Snub dodecahedron

Cartesian coordinates

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

(±2α, ±2, ±2β),(±(α + β/φ + φ), ±(−αφ + β + 1/φ), ±(α/φ + βφ − 1)),(±(α + β/φ − φ), ±(αφ − β + 1/φ), ±(α/φ + βφ + 1)),(±(−α/φ + βφ + 1), ±(−α + β/φ − φ), ±(αφ + β − 1/φ)) and(±(−α/φ + βφ − 1), ±(α − β/φ − φ), ±(αφ + β + 1/φ)),

with an even number of plus signs, where

α = ξ − 1/ξ

and

β = ξφ + φ² + φ/ξ,

where φ = 1 + √5/2 is the golden ratio and ξ is the real solution to ξ³ − 2ξ = φ, which is thenumber:

ξ = φ 2 + 1 2 φ − 5 27 3 + φ 2 − 1 2 φ − 5 27 3 ≈ 1.715 5615.

This snub dodecahedron has an edge length of approximately 7000604373808410000♠6.0437380841.

Taking the odd permutations of the above coordinates with an even number of plus signs gives another form, the enantiomorph of the other one. Though it may not be immediately obvious, the figure obtained by taking the even permutations with an even number of plus signs is the same as that obtained by taking the odd permutations with an odd number of plus signs. Similarly, the mirror image has either an odd permutation with an even number of plus signs or an even permutation with an odd number of plus signs.

Surface area and volume

For a snub dodecahedron whose edge length is 1, the surface area is

A = 20 3 + 3 25 + 10 5 ≈ 55.286 744 958 445 15

and the volume is

V = 12 ξ 2 ( 3 φ + 1 ) − ξ ( 36 φ + 7 ) − ( 53 φ + 6 ) 6 3 − ξ 2 3 ≈ 37.616 649 962 733 36

where φ is the golden ratio.

The snub dodecahedron has the highest sphericity (about 0.982) of all Archimedean solids.

Orthogonal projections

The snub dodecahedron has two special orthogonal projections, centered on two types of faces: triangles and pentagons, corresponding to the A₂ and H₂ Coxeter planes.

Geometric relations

The snub dodecahedron can be generated by taking the twelve pentagonal faces of the dodecahedron and 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.

The snub dodecahedron can also be derived from the truncated icosidodecahedron by the process of alternation. Sixty of the vertices of the truncated icosidodecahedron form a polyhedron topologically equivalent to one snub dodecahedron; the remaining sixty form its mirror-image. The resulting polyhedron is vertex-transitive but not uniform, because its edges are of unequal lengths; some deformation is required to transform it into a uniform polyhedron.

Related polyhedra and tilings

This semiregular polyhedron is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n = 6, and hyperbolic plane for any higher n. The series can be considered to begin with n = 2, with one set of faces degenerated into digons.

Snub dodecahedral graph

In the mathematical field of graph theory, a snub dodecahedral graph is the graph of vertices and edges of the snub dodecahedron, one of the Archimedean solids. It has 60 vertices and 150 edges, and is an Archimedean graph.