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{{Semireg polyhedra db|Semireg polyhedron stat table|nID}}
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 any convex uniform polyhedron other than [[Prism (geometry)|prism]]s and [[antiprism]]s), of which 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.


==Cartesian coordinates==
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[[Cartesian coordinates]] for the vertices of a snub dodecahedron are all the [[even permutation]]s of
:(±2α, ±2, ±2β),
:(±(α+β/τ+τ), ±(−ατ+β+1/τ), ±(α/τ+βτ−1)),
:(±(−α/τ+βτ+1), ±(−α+β/τ−τ), ±(ατ+β−1/τ)),
:(±(−α/τ+βτ−1), ±(α−β/τ−τ), ±(ατ+β+1/τ)) and
:(±(α+β/τ−τ), ±(ατ−β+1/τ), ±(α/τ+βτ+1)),
 
with an even number of plus signs, where
:α = ξ − 1 / ξ
and
:β = ξτ + τ<sup>2</sup> + τ /ξ,
where τ = (1 + √5) / 2 is the [[golden ratio]] and ξ is the real solution to ξ<sup>3</sup> − 2ξ = τ, which is the number:
:<math>\xi = \sqrt[3]{\frac{\tau}{2} + \frac{1}{2}\sqrt{\tau - \frac{5}{27}}} + \sqrt[3]{\frac{\tau}{2} - \frac{1}{2}\sqrt{\tau - \frac{5}{27}}}</math>
 
or approximately 1.7155615.
 
This snub dodecahedron has an edge length of approximately 6.0437380841.
 
Taking the [[even permutation]]s of the above coordinates with an odd number of plus signs gives another form, the [[Chirality (mathematics)|enantiomorph]] of the other one.
 
==Surface area and volume==
For a snub dodecahedron whose edge length is 1, the surface area is
:<math>A = 20\sqrt{3} + 3\sqrt{25+10\sqrt{5}} \approx 55.28674495844515</math>
and the volume is
:<math>V= \frac{12\xi^2(3\tau+1)-\xi(36\tau+7)-(53\tau+6)}{6\sqrt{3-\xi^2}^3} \approx 37.61664996273336</math>
where τ is the [[golden ratio]].
 
The snub dodecahedron has the highest [[sphericity]] of all Archimedean solids.
 
==Orthogonal projections==
The ''snub dodecahedron '' has two special [[orthogonal projection]]s, centered, on two types of faces: triangles, and pentagons, correspond 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
|-
!Image
|[[Image:Snub dodecahedron A2.png|160px]]
|[[Image:Snub dodecahedron H2.png|160px]]
|[[Image:Snub dodecahedron e1.png|160px]]
|- align=center
!Projective<BR>symmetry
|[3]<sup>+</sup>
|[5]<sup>+</sup>
|[2]
|}
 
==Geometric relations==
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, only add the triangle faces and leave the square gaps empty. 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.
{|class="wikitable"
|[[Image:Dodecahedron.png|120px]]<br>Dodecahedron
|[[Image:Small rhombicosidodecahedron.png|120px]]<br>Rhombicosidodecahedron<br>(''Expanded dodecahedron'')
|[[Image:Snub dodecahedron cw.png|120px]]<br>Snub dodecahedron
|}
 
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, because its edges are of unequal lengths; some deformation is required to transform it into a uniform polyhedron.
 
[[Archimedes]], an ancient Greek who showed major interest in polyhedral shapes, wrote a treatise on thirteen semi-regular solids. The snub dodecahedron is one of them.
 
==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 (n32) rotational [[Orbifold notation|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 [[digon]]s.
 
{{Snub table}}
 
==See also==
*[[:Image:Snubdodecahedronccw.gif|ccw]] and [[:Image:Snubdodecahedroncw.gif|cw]] spinning snub dodecahedron
 
==References==
*{{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}}
*{{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}}
*{{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
*[http://www.mocpages.com/moc.php/253958 The Snub Dodecahedron made with LEGO] by Antonio Nicassio (ITALY)
 
{{Archimedean solids}}
{{Polyhedron navigator}}
 
[[Category:Chiral polyhedra]]
[[Category:Uniform polyhedra]]
[[Category:Archimedean solids]]

Latest revision as of 21:34, 29 November 2014


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