Cube
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222032189
2008-06-27T06:37:17Z
RobertCWebb
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:''This article is about the geometric shape. For other meanings of the word "cube", see [[cube (disambiguation)]].''
{{Reg polyhedra db|Reg polyhedron stat table|C}}
A '''cube'''<ref>English ''cube'' from Old French < Latin ''cubus'' < Greek ''kubos'', "a cube, a die, vertebra". In turn from [[PIE]] ''*keu(b)-'', "to bend, turn".</ref> is a [[three-dimensional space|three-dimensional]] solid object bounded by six [[square (geometry)|square]] faces, facets or sides, with three meeting at each [[wikt:vertex|vertex]]. The cube can also be called a '''regular [[hexahedron]]''' and is one of the five [[Platonic solid]]s. It is a special kind of square [[prism (geometry)|prism]], of rectangular [[parallelepiped]] and of 3-sided [[trapezohedron]]. The cube is [[dual polyhedron|dual]] to the [[octahedron]]. It has cubical symmetry (also called [[octahedral symmetry]]). A cube is the three-dimensional case of the more general concept of a [[hypercube]], which exists in any dimension.
== Cartesian coordinates ==
For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the [[Cartesian coordinates]] of the vertices are
: (±1,±1,±1)
while the interior consists of all points (x<sub>0</sub>, x<sub>1</sub>, x<sub>2</sub>) with -1 < x<sub>i</sub> < 1.
== Formulae ==
For a cube of edge length <math>a</math>,
{|class="wikitable"
|[[area (mathematics)|surface area]]
|align=center|<math>6 a^2</math>
|-
|[[volume]]
|align=center|<math>a^3</math>
|-
|radius of circumscribed sphere
|align=center|<math>\frac{{\sqrt 3} a}{2}</math>
|-
|radius of sphere tangent to edges
|align=center|<math>\frac{a}{\sqrt 2}</math>
|-
|radius of inscribed sphere
|align=center|<math>\frac{a}{2}</math>
|}
As the volume of a cube is the third power of its sides ''a''×''a''×''a'', [[third power]]s are called ''[[cube (algebra)|cube]]s'', by analogy with [[square (algebra)|square]]s and second powers.
A cube has the largest volume among [[cuboid]]s (rectangular boxes) with a given [[surface area]]. Also, a cube has the largest volume among cuboids with the same total linear size (length + width + height).
== Symmetry ==
The cube has 3 classes of symmetry, which can be represented by [[vertex-transitive]] coloring the faces. The highest [[octahedral symmetry]] O<sub>h</sub> has all the faces the same color. The [[Dihedral symmetry in three dimensions|dihedral symmetry]] D<sub>4h</sub> comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D<sub>2h</sub> is also a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different [[Wythoff symbol]].
{| class="prettytable"
|align="center"|[[Image:Uniform polyhedron 222-t012.png|160px]]<BR>(3 colors)<BR>'''| 2 2 2'''<BR>D<sub>2h</sub>
|align="center"|[[Image:Tetragonal prism.png|160px]]<BR>(2 colors)<BR>'''4 2 | 2'''<BR>D<sub>4h</sub>
|align="center"|[[Image:Hexahedron.png|160px]]<BR>(1 color)<BR>'''3 | 4 2'''<BR>O<sub>h</sub>
|}
== Geometric relations ==
[[Image:Stone Dice 17.JPG|right|thumb|150px|The familiar six-sided [[dice]] are cube shaped]]
The cube is unique among the Platonic solids for being able to tile space regularly. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a [[zonohedron]] (every face has point symmetry).
<br clear="all" />
== Other dimensions ==
The analogue of a cube in four-dimensional [[Euclidean space]] has a special name — a [[tesseract]] or (rarely) [[hypercube]].
The analogue of the cube in ''n''-dimensional Euclidean space is called a [[hypercube]] or '''''n''-dimensional cube''' or simply '''''n''-cube'''. It is also called a ''measure polytope''.
There are analogues of the cube in lower dimensions too: a [[Point (geometry)|point]] in dimension 0, a [[segment (mathematics)|segment]] in one dimension and a [[Square (geometry)|square]] in two dimensions.
==Related polyhedra==
The vertices of a cube can be grouped into two groups of four, each forming a regular [[tetrahedron]]. These two together form a regular [[polyhedral compound|compound]], the [[stella octangula]]. The intersection of the two forms a regular [[octahedron]]. The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself; the other symmetries of the cube map the two to each other.
One such regular tetrahedron has a volume of ⅓ of that of the cube. The remaining space consists of four equal irregular polyhedra with a volume of 1/6 of that of the cube, each.
The [[Rectification (geometry)|rectified]] cube is the [[cuboctahedron]]. If smaller corners are cut off we get a polyhedron with 6 [[octagon]]al faces and 8 triangular ones. In particular we can get regular octagons ([[truncated cube]]). The [[rhombicuboctahedron]] is obtained by cutting off both corners and edges to the correct amount.
A cube can be inscribed in a [[dodecahedron]] so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular [[polyhedral compound|compound]] of five cubes.
<gallery>
Image:Stella octangula.png|The tetrahedra in the cube ([[stella octangula]])
Image:Cuboctahedron.svg|The [[Rectification (geometry)|rectified]] cube ([[cuboctahedron]])
Image:Truncatedhexahedron.jpg|[[Truncated cube]]
Image:Rhombicuboctahedron.jpg|[[Rhombicuboctahedron]]
Image:UC08-3 cubes.png|[[Compound of three cubes]]
Image:Alternate truncated cube.png|An alternately truncated cube
</gallery>
All but the last of the figures shown have the same symmetries as the cube (see [[octahedral symmetry]]).
== Combinatorial cubes ==
A different kind of cube is the '''cube graph''', which is the graph of vertices and edges of the geometrical cube. It is a special case of the [[hypercube graph]].
An extension is the 3-dimensional ''k''-ary [[Hamming graph]], which for ''k'' = 2 is the cube graph.
Graphs of this sort occur in the theory of [[parallel computing|parallel processing]] in computers.
==See also==
* [[Unit cube]]
* [[Kaaba]], a large masonry structure roughly the shape of a cube
==References==
{{reflist}}
==External links==
{{Spoken Wikipedia|Cube.ogg|2006-07-07}}
* {{mathworld | urlname = Cube | title = Cube}}
*[http://www.software3d.com/Cube.php Paper model of the cube]
*[http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra]
*[http://polyhedra.org/poly/show/1/cube Cube: Interactive Polyhedron Model] -- works right in your web browser
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra]
*[http://www.korthalsaltes.com/ Paper Models of Polyhedra]
*[http://www.mathopenref.com/cubevolume.html Volume of a cube] With interactive animation
[[Category:Platonic solids]]
[[Category:Polyhedra]]
[[Category:Regular polyhedra]]
[[Category:Prismatoid polyhedra]]
[[Category:Volume]]
[[Category:Zonohedra]]
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