dual polyhedron Snub dodecadodecahedron
3D model of a medial pentagonal hexecontahedron

The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.

See also

References

  1. ^ Maeder, Roman. "40: snub dodecadodecahedron". MathConsult.

External links

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Kepler-Poinsot
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regular polyhedra)Uniform truncations
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