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| {{more footnotes|date=January 2012}}
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| {{Semireg polyhedra db|Semireg polyhedron stat table|nCO}}
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| In [[geometry]], the '''snub cube''', or '''snub cuboctahedron''', is an [[Archimedean solid]] with 38 faces: 6 [[square (geometry)|square]]s and 32 [[equilateral triangle]]s. It has 60 [[edge (geometry)|edges]] and 24 [[vertex (geometry)|vertices]]. It is a [[chirality (mathematics)|chiral]] polyhedron, that is, it has two distinct forms, which are [[mirror image]]s (or "[[Chirality (mathematics)|enantiomorphs]]") of each other. The only other chiral Archimedean solid is the [[snub dodecahedron]].
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| ==Dimensions==
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| For a snub cube with edge length 1, its surface area is <math>\scriptstyle{6+8\sqrt{3}}</math> and its volume is <math>\sqrt{\tfrac{613t+203}{9(35t-62)}}</math>, where ''t'' is the [[Generalizations of Fibonacci numbers|tribonacci constant]] <math>\tfrac{1}{3}\scriptstyle{\left(1+\sqrt[3]{19-3\sqrt{33}}+\sqrt[3]{19+3\sqrt{33}}\right) \approx 1.83929}</math>.
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| If the original snub cube has edge length 1, its dual [[pentagonal icositetrahedron]] has side lengths <math>\tfrac{1}{\sqrt{t+1}} \scriptstyle{\approx 0.593465}</math> and <math>\tfrac{1}{2}\scriptstyle{\sqrt{t+1} \approx 0.842509}</math>.
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| ==Cartesian coordinates==
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| [[Cartesian coordinates]] for the [[vertex (geometry)|vertices]] of a snub cube are all the [[even permutation]]s of
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| :(±1, ±ξ, ±1/ξ)
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| with an even number of plus signs, along with all the [[odd permutation]]s with an odd number of plus signs, where ξ is the real solution to
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| :<math>\xi^3+\xi^2+\xi=1, \,</math>
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| which can be written
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| :<math>\xi = \frac{1}{3}\left(\sqrt[3]{17+3\sqrt{33}} - \sqrt[3]{-17+3\sqrt{33}} - 1\right)</math>
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| or approximately 0.543689. ξ is the reciprocal of the [[Generalizations of Fibonacci numbers|tribonacci constant]]. Taking the even permutations with an odd number of plus signs, and the odd permutations with an even number of plus signs, gives a different snub cube, the mirror image.
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| This snub cube has edges of length α, a number which satisfies the equation
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| :<math>\alpha^6-4\alpha^4+16\alpha^2-32=0, \,</math>
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| and can be written as | |
| :<math>\alpha = \sqrt{\frac{4}{3}-\frac{32}{6\sqrt[3]{2}\beta}+\frac{6\sqrt[3]{2}\beta}{9}}\approx1.60972</math>
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| :<math>\beta = \sqrt[3]{13+3\sqrt{33}}</math>
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| For a snub cube with unit edge length, use all the even permutations of
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| :<math>(\pm C_1,\pm C_2,\pm C_3)</math>
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| having an even number of plus signs, along with all the odd permutations having an odd number of plus signs.
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| :<math>C_1=\sqrt{\frac{4 - c_1 + c_2}{12}}\approx 0.337754</math> | |
| :<math>C_2=\sqrt{\frac{2 + c_1 - c_2}{12}}\approx 0.621226</math>
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| :<math>C_3=\sqrt{\frac{4 + c_3 + c_4}{12}}\approx 1.14261</math>
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| :<math>c_1=\sqrt[3]{3\sqrt{33}+17}</math>
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| :<math>c_2=\sqrt[3]{3\sqrt{33}-17}</math>
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| :<math>c_3=\sqrt[3]{199+3\sqrt{33}}</math>
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| :<math>c_4=\sqrt[3]{199-3\sqrt{33}}</math>
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| ==Orthogonal projections==
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| The ''snub cube'' has two special [[orthogonal projection]]s, centered, on two types of faces: triangles, and squares, correspond to the A<sub>2</sub> and B<sub>2</sub> [[Coxeter plane]]s.
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| {|class=wikitable width=500
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| |+ Orthogonal projections
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| |-
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| !Centered by
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| !Face<br>Triangle
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| !Face<br>Square
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| !Edge
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| |-
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| !Image
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| |[[Image:Snub cube_A2.png|160px]]
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| |[[Image:Snub cube_B2.png|160px]]
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| |[[Image:Snub cube_e1.png|160px]]
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| |- align=center
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| !Projective<BR>symmetry
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| |[3]<sup>+</sup>
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| |[4]<sup>+</sup>
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| |[2]
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| |}
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| ==Geometric relations==
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| The snub cube can be generated by taking the six faces of the cube, [[Expansion (geometry)|pulling them outward]] so they no longer touch, then giving them each a small rotation on their centers (all clockwise or all counter-clockwise) until the spaces between can be filled with [[equilateral triangle]]s.
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| {|class="wikitable"
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| |align=center|[[Image:Hexahedron.png|120px]]<br>Cube
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| |align=center|[[Image:Small rhombicuboctahedron.png|120px]]<br>[[Rhombicuboctahedron]]<br>(''Expanded cube'')
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| |align=center|[[Image:Snub hexahedron.png|120px]]<br>Snub cube
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| |}
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| It can also be constructed as an [[Alternation (geometry)|alternation]] of a nonuniform [[truncated cuboctahedron|omnitruncated cube]], deleting every other vertex and creating new triangles at the deleted vertices. A properly proportioned ([[Uniform polyhedron|nonuniform]]) great rhombicuboctahedron will create equilateral triangles at the deleted vertices. Depending on which set of vertices are alternated, the resulting snub cube can have a clockwise or counterclockwise twist.
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| :[[Image:Snubcubes in grCO.svg|425px]]
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| A "improved" snub cube, with a slightly smaller square face and slightly larger triangular faces compared to Archimedes' uniform snub cube, is useful as a [[spherical design]].<ref>[http://neisloane.com/sphdesigns/ "Spherical Designs"] by R.H. Hardin and [[Neil Sloane|N.J.A. Sloane]]</ref>
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| ==Related polyhedra and tilings==
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| The snub cube is one of a family of uniform polyhedra related to the cube and regular octahedron.
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| {{Octahedral truncations}}
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| 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.
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| {{Snub table}}
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| The ''snub cube'' is second in a series of snub polyhedra and tilings with [[vertex figure]] 3.3.4.3.''n''. | |
| {{Snub4 table}}
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| ==See also==
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| *[[Truncated cube]]
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| *[[Snub square tiling]]
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| ==References==
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| {{reflist}}
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| *{{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}}
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| *{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
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| *{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79-86 ''Archimedean solids''|isbn=0-521-55432-2}}
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| ==External links==
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| *{{mathworld2 |urlname=SnubCube |title=Snub cube |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
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| *{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|s3s4s - snic}}
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| *[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
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| *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
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| *[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=KPFQTjUF59q9qFlmEqGmbfyT4Ykrpg7vn7pPKHBbttGwDk2Z6dABBNQuTy7b46U3TTtKxWPq6lgrdE2qYMNpS5ceb5Ie9K4gQt25UcMlwmW6OKK3HtK2QvnmOLGTZFLfHD7hM4GN1modJYJ5PjowXOUDwYnjnCRQFA0vsrVIwFkFiIy7Pi9foWycmqdJAnWMMpuCxwRrcdA49hnAjViEzr&name=Snub+Cube#applet Editable printable net of a Snub Cube with interactive 3D view]
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| {{Archimedean solids}}
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| {{Polyhedron navigator}}
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| [[Category:Chiral polyhedra]]
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| [[Category:Uniform polyhedra]]
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| [[Category:Archimedean solids]]
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