Snub dodecahedron

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Snub dodecahedron
(Click here for rotating model)
Type	Archimedean solid
Elements	F=92, E=150, V=60 (χ=2)
Faces by sides	(20+60){3}+12{5}
Schläfli symbol	$s\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}$
Wythoff symbol	\| 2 3 5
Coxeter-Dynkin
Symmetry	I
References	U₂₉, C₃₂, W₁₈
Properties	Semiregular convex chiral
Colored faces	3.3.3.3.5 (Vertex figure)
Pentagonal hexecontahedron (dual polyhedron)	Net

The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid.

The snub dodecahedron has 92 faces, of which 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other.

1 Geometric relations
2 Cartesian coordinates
3 See also
4 References
5 External links

[edit] Geometric relations

The snub dodecahedron can be generated by taking the twelve pentagonal faces of the dodecahedron, pulling them outward so they no longer touch. This creates the rhombicosidodecahedron. Then give them all a small rotation on their centers (all clockwise or all counter-clockwise) until the space between can be filled by equilateral triangles.

Dodecahedron

Rhombicosidodecahedron
(Expanded dodecahedron)

Archimedes, an ancient Greek who showed major interest in polyhedral shapes wrote a treatise on thirteen semi-regular solids. Snub-Dodecahedron belongs to the thirteen semi-regular solids.

[edit] Cartesian coordinates

Cartesian coordinates for the vertices of a snub dodecahedron are all the even permutations of

(±2α, ±2, ±2β),

(±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)),

(±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)),

(±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and

(±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)),

with an even number of plus signs, where

α = ξ-1/ξ

and

β = ξτ+τ²+τ/ξ,

where τ = (1+√5)/2 is the golden mean and ξ is the real solution to ξ³-2ξ=τ, which is the beautiful number

$\xi = \sqrt[3]{\frac{\tau}{2} + \frac{1}{2}\sqrt{\tau - \frac{5}{27}}} + \sqrt[3]{\frac{\tau}{2} - \frac{1}{2}\sqrt{\tau - \frac{5}{27}}}$

or approximately 1.7155615. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.

[edit] See also

ccw and cw spinning snub dodecahedron
Snub cube
Snub hexagonal tiling

[edit] References

Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)