View source for Snub dodecahedron

{{Short description|Archimedean solid with 92 faces}}
{{Semireg polyhedra db|Semireg polyhedron stat table|nID}}
[[File:Snub dodecahedron.stl|thumb|3D model of a snub dodecahedron]]
In [[geometry]], the '''snub dodecahedron''', or '''snub icosidodecahedron''', is an [[Archimedean solid]], one of thirteen convex [[Isogonal figure|isogonal]] nonprismatic solids constructed by two or more types of [[regular polygon]] [[Face (geometry)|face]]s.

The snub dodecahedron has 92 faces (the most of the 13 Archimedean solids): 12 are [[pentagon]]s and the other 80 are [[equilateral triangle]]s. It also has 150 edges, and 60 vertices.

It has two distinct forms, which are [[mirror image]]s (or "[[Chirality (mathematics)|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 {{math|sr{5,3}.}}

==Cartesian coordinates==
Let {{math|''ξ'' ≈ {{val|0.94315125924}}}} be the real zero of the [[cubic polynomial]] {{math|''x''<sup>3</sup> + 2''x''<sup>2</sup> − ''φ''<sup>2</sup>}}, where {{mvar|φ}} is the [[golden ratio]]. Let the point {{mvar|p}} 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 matrix|rotation matrices]] {{math|''M''<sub>1</sub>}} and {{math|''M''<sub>2</sub>}} 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>
{{math|''M''<sub>1</sub>}} represents the rotation around the axis {{math|(0, 1, ''φ'')}} through an angle of {{sfrac|2{{pi}}|5}} counterclockwise, while {{math|''M''<sub>2</sub>}} being a cyclic shift of {{math|(''x'', ''y'', ''z'')}} represents the rotation around the axis {{math|(1, 1, 1)}} through an angle of {{sfrac|2{{pi}}|3}}. Then the 60 vertices of the snub dodecahedron are the 60 images of point {{mvar|p}} under repeated multiplication by {{math|''M''<sub>1</sub>}} and/or {{math|''M''<sub>2</sub>}}, iterated to convergence. (The matrices {{math|''M''<sub>1</sub>}} and {{math|''M''<sub>2</sub>}} [[Generating set of a group|generate]] the 60 rotation matrices corresponding to [[icosahedral group|the 60 rotational symmetries]] of a [[regular icosahedron]].) The coordinates of the vertices are integral linear combinations of {{math|1, ''φ'', ''ξ'', ''φξ'', ''ξ''<sup>2</sup>}} and {{math|''φξ''<sup>2</sup>}}. 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 {{math|''V''<sub>3</sub>}} 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 {{math|''V''<sub>5</sub>}} 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 [[Midsphere|midradius]] equals {{mvar|ξ}}. This gives an interesting geometrical interpretation of the number {{mvar|ξ}}. 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 {{mvar|ξ}} 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 properties==
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, ''r''<sub>3</sub>  can be expressed as a real root of this polynomial:

2985984 x<sup>12</sup> - 14183424 x<sup>10</sup> + 5723136 x<sup>8</sup> - 478656 x<sup>6</sup> + 12528 x<sup>4</sup> - 360 x<sup>2</sup> + 1 

The four positive real roots of the [[sextic]] equation in ''R''<sup>2</sup>
<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 (''U''<sub>29</sub>), [[great snub icosidodecahedron]] (''U''<sub>57</sub>), [[great inverted snub icosidodecahedron]] (''U''<sub>69</sub>), and [[great retrosnub icosidodecahedron]] (''U''<sub>74</sub>).

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{{pi}} (where this constant makes the sphericity of a sphere equal to 1), the sphericity of the snub dodecahedron is about 0.947.<ref>{{citation
 | last = Aravind | first = P. K.
 | title = How Spherical Are the Archimedean Solids and Their Duals?
 | date = March 2011
 | journal = The College Mathematics Journal
 | volume = 42
 | issue = 2
 | pages = 98&ndash;107
 | doi = 10.4169/college.math.j.42.2.098
}}</ref>

==Orthogonal projections==
[[File:Polyhedron snub 12-20 left from vertex.png|thumb|The snub dodecahedron has no [[Point reflection|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 projection]]s as shown below, centered on two types of faces: triangles and pentagons, corresponding to the A<sub>2</sub> and H<sub>2</sub> [[Coxeter plane]]s.
{|class=wikitable width=500
|+ Orthogonal projections
|-
!Centered by
!Face<br>Triangle
!Face<br>Pentagon
!Edge
|-
!Solid
|[[File:Polyhedron snub 12-20 left from yellow max.png|120px]]
|[[File:Polyhedron snub 12-20 left from red max.png|120px]]
|[[File:Polyhedron snub 12-20 left from blue max.png|120px]]
|-
!Wireframe
|[[Image:Snub dodecahedron A2.png|120px]]
|[[Image:Snub dodecahedron H2.png|120px]]
|[[Image:Snub dodecahedron e1.png|120px]]
|- align=center
!Projective<BR>symmetry
|[3]
|[5]
|[2]
|-
!Dual
|[[Image:Dual snub dodecahedron A2.png|120px]]
|[[Image:Dual snub dodecahedron H2.png|120px]]
|[[Image:Dual snub dodecahedron e1.png|120px]]
|}

==Geometric relations==
{{multiple image
 | align = right| total_width = 350
 | image1 = Polyhedron 12 max.png |width1=3652 |height1=3960
 | image2 = Polyhedron small rhombi 12-20 max.png |width2=3973 |height2=4000
 | image3 = Polyhedron snub 12-20 left max.png |width3=3966 |height3=4000
 | footer = Dodecahedron, rhombicosidodecahedron and snub dodecahedron <small>(animated [[:File:P4-A11-P5.gif|expansion]] and [[:File:A11-A13.gif|twisting]])</small>
}}
{{multiple image
 | align = right  | total_width = 350
 | image1 = Polyhedron great rhombi 12-20 subsolid snub left big.png
 | image2 = Polyhedron great rhombi 12-20 big.png
 | image3 = Polyhedron great rhombi 12-20 subsolid snub right big.png
 | footer = Uniform alternation of a truncated icosidodecahedron
}}

The ''snub dodecahedron'' can be generated by taking the twelve [[pentagon]]al faces of the [[dodecahedron]] and [[Expansion (geometry)|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 (geometry)|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 [[Snub_dodecahedron#Cartesian_coordinates|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 permutation]]s contained in the five sets of [[Truncated_icosidodecahedron#Cartesian_coordinates|truncated icosidodecahedron Cartesian coordinates]]). The alternations are those with an odd number of minus signs in these three sets: 

{{multiple image
 | align = right  | total_width = 350
 | image1 = Truncated_icosidodecahedron_comparing_regular_vs_quasi-regular_overlay_of_regular_and_quasi-regular_constructions.svg
 | image2 = Snub_decahedron_comparing_regular_vs_quasi-regular.svg
 | footer = Overlay of  regular and semiregular truncated icosidodecahedra and snub dodecahedra
}}
<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 tilings==
{{Icosahedral truncations}}

This semiregular polyhedron is a member of a sequence of [[Snub (geometry)|snubbed]] polyhedra and tilings with vertex figure (3.3.3.3.''n'') and [[Coxeter–Dynkin diagram]] {{CDD|node_h|n|node_h|3|node_h}}. These figures and their duals have (''n''32) rotational [[Orbifold notation|symmetry]], being in the Euclidean plane for ''n''&nbsp;=&nbsp;6, and hyperbolic plane for any higher ''n''. The series can be considered to begin with ''n''&nbsp;=&nbsp;2, with one set of faces degenerated into [[digon]]s.

{{Snub table}}

== Snub dodecahedral graph ==
{{Infobox graph
 | name = Snub dodecahedral graph
 | image = [[File:Snub dodecahedral graph.png|240px]]
 | image_caption = 5-fold symmetry [[Schlegel diagram]]
 | namesake = 
 | vertices = 60
 | edges = 150
 | automorphisms = 60
 | radius = 
 | diameter = 
 | girth = 
 | chromatic_number =
 | chromatic_index = 
 | fractional_chromatic_index = 
 | properties = [[hamiltonian graph|Hamiltonian]], [[regular graph|regular]]
}}
In the [[mathematics|mathematical]] field of [[graph theory]], a '''snub dodecahedral graph''' is the [[1-skeleton|graph of vertices and edges]] of the snub dodecahedron, one of the [[Archimedean solid]]s. It has 60 [[Vertex (graph theory)|vertices]] and 150 edges, and is an [[Archimedean graph]].<ref>{{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=[[Oxford University Press]]|year= 1998|page=269}}</ref>

==See also==
* Planar polygon to polyhedron transformation [[:Image:Snub Dodecahedron Animaton.gif| animation]]
* [[:Image:Snubdodecahedronccw.gif|ccw]] and [[:Image:Snubdodecahedroncw.gif|cw]] spinning snub dodecahedron

==References==
{{Reflist}}
*{{cite journal |last=Jayatilake |first=Udaya |title=Calculations on face and vertex regular polyhedra |journal=Mathematical Gazette|date=March 2005 |volume=89 |issue=514 |pages=76–81|doi=10.1017/S0025557200176818 |s2cid=125675814 }}
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
*{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79–86 ''Archimedean solids''|isbn=0-521-55432-2}}

==External links==
*{{mathworld2 |urlname=SnubDodecahedron |title=Snub dodecahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
**{{mathworld |urlname=SnubDodecahedralGraph |title=Snub dodecahedral graph}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|s3s5s - snid}}
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=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&name=Snub+Dodecahedron#applet Editable printable net of a Snub Dodecahedron with interactive 3D view]
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
*Mark S. Adams and Menno T. Kosters. [https://nbviewer.jupyter.org/url/archive.org/download/musing_math/Volume_Solutions_To_Snub_Dodecahedron.ipynb Volume Solutions to the Snub Dodecahedron]

{{Archimedean solids}}
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[[Category:Uniform polyhedra]]
[[Category:Archimedean solids]]
[[Category:Snub tilings]]