View source for Snub cube

{{Short description|Archimedean solid with 38 faces}}
{{infobox polyhedron
 | name = Snub cube
 | image = [[File:Snubhexahedroncw.jpg|130px]][[File:Snubhexahedronccw.jpg|130px]]
 | caption = Two different forms of a snub cube
 | type = [[Archimedean solid]]
 | faces = 38
 | edges = 60
 | vertices = 24
 | symmetry = Rotational [[octahedral symmetry]] <math> \mathrm{O} </math>
 | angle = triangle-to-triangle: 153.23° <br> triangle-to-square: 142.98°
 | dual = [[Pentagonal icositetrahedron]]
 | properties = [[Convex set|convex]], [[Chirality (mathematics)|chiral]]
 | vertex_figure = Polyhedron snub 6-8 left vertfig.svg
 | net = Polyhedron snub 6-8 left net.svg
}}
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]]. [[Kepler]] first named it in [[Latin]] as ''cubus simus'' in 1619 in his [[Harmonices Mundi]].{{r|cbg}} [[H. S. M. Coxeter]], noting it could be derived equally from the octahedron as the cube, called it '''snub cuboctahedron''', with a vertical extended [[Schläfli symbol]] <math>s \scriptstyle\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}</math>, and representing an [[Alternation (geometry)|alternation]] of a [[truncated cuboctahedron]], which has Schläfli symbol <math>t \scriptstyle\begin{Bmatrix} 4 \\ 3 \end{Bmatrix}</math>.

== Construction ==
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.{{r|holme}}

[[File:A5-A7.gif|thumb|left|150px|Process of snub cube's construction by rhombicuboctahedron]]
The snub cube may also be constructed from a [[rhombicuboctahedron]]. It started by twisting its square face (in blue), allowing its triangles (in red) to be automatically twisted in opposite directions, forming other square faces (in white) to be skewed quadrilaterals that can be filled in two equilateral triangles.{{sfnp|Conway|Burgiel|Goodman-Struss|2008|p=287&ndash;288}}

The snub cube can also be derived from the [[truncated cuboctahedron]] by the process of [[alternation (geometry)|alternation]]. 24 vertices of the truncated cuboctahedron form a polyhedron topologically equivalent to the snub cube; the other 24 form its mirror-image. The resulting polyhedron is [[vertex-transitive]] but not uniform.
{{multiple image
 | align = center  | total_width = 450
 | image1 = Polyhedron great rhombi 6-8 subsolid snub left maxmatch.png
 | image2 = Polyhedron great rhombi 6-8 max.png
 | image3 = Polyhedron great rhombi 6-8 subsolid snub right maxmatch.png
 | footer = Uniform alternation of a truncated cuboctahedron
}}

=== Cartesian coordinates ===
[[Cartesian coordinates]] for the [[vertex (geometry)|vertices]] of a snub cube are all the [[even permutation]]s of
<math display="block"> \left(\pm 1, \pm \frac{1}{t}, \pm t \right), </math>
with an even number of plus signs, along with all the [[odd permutation]]s with an odd number of plus signs, where <math> t \approx 1.83929 </math> is the [[Generalizations of Fibonacci numbers#Tribonacci numbers|tribonacci constant]].{{r|collins}} 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. Taking them together yields the [[compound of two snub cubes]].

This snub cube has edges of length <math>\alpha = \sqrt{2+4t-2t^2}</math>, a number which satisfies the equation
<math display="block">\alpha^6-4\alpha^4+16\alpha^2-32=0, </math>
and can be written as
<math display="block">\begin{align} \alpha &= \sqrt{\frac{4}{3}-\frac{16}{3\beta}+\frac{2\beta}{3}}\approx1.609\,72 \\ \beta &= \sqrt[3]{26+6\sqrt{33}}. \end{align}</math>
To get a snub cube with unit edge length, divide all the coordinates above by the value ''α'' given above.

== Properties ==
[[File:Snub cube.stl|thumb|3D model of a snub cube]]
For a snub cube with edge length <math>a</math>, its surface area and volume are:{{r|berman}}
<math display="block"> \begin{align}
 A &= \left(6+8\sqrt{3}\right)a^2 &\approx 19.856a^2 \\
 V &= \frac{8t+6}{3\sqrt{2(t^2-3)}}a^3 &\approx 7.889a^3.
\end{align}</math>

The snub cube is an [[Archimedean solid]], meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.{{r|diudea}} It is [[Chirality (mathematics)|chiral]], meaning there are two distinct forms whenever being [[Mirror image|mirrored]]. Therefore, the snub cube has the rotational [[octahedral symmetry]] <math> \mathrm{O} </math>.{{r|kocakoca|cromwell}} The polygonal faces that meet for every vertex are four equilateral triangles and one square, and the [[vertex figure]] of a snub cube is <math> 3^4 \cdot 4 </math>. The [[dual polyhedron]] of a snub cube is [[pentagonal icositetrahedron]], a [[Catalan solid]].{{r|williams}}

== Graph ==
[[File:Snub cubic graph.png|thumb|The graph of a snub cube]]
The [[Skeleton (topology)|skeleton]] of a snub cube can be represented as a [[Graph (discrete mathematics)|graph]] with 24 [[Vertex (graph theory)|vertices]] and 60 edges, 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>

==References==
{{reflist|refs=

<ref name=berman>{{cite journal
 | last = Berman | first = Martin
 | year = 1971
 | title = Regular-faced convex polyhedra
 | journal = Journal of the Franklin Institute
 | volume = 291
 | issue = 5
 | pages = 329–352
 | doi = 10.1016/0016-0032(71)90071-8
 | mr = 290245
}}</ref>

<ref name=cbg>{{cite book
 | last1 = Conway | first1 = John H.
 | last2 = Burgiel | first2 = Heidi
 | last3 = Goodman-Struss | first3 = Chaim
 | year = 2008
 | title = The Symmetries of Things
 | page = 287
 | url = https://books.google.com/books?id=Drj1CwAAQBAJ&pg=PA287
 | publisher = [[CRC Press]]
| isbn = 978-1-4398-6489-0
 }}</ref>

<ref name=collins>{{cite book
 | last = Collins | first = Julian
 | year = 2019
 | title = Numbers in Minutes
 | url = https://books.google.com/books?id=azKKDwAAQBAJ&pg=PA96
 | page = 36&ndash;37
 | publisher = Hachette
| isbn = 978-1-78747-730-8
 }}</ref>

<ref name=cromwell>{{cite book
 | last = Cromwell | first = Peter R.
 | title = Polyhedra
 | year = 1997
 | url = https://archive.org/details/polyhedra0000crom/page/386/mode/1up
 | publisher = [[Cambridge University Press]]
 | isbn = 978-0-521-55432-9
 | page = 386
}}</ref>

<ref name=diudea>{{cite book
 | last = Diudea | first = M. V.
 | year = 2018
 | title = Multi-shell Polyhedral Clusters
 | publisher = [[Springer Science+Business Media|Springer]]
 | isbn = 978-3-319-64123-2
 | doi = 10.1007/978-3-319-64123-2
 | url = https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39
 | page = 39
}}</ref>

<ref name=holme>{{cite book
 | last = Holme | first = A.
 | year = 2010
 | title = Geometry: Our Cultural Heritage
 | publisher = [[Springer Science+Business Media|Springer]]
 | url = https://books.google.com/books?id=zXwQGo8jyHUC
 | isbn = 978-3-642-14441-7
 | doi = 10.1007/978-3-642-14441-7
}}</ref>

<ref name=kocakoca>{{cite book
 | last1 = Koca | first1 = M.
 | last2 = Koca | first2 = N. O.
 | year = 2013
 | title = Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27&ndash;31 October 2010
 | contribution = Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes
 | contribution-url = https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA49
 | page = 49
 | publisher = World Scientific
}}</ref>

<ref name=williams>{{cite book
 | last = Williams | first = Robert | authorlink = Robert Williams (geometer)
 | year = 1979
 | title = The Geometrical Foundation of Natural Structure: A Source Book of Design
 | page = 85
 | publisher = Dover Publications, Inc.
 | isbn = 978-0-486-23729-9 | url = https://archive.org/details/geometricalfound00will
}}</ref>

}}
*{{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)

==External links==
*{{mathworld2 |urlname=SnubCube |title=Snub cube |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
**{{mathworld |urlname=SnubCubicalGraph |title=Snub cubical graph}}
*{{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]]
[[Category:Snub tilings]]