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 $s\begin{Bmatrix} 5 \\ 3 \end{Bmatrix}$
	ht0,1,2{5,3}
Wythoff symbol	| 2 3 5
Coxeter diagram
Symmetry group	I, ½H3, [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

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 (the most of the 13 Archimedean solids): 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. The union of both forms is a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron.

Kepler first named it in Latin as dodecahedron simum in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from either the dodecahedron or the icosahedron, called it snub icosidodecahedron, with a vertical extended Schläfli symbol $s\begin{Bmatrix} 5 \\ 3 \end{Bmatrix}$.

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 the number:

$\xi = \sqrt[3]{\frac{\phi}{2} + \frac{1}{2}\sqrt{\phi - \frac{5}{27}}} + \sqrt[3]{\frac{\phi}{2} - \frac{1}{2}\sqrt{\phi - \frac{5}{27}}}$

or approximately 1.7155615.

This snub dodecahedron has an edge length of approximately 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

Transformation from rhombicosidodecahedron to snub dodecahedron

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

$A = 20\sqrt{3} + 3\sqrt{25+10\sqrt{5}} \approx 55.28674495844515$

and the volume is

$V= \frac{12\xi^2(3\phi+1)-\xi(36\phi+7)-(53\phi+6)}{6\sqrt{3-\xi^2}^3} \approx 37.61664996273336$

where ϕ is the golden ratio.

The snub dodecahedron has the highest sphericity of all Archimedean solids.

Orthogonal projections

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

Orthogonal projections

Centered by	Face Triangle	Face Pentagon	Edge
Image
Projective symmetry	[3]	[5]+	[2]
Dual image

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.

Dodecahedron

Rhombicosidodecahedron (Expanded dodecahedron)

Snub dodecahedron

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

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532)							[5,3]+, (532)
{5,3}	t{5,3}	r{5,3}	t{3,5}	{3,5}	rr{5,3}	tr{5,3}	sr{5,3}
Duals to uniform polyhedra
V5.5.5	V3.10.10	V3.5.3.5	V5.6.6	V3.3.3.3.3	V3.4.5.4	V4.6.10	V3.3.3.3.5

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.

n32 symmetry mutations of snub tilings: 3.3.3.3.n
Symmetry n32	Spherical				Euclidean	Compact hyperbolic		Paracomp.
	232	332	432	532	632	732	832	∞32
Snub figures
Config.	3.3.3.3.2	3.3.3.3.3	3.3.3.3.4	3.3.3.3.5	3.3.3.3.6	3.3.3.3.7	3.3.3.3.8	3.3.3.3.∞
Gryro figures
Config.	V3.3.3.3.2	V3.3.3.3.3	V3.3.3.3.4	V3.3.3.3.5	V3.3.3.3.6	V3.3.3.3.7	V3.3.3.3.8	V3.3.3.3.∞

Snub dodecahedral graph

Snub dodecahedral graph
5-fold symmetry Schlegel diagram
Vertices	60
Edges	150
Automorphisms	60
Properties	Hamiltonian, regular

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.

Orthogonal projections

See also

• Planar polygon to polyhedron transformation animation
• ccw and cw spinning snub dodecahedron

References

• (Section 3-9)

External links

• Eric W. Weisstein, Snub dodecahedron (Archimedean solid) at MathWorld.
  • Weisstein, Eric W., "Snub dodecahedral graph", MathWorld.
• Richard Klitzing, 3D convex uniform polyhedra, s3s5s - snid
• Editable printable net of a Snub Dodecahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• The Snub Dodecahedron made with LEGO by Antonio Nicassio (ITALY)