Snub dodecahedron

Let ${\displaystyle \xi \approx 0.943\,151\,259\,24}$ be the real zero of the polynomial ${\displaystyle x^{3}+2x^{2}-\phi ^{2}}$, where ${\displaystyle \phi }$ is the golden ratio. Let the point ${\displaystyle p}$ be given by

${\displaystyle p={\begin{pmatrix}\phi ^{2}-\phi ^{2}\xi \\-\phi ^{3}+\phi \xi +2\phi \xi ^{2}\\\xi \end{pmatrix}}}$.

Let the matrix ${\displaystyle M}$ be given by

${\displaystyle M={\begin{pmatrix}1/(2\phi )&-\phi /2&1/2\\\phi /2&1/2&1/(2\phi )\\-1/2&1/(2\phi )&\phi /2\end{pmatrix}}}$.

${\displaystyle M}$ is the rotation around the axis ${\displaystyle (0,1,\phi )}$ through an angle of ${\displaystyle 2\pi /5}$, counterclockwise. Let the linear transformations ${\displaystyle T_{0},\ldots ,T_{11}}$ be the transformations which send a point ${\displaystyle (x,y,z)}$ to the even permutations of ${\displaystyle (\pm x,\pm y,\pm z)}$ with an even number of minus signs. The transformations ${\displaystyle T_{i}}$ constitute the group of rotational symmetries of a regular tetrahedron. The transformations ${\displaystyle T_{i}M^{j}}$ ${\displaystyle (i=0,\ldots ,11}$, ${\displaystyle j=0,\ldots ,4)}$ constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points ${\displaystyle T_{i}M^{j}p}$ are the vertices of a snub dodecahedron. The coordinates of the vertices are integral linear combinations of ${\displaystyle 1}$, ${\displaystyle \phi }$, ${\displaystyle \xi }$, ${\displaystyle \phi \xi }$, ${\displaystyle \xi ^{2}}$ and ${\displaystyle \phi \xi ^{2}}$. The edge length equals ${\displaystyle 2\xi {\sqrt {1-\xi }}\approx 0.449\,750\,618\,41}$. Negating all coordinates gives the mirror image of this snub dodecahedron.

As a volume, the snub dodecahedron consists of 80 triangular and 12 pentagonal pyramids. The volume ${\displaystyle V_{3}}$ of one triangular pyramid is given by:

${\displaystyle V_{3}={\frac {1}{3}}\phi (3\xi ^{2}-\phi ^{2})\approx 0.027\,274\,068\,85,}$

and the volume ${\displaystyle V_{5}}$ of one pentagonal pyramid by:

${\displaystyle V_{5}={\frac {1}{3}}(3\phi +1)(\phi +3-2\xi -3\xi ^{2})\xi ^{3}\approx 0.103\,349\,665\,04.}$

The total volume is ${\displaystyle 80V_{3}+12V_{5}\approx 3.422\,121\,488\,76}$.

The circumradius equals ${\displaystyle {\sqrt {4\xi ^{2}-\phi ^{2}}}\approx 0.969\,589\,192\,65}$. The midradius equals ${\displaystyle \xi }$. This gives an interesting geometrical interpretation of the number ${\displaystyle \xi }$. The 20 "icosahedral" triangles of the snub dodecahedron described above are coplanar with the faces of a regular icosahedron. The midradius of this "circumscribed" icosahedron equals ${\displaystyle 1}$. This means that ${\displaystyle \xi }$ is the ratio between the midradii of a snub dodecahedron and the icosahedron in which it is inscribed.

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

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

Its volume is, putting ${\displaystyle \eta =\varphi /\xi }$,

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

Its circumradius is

${\displaystyle R={\frac {1}{2}}{\sqrt {\frac {2-\xi }{1-\xi }}}\approx 2.155\,837\,375}$.

The four positive real roots of the sextic in ${\displaystyle R^{2}}$

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

are the circumradii 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 of all Archimedean solids. If sphericity is defined as the ratio of volume squared over surface area cubed, multiplied by a constant of 36 times pi (where this constant makes the sphericity of a sphere equal to 1), the sphericity of the snub dodecahedron is about 0.947.[1]

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

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.

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.∞
Gyro 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
5-fold symmetry Schlegel diagram
Vertices	60
Edges	150
Automorphisms	60
Properties	Hamiltonian, regular
Table of graphs and parameters

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]

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

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• 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.

• 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