Snub dodecahedron

Template:Short description Template:Semireg polyhedra db

3D model of a snub dodecahedron

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 <math>s \scriptstyle\begin{Bmatrix} 5 \\ 3 \end{Bmatrix}</math> and flat Schläfli symbol Template:Math

Cartesian coordinatesEdit

Let Template:Math be the real zero of the cubic polynomial Template:Math, where Template:Mvar is the golden ratio. Let the point Template:Mvar be given by <math display=block>p= \begin{pmatrix}

       \varphi^2-\varphi^2\xi \\
       -\varphi^3+\varphi\xi+2\varphi\xi^2 \\
       \xi

\end{pmatrix}. </math> Let the rotation matrices Template:Math and Template:Math be given by <math display=block> M_1= \begin{pmatrix}

        \frac{1}{2\varphi}  & -\frac{\varphi}{2} & \frac{1}{2} \\
        \frac{\varphi}{2}   & \frac{1}{2}     & \frac{1}{2\varphi} \\
        -\frac{1}{2}     & \frac{1}{2\varphi} & \frac{\varphi}{2}

\end{pmatrix}, \quad M_2= \begin{pmatrix}

        0  & 0 & 1 \\
        1  & 0 & 0 \\
        0  & 1 & 0

\end{pmatrix}. </math> Template:Math represents the rotation around the axis Template:Math through an angle of Template:Sfrac counterclockwise, while Template:Math being a cyclic shift of Template:Math represents the rotation around the axis Template:Math through an angle of Template:Sfrac. Then the 60 vertices of the snub dodecahedron are the 60 images of point Template:Mvar under repeated multiplication by Template:Math and/or Template:Math, iterated to convergence. (The matrices Template:Math and Template:Math generate the 60 rotation matrices corresponding to the 60 rotational symmetries of a regular icosahedron.) The coordinates of the vertices are integral linear combinations of Template:Math and Template:Math. The edge length equals <math display=block>2\xi\sqrt{1-\xi}\approx 0.449\,750\,618\,41.</math> 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 Template:Math of one triangular pyramid is given by: <math display=block>

   V_3 = \frac{1}{3}\varphi\left(3\xi^2-\varphi^2\right) \approx 0.027\,274\,068\,85,

</math> and the volume Template:Math of one pentagonal pyramid by: <math display=block>

   V_5 = \frac{1}{3}(3\varphi+1)\left(\varphi+3-2\xi-3\xi^2\right)\xi^3 \approx 0.103\,349\,665\,04.

</math> The total volume is <math display=block>80V_3+12V_5 \approx 3.422\,121\,488\,76.</math>

The circumradius equals <math display=block>\sqrt{4\xi^2-\varphi^2} \approx 0.969\,589\,192\,65.</math> The midradius equals Template:Mvar. This gives an interesting geometrical interpretation of the number Template:Mvar. 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 1. This means that Template:Mvar is the ratio between the midradii of a snub dodecahedron and the icosahedron in which it is inscribed.

The triangle–triangle dihedral angle is given by <math display=block>

   \theta_{33} = 180^\circ - \arccos\left(\frac23\xi+\frac13\right) \approx 164.175\,366\,056\,03^\circ.

</math>

The triangle–pentagon dihedral angle is given by <math display=block>\begin{align}

   \theta_{35} &= 180^\circ - \arccos\sqrt{\frac{-(4\varphi + 8)\xi^2 - (4\varphi + 8)\xi + 12\varphi + 19}{15}} \\[2pt]
   &\approx 152.929\,920\,275\,84^\circ.

\end{align}</math>

Metric propertiesEdit

For a snub dodecahedron whose edge length is 1, the surface area is <math display=block>A = 20\sqrt{3} + 3\sqrt{25+10\sqrt{5}} \approx 55.286\,744\,958\,445\,15.</math> Its volume is <math display=block>V= \frac{(3\varphi+1)\xi^2+(3\varphi+1)\xi-\varphi/6-2}{\sqrt{3\xi^2-\varphi^2}}

\approx 37.616\,649\,962\,733\,36.</math>

Alternatively, this volume may be written as <math display=block>\begin{align}

 V &= \frac{5+5\sqrt{5}}{6\sqrt{3}}\sqrt{{{18+6\sqrt{5}}} + {a \left({3+3\sqrt{5}} + a \right)} }

+\frac{5+3\sqrt{5}}{24\sqrt{2}}\sqrt{ 72 + {\left({5+\sqrt{5}}\right)}a \left({{3+3\sqrt{5}}} + a \right)} \\[2pt]

 &\approx 37.616\,649\,962\,733\,36,

\end{align}</math> where <math display=block>\begin{align}

 a &= \sqrt[3]{54(1+\sqrt{5})+6\sqrt{102+162\sqrt{5}}}+\sqrt[3]{54(1+\sqrt{5})-6\sqrt{102+162\sqrt{5}}} \\[2pt]
 &\approx 10.293\,368\,998\,184\,21.

\end{align}</math> Its circumradius is <math display=block>R = \frac12\sqrt{\frac{2-\xi}{1-\xi}} \approx 2.155\,837\,375.</math> Its midradius is <math display=block>r=\frac{1}{2}\sqrt{\frac{1}{1-\xi}}\approx 2.097\,053\,835\,25.</math>

There are two inscribed spheres, one touching the triangular faces, and one, slightly smaller, touching the pentagonal faces. Their radii are, respectively: <math display=block>\begin{align}

 r_3 &= \frac{\varphi\sqrt3}{6\xi}\sqrt{\frac{1}{1-\xi}}\approx 2.077\,089\,659\,74 \\[4pt]
 r_5 &= \frac12\sqrt{\varphi^2\xi^2 + 3\varphi^2\xi + \frac{11}{5}\varphi + \frac{12}{5}}\approx 1.980\,915\,947\,28.

\end{align}</math>Alternatively, r₃ can be expressed as a real root of this polynomial:

2985984 x¹² - 14183424 x¹⁰ + 5723136 x⁸ - 478656 x⁶ + 12528 x⁴ - 360 x² + 1

The four positive real roots of the sextic equation in R² <math display="block">4096R^{12} - 27648R^{10} + 47104R^8 - 35776R^6 + 13872R^4 - 2696R^2 + 209 = 0</math> 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 36Template:Pi (where this constant makes the sphericity of a sphere equal to 1), the sphericity of the snub dodecahedron is about 0.947.<ref>Template:Citation</ref>

Orthogonal projectionsEdit

File:Polyhedron snub 12-20 left from vertex.png

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	File:Polyhedron snub 12-20 left from yellow max.png	File:Polyhedron snub 12-20 left from red max.png	File:Polyhedron snub 12-20 left from blue max.png
Wireframe	File:Snub dodecahedron A2.png	File:Snub dodecahedron H2.png	File:Snub dodecahedron e1.png
Projective symmetry	[3]	[5]	[2]
Dual	File:Dual snub dodecahedron A2.png	File:Dual snub dodecahedron H2.png	File:Dual snub dodecahedron e1.png

Geometric relationsEdit

Template:Multiple image Template:Multiple image

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.

Alternatively, combining the vertices of the snub dodecahedron given by the Cartesian coordinates (above) and its mirror will form a semiregular truncated icosidodecahedron. The comparisons between these regular and semiregular polyhedrons is shown in the figure to the right.

Cartesian coordinates for the vertices of this alternative snub dodecahedron are obtained by selecting sets of 12 (of 24 possible even permutations contained in the five sets of truncated icosidodecahedron Cartesian coordinates). The alternations are those with an odd number of minus signs in these three sets:

Template:Multiple image <math display=block>\begin{array}{ccccccc}

 \Bigl(& \pm \tfrac{1}{\varphi} &,& \pm \tfrac{1}{\varphi} &,& \pm [3 + \varphi] & \Bigr), \\[2pt]
 \Bigl(& \pm \tfrac{1}{\varphi} &,& \pm\,\varphi^2 &,& \pm [3\varphi - 1] & \Bigr),  \\[2pt]
 \Bigl(& \pm [2\varphi - 1] &,& \pm\,2 &,& \pm [2 + \varphi] & \Bigr),

\end{array}</math> and an even number of minus signs in these two sets: <math display=block>\begin{array}{ccccccc}

 \Bigl(& \pm \tfrac{2}{\varphi} &,& \pm\,\varphi &,& \pm [1 + 2\varphi] & \Bigr), \\[2pt]
 \Bigl(& \pm\,\varphi &,& \pm\,3 &,& \pm\,2\varphi & \Bigr),

\end{array}</math>

where <math>\varphi = \tfrac{1 + \sqrt 5}{2}</math> is the golden ratio. The mirrors of both the regular truncated icosidodecahedron and this alternative snub dodecahedron are obtained by switching the even and odd references to both sign and position permutations.

Related polyhedra and tilingsEdit

Template:Icosahedral truncations

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 Template:CDD. 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.

Template:Snub table