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{{more footnotes|date=January 2012}}
{{Semireg polyhedra db|Semireg polyhedron stat table|nCO}}
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]].


==Dimensions==
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>.


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==
[[Cartesian coordinates]] for the [[vertex (geometry)|vertices]] of a snub cube are all the [[even permutation]]s of
:(±1, ±ξ, ±1/ξ)
 
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
:<math>\xi^3+\xi^2+\xi=1, \,</math>
 
which can be written
:<math>\xi = \frac{1}{3}\left(\sqrt[3]{17+3\sqrt{33}} - \sqrt[3]{-17+3\sqrt{33}} - 1\right)</math>
 
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.
 
This snub cube has edges of length α, a number which satisfies the equation
:<math>\alpha^6-4\alpha^4+16\alpha^2-32=0, \,</math>
 
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>
:<math>\beta = \sqrt[3]{13+3\sqrt{33}}</math>
 
For a snub cube with unit edge length, use all the even permutations of
:<math>(\pm C_1,\pm C_2,\pm C_3)</math>
having an even number of plus signs, along with all the odd permutations having an odd number of plus signs.
:<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>
:<math>C_3=\sqrt{\frac{4 + c_3 + c_4}{12}}\approx 1.14261</math>
:<math>c_1=\sqrt[3]{3\sqrt{33}+17}</math>
:<math>c_2=\sqrt[3]{3\sqrt{33}-17}</math>
:<math>c_3=\sqrt[3]{199+3\sqrt{33}}</math>
:<math>c_4=\sqrt[3]{199-3\sqrt{33}}</math>
 
==Orthogonal projections==
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.
{|class=wikitable width=500
|+ Orthogonal projections
|-
!Centered by
!Face<br>Triangle
!Face<br>Square
!Edge
|-
!Image
|[[Image:Snub cube_A2.png|160px]]
|[[Image:Snub cube_B2.png|160px]]
|[[Image:Snub cube_e1.png|160px]]
|- align=center
!Projective<BR>symmetry
|[3]<sup>+</sup>
|[4]<sup>+</sup>
|[2]
|}
 
==Geometric relations==
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.
{|class="wikitable"
|align=center|[[Image:Hexahedron.png|120px]]<br>Cube
|align=center|[[Image:Small rhombicuboctahedron.png|120px]]<br>[[Rhombicuboctahedron]]<br>(''Expanded cube'')
|align=center|[[Image:Snub hexahedron.png|120px]]<br>Snub cube
|}
 
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.
:[[Image:Snubcubes in grCO.svg|425px]]
 
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>
 
==Related polyhedra and tilings==
The snub cube is one of a family of uniform polyhedra related to the cube and regular octahedron.
 
{{Octahedral 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}}
 
The ''snub cube'' is second in a series of snub polyhedra and tilings with [[vertex figure]] 3.3.4.3.''n''.
{{Snub4 table}}
 
==See also==
*[[Truncated cube]]
*[[Snub square tiling]]
 
==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}}
*{{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=SnubCube |title=Snub cube |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|s3s4s - snic}}
*[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.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]
 
{{Archimedean solids}}
{{Polyhedron navigator}}
 
[[Category:Chiral polyhedra]]
[[Category:Uniform polyhedra]]
[[Category:Archimedean solids]]

Latest revision as of 23:22, 5 January 2015


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