| 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 |
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
![{\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)
with an even number of plus signs, where
is the
golden ratio, and α is the positive real
root of
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 |
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
- ^ Maeder, Roman. "40: snub dodecadodecahedron". MathConsult.
- Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi: 10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links
- Weisstein, Eric W. "Medial pentagonal hexecontahedron". MathWorld.
- Weisstein, Eric W. "Snub dodecadodecahedron". MathWorld.
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