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.

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 horrible number

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

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.

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.

The snub dodecahedron should not be confused with the truncated dodecahedron.

See also

---

• Snubdodecahedronccw.gif and Snubdodecahedronccw.gif spinning snub dodecahedron
• snub cube

External links

---

• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Irregular Snub Dodecahdron Wire Model Math Models, Vincent Herr

Chiral polyhedra | Uniform polyhedra | Archimedean solids

Icosidodecaedro snub | Stompe dodecaëder | 変形十二面体 | Dwunastościan przycięty | Dodecaedro snub