Snub dodecadodecahedron

Short description: 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}
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

File:Snub dodecadodecahedron.stl

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. 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 $$\begin{array}{crrrc}(& \pm 2\alpha ,& \pm 2,& \pm 2\beta & ),\\ (& \pm [\alpha +\frac{\beta }{\phi }+\phi ],& \pm [-\alpha \phi +\beta +\frac{1}{\phi }],& \pm [\frac{\alpha }{\phi }+\beta \phi -1]& ),\\ (& \pm [-\frac{\alpha }{\phi }+\beta \phi +1],& \pm [-\alpha +\frac{\beta }{\phi }-\phi ],& \pm [\alpha \phi +\beta -\frac{1}{\phi }]& ),\\ (& \pm [-\frac{\alpha }{\phi }+\beta \phi -1],& \pm [\alpha -\frac{\beta }{\phi }-\phi ],& \pm [\alpha \phi +\beta +\frac{1}{\phi }]& ),\\ (& \pm [\alpha +\frac{\beta }{\phi }-\phi ],& \pm [\alpha \phi -\beta +\frac{1}{\phi }],& \pm [\frac{\alpha }{\phi }+\beta \phi +1]& ),\end{array}$$

with an even number of plus signs, where $$\beta =\frac{\frac{{\alpha }^2}{\phi }+\phi }{\alpha \phi -\frac{1}{\phi }},$$ $\phi =\frac{1+\sqrt{5}}{2}$ is the golden ratio, and α is the positive real root of $$\phi {\alpha }^4-{\alpha }^3+2{\alpha }^2-\alpha -\frac{1}{\phi }⟹\alpha \approx 0.7964421.$$ 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

File:Medial pentagonal hexecontahedron.stl 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 

External links

• Weisstein, Eric W.. "Medial pentagonal hexecontahedron". http://mathworld.wolfram.com/MedialPentagonalHexecontahedron.html. 
• Weisstein, Eric W.. "Snub dodecadodecahedron". http://mathworld.wolfram.com/SnubDodecadodecahedron.html.