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Snub dodecahedron
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 CDW hole.pngCDW 5.pngCDW hole.pngCDW 3.pngCDW hole.png
Symmetry I
or (532)
References U29, C32, W18
Properties Semiregular convex chiral
Snub dodecahedron color
Colored faces
Snub dodecahedron
3.3.3.3.5
(Vertex figure)
Pentagonalhexecontahedronccw.jpg
Pentagonal hexecontahedron
(dual polyhedron)
Snub dodecahedron Net
Net

In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

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.

Contents

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. At a proper distance this can create the rhombicosidodecahedron by filling in square faces between the divided edges and triangle faces between the divided vertices. But for the snub form, only add the triangle faces and leave the square gaps empty. Then apply an equal rotation to the centers of the pentagons and triangles, continuing the rotation until the gaps can be filled by two equilateral triangles.

Dodecahedron.png
Dodecahedron
Small rhombicosidodecahedron.png
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.

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 ratio and ξ is the real solution to ξ3-2ξ=τ, which is the 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.

See also

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)

External links

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