Snub dodecahedron

Snub dodecahedral graph
	5-fold symmetry Schlegel diagram
Vertices	60
Edges	150
Automorphisms	60
Properties	Hamiltonian, regular
	Table of graphs and parameters

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 $s{\begin{Bmatrix}5\\3\end{Bmatrix}}$
Schläfli symbols	ht_0,1,2{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

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 $s\scriptstyle {\begin{Bmatrix}5\\3\end{Bmatrix}}$ and flat Schläfli symbol sr{5,3}.

Cartesian coordinates

Coordinate values

phi is the golden ratio, sqrt is the square root, and cbrt is the cube root.

phi = (1 + sqrt(5)) / 2
x = cbrt((phi + sqrt(phi-5/27))/2) + cbrt((phi - sqrt(phi-5/27))/2)

c0  = phi * sqrt(3 - (x^2)) / 2
c1  = x * phi * sqrt(3 - (x^2)) / 2
c2  = phi * sqrt((x - 1 - (1/x)) * phi) / 2
c3  = (x^2) * phi * sqrt(3 - (x^2)) / 2
c4  = x * phi * sqrt((x - 1 - (1/x)) * phi) / 2
c5  = phi * sqrt(1 - x + (phi + 1) / x) / 2
c6  = phi * sqrt(x - phi + 1) / 2
c7  = (x^2) * phi * sqrt((x - 1 - (1/x)) * phi) / 2
c8  = x * phi * sqrt(1 - x + (phi + 1) / x) / 2
c9  = sqrt((x + 2) * phi + 2) / 2
c10 = x * sqrt(x * (phi + 1) - phi) / 2
c11 = sqrt((x^2) * (2 * phi + 1) - phi) / 2
c12 = phi * sqrt((x^2) + x) / 2
c13 = (phi^2) * sqrt(x * (x + phi) + 1) / (2 * x)
c14 = phi * sqrt(x * (x + phi) + 1) / 2

With the 15 absolute values given, the coordinates of a snub dodecahedron with edge length 1 are the even permutations of

(c2, c1, c14), (c0, c8, c12) and (c7, c6, c11) with an even number of plus signs
(c3, c4, c13) and (c9, c5, c10) with an odd number of plus signs. ^[1]

Surface area and volume

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

A=20{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\approx 55.286\,744\,958\,445\,15

and the volume is

V={\frac {12\xi ^{2}(3\varphi +1)-\xi (36\varphi +7)-(53\varphi +6)}{6{\sqrt {3-\xi ^{2}}}^{3}}}\approx 37.616\,649\,962\,733\,36

and circumradius is

R={\frac {1}{2}}{\sqrt {\frac {2-x}{1-x}}}=2.15584\dots

where $x$ is the appropriate root of $x^{3}+2x^{2}={\Big (}{\tfrac {1\pm {\sqrt {5}}}{2}}{\Big )}^{2}$ and $\varphi$ is the golden ratio. The four positive real roots of the sextic in $R^{2}$

4096R^{12}-27648R^{10}+47104R^{8}-35776R^{6}+13872R^{4}-2696R^{2}+209=0

is the circumradius of the snub dodecahedron (U₂₉), great snub icosidodecahedron (U₅₇), great inverted snub icosidodecahedron (U₆₉), and great retrosnub icosidodecahedron (U₇₄).

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

Orthogonal projections

The snub dodecahedron has no point symmetry, so the vertex in the front does not correspond to an opposite vertex in the back.

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

Orthogonal projections
Centered by	Face Triangle	Face Pentagon	Edge
Solid
Wireframe
Projective symmetry	[3]	[5]⁺	[2]
Dual

Geometric relations

Dodecahedron, rhombicosidodecahedron and snub dodecahedron (animated expansion and twisting)

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, pull the pentagonal faces out slightly less, only add the triangle faces and leave the other gaps empty (the other gaps are rectangles at this point). 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 fact that the proper amount to pull the faces out is less in the case of the snub dodecahedron can be seen in either of two ways: the circumradius of the snub dodecahedron is smaller than that of the icosidodecahedron; or, the edge length of the equilateral triangles formed by the divided vertices increases when the pentagonal faces are rotated.)

Uniform alternation of a truncated icosidodecahedron

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.

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.
Symmetry n32	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

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.^[2]

References

↑ The full list of coordinates can be found here: http://dmccooey.com/polyhedra/RsnubDodecahedron.html (And here are those for the enantiomorph.)
↑ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269

Jayatilake, Udaya (March 2005). "Calculations on face and vertex regular polyhedra". Mathematical Gazette. 89 (514): 76–81.
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)
Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.

External links

Eric W. Weisstein, Snub dodecahedron (Archimedean solid) at MathWorld.
- Weisstein, Eric W. "Snub dodecahedral graph". MathWorld.
Klitzing, Richard. "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

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[1] The full list of coordinates can be found here: http://dmccooey.com/polyhedra/RsnubDodecahedron.html (And here are those for the enantiomorph.)

[2] Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269