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

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

with an even number of plus signs, where

$${\displaystyle \beta ={\frac {\ \ {\frac {\alpha ^{2}}{\varphi }}+\varphi \ \ }{\ \alpha \varphi -{\frac {1}{\varphi }}}}\ ,}$$

${\displaystyle \varphi ={\tfrac {1+{\sqrt {5}}}{2}}}$ is the golden ratio, and α is the positive real root of

$${\displaystyle \varphi \alpha ^{4}-\alpha ^{3}+2\alpha ^{2}-\alpha -{\frac {1}{\varphi }}\quad \implies \quad \alpha \approx 0.7964421.}$$

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

• List of uniform polyhedra
• Inverted snub dodecadodecahedron

References

• 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.