Snub dodecahedron

phi = (1 + sqrt(5)) / 2
x = cbrt((phi + sqrt(phi-5/27))/2) + cbrt((phi - sqrt(phi-5/27))/2) (solution of x^3=2x+phi) 

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]

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

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

and the volume is

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

and circumradius is

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

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

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

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 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.[2]

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.

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.)

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.

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.

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.[3]

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

• 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.

• 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