Snub dodecahedron
| Snub dodecahedron | |
|---|---|
 (Click here for rotating model) | |
| Type | Archimedean solid Uniform polyhedron |
| Elements | F = 92, E = 150, V = 60 (χ = 2) |
| Faces by sides | (20+60){3}+12{5} |
| Conway notation | sD |
| Schläfli symbols | sr{5,3} or |
| ht0,1,2{5,3} | |
| Wythoff symbol | | 2 3 5 |
| Coxeter diagram |   
|
| Symmetry group | I, 1/2H3, [5,3]+, (532), order 60 |
| Rotation group | I, [5,3]+, (532), order 60 |
| Dihedral angle | 3-3: 164°10′31″ (164.18°) 3-5: 152°55′53″ (152.93°) |
| 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.
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. 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.
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
![{\displaystyle \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}}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee26fccef1bd02dc31741e4504f635046ec8d2a7)
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
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- Irregular Snub Dodecahdron Wire Model Math Models, Vincent Herr