Snub dodecahedron
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| Snub dodecahedron | |
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 Click on picture for large version. Click here for spinning version. | |
| Missing image Snubdodecahedroncw.jpg Snub dodecahedron, clockwise twist Click on picture for large version. Click here for spinning version. | |
| Type | Archimedean |
| Faces | 80 triangles 12 pentagons |
| Edges | 150 |
| Vertices | 60 |
| Vertex configuration | 3,3,3,3,5 |
| Symmetry group | icosahedral (I) |
| Dual polyhedron | pentagonal hexecontahedron |
| Properties | convex, semi-regular (vertex-uniform), chiral |
The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a dodecahedron or an icosahedron.
Missing image
Snub_dodecahedron_flat.png
image:Snub dodecahedron flat.png
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. In three-dimensional space, it has two distinct forms, which are mirror images (or "enantiomorphs") of each other. In higher-dimensional spaces, these are congruent.
Canonical coordinates for 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 horrible number
- <math>\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}}}<math>
or approximately 1.7155615.
The snub dodecahedron should not be confused with the truncated dodecahedron.
See also
External links
- The Uniform Polyhedra (http://www.mathconsult.ch/showroom/unipoly/)
- Virtual Reality Polyhedra (http://www.georgehart.com/virtual-polyhedra/vp.html) The Encyclopedia of Polyhedra
- Irregular Snub Dodecahdron Wire Model (http://www.vincentherr.com/art/wire/dodec6866.htm) Math Models, Vincent Herrnl:Stompe dodecaëder