View source for Cube

{{Short description|Solid object with six equal square faces}}
{{mergefrom|Sum of two cubes|date=November 2023}}
{{Reg polyhedra db|Reg polyhedron stat table|C}}
[[File:Hexahedron.stl|thumb|3D model of a cube]]

In [[Geometry|geometry]], a '''cube'''{{efn|{{ety|la|cubus}}, {{ety|el|''κύβος'' (kubos)|a cube, a die, vertebra}}. In turn from [[Social:Proto-Indo-European language|Proto-Indo-European]] ''*keu(b)-'', "to bend, turn".}} is a [[Three-dimensional space|three-dimensional]] solid object bounded by six square faces, [[Facet (geometry)|facets]], or sides, with three meeting at each [[Vertex (geometry)|vertex]]. Viewed from a corner, it is a [[Hexagon|hexagon]] and its [[Net (polyhedron)|net]] is usually depicted as a cross.<ref>{{cite web | url=https://www.math-only-math.com/nets-of-solids.html | title=Nets of a Solids &#124; Geometry &#124;Nets of a Cube &#124;Nets of a Cone & Cylinder }}</ref>

The cube is the  only [[Regular polyhedron|regular]] [[Hexahedron|hexahedron]] and is one of the five [[Platonic solid]]s. It has 6 faces, 12 edges, and 8 vertices.

The cube is also a square [[Parallelepiped|parallelepiped]], an equilateral [[Cuboid|cuboid]], a right [[Rhombohedron|rhombohedron]], and a '''3'''-[[Zonohedron|zonohedron]]. It is a regular square [[Prism (geometry)|prism]] in three orientations, and a [[Trigonal trapezohedron|trigonal trapezohedron]] in four orientations.

The cube is [[Dual polyhedron|dual]] to the [[Octahedron|octahedron]]. It has cubical or [[Octahedral symmetry|octahedral symmetry]], and is the only convex polyhedron whose faces are all [[Square|square]]s. Its generalization for higher-dimensional spaces is called a ''[[Hypercube|hypercube]]''.

==Orthogonal projections==
The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its [[Vertex figure|vertex figure]]. The first and third correspond to the A<sub>2</sub> and B<sub>2</sub> Coxeter planes.
{|class=wikitable width=360
|+ Orthogonal projections
|-
!Centered by
!Face
!Vertex
|- align=center
!Coxeter planes
|'''B<sub>2</sub>'''<BR>[[File:2-cube.svg|100px]]
|'''A<sub>2</sub>'''<BR>[[File:3-cube t0.svg|100px]]
|- align=center
!Projective<BR>symmetry
|[4]
|[6]
|-
!Tilted views
|[[File:Cube t0 e.png|100px]]
|[[File:Cube t0 fb.png|100px]]
|}

==Spherical tiling==
The cube can also be represented as a spherical tiling, and projected onto the plane via a [[Stereographic projection|stereographic projection]]. This projection is [[Conformal map|conformal]], preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
{|class=wikitable
|- align=center valign=top
|[[File:Uniform tiling 432-t0.png|160px]]
|[[File:Cube stereographic projection.svg|160px]]
|-
![[Orthographic projection]]
![[Stereographic projection]]
|}

==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 for all ''i''.

=== As a configuration ===
This [[Platonic solid#As a configuration|configuration matrix]] represents the cube. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.{{Sfn|Coxeter|1973|loc=§1.8 Configurations|p=12}} For example, the 2 in the first column of the middle row indicates that there are 2 vertices in (i.e., at the extremes of) each edge; the 3 in the middle column of the first row indicates that 3 edges meet at each vertex.

<math>\begin{bmatrix}\begin{matrix}8 & 3 & 3 \\ 2 & 12 & 2 \\ 4 & 4 & 6 \end{matrix}\end{bmatrix}</math>

==Equation in three dimensional space==
In [[Analytic geometry|analytic geometry]], a cube's surface with center (''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>) and edge length of ''2a'' is the [[Locus (mathematics)|locus]] of all points (''x'', ''y'', ''z'') such that
:<math> \max\{ |x-x_0|,|y-y_0|,|z-z_0| \} = a.</math>

A cube can also be considered the limiting case of a 3D [[Superellipsoid|superellipsoid]] as all three exponents approach infinity.

==Formulas==
For a cube of edge length <math>a</math>:
{|class="wikitable"
|-
|surface area
|align=center|<math>6 a^2\,</math>
|[[Physics:Volume|volume]]
|align=center|<math>a^3\,</math>
|-
|[[Face diagonal|face diagonal]]
|align=center|<math>\sqrt 2a</math>
|[[Space diagonal|space diagonal]]
|align=center|<math display="inline">\sqrt 3a</math>
|-
|radius of [[Circumscribed sphere|circumscribed sphere]]
|align=center|<math>\frac{\sqrt 3}{2} a</math>
|radius of sphere tangent to edges
|align=center|<math>\frac{a}{\sqrt 2}</math>
|-
|radius of [[Inscribed sphere|inscribed sphere]]
|align=center|<math>\frac{a}{2}</math>
|[[Dihedral angle|angles between faces]] (in [[Radian|radian]]s)
|align=center|<math>\frac{\pi}{2}</math>
|}

As the volume of a cube is the third power of its sides <math>a \times a \times a</math>, third powers are called ''[[Cube (algebra)|cube]]s'', by analogy with [[Square (algebra)|square]]s and second powers.

A cube has the largest volume among [[Cuboid|cuboid]]s (rectangular boxes) with a given [[Surface area|surface area]]. Also, a cube has the largest volume among cuboids with the same total linear size (length+width+height).

===Point in space===
For a cube whose circumscribing sphere has radius ''R'', and for a given point in its 3-dimensional space with distances ''d<sub>i</sub>'' from the cube's eight vertices, we have:<ref>Park, Poo-Sung. "Regular polytope distances", [[Organization:Forum Geometricorum|Forum Geometricorum]] 16, 2016, 227-232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf {{Webarchive|url=https://web.archive.org/web/20161010184811/http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf |date=2016-10-10 }}</ref>

:<math>\frac{\sum_{i=1}^8 d_i^4}{8} + \frac{16R^4}{9} = \left(\frac{\sum_{i=1}^8 d_i^2}{8} + \frac{2R^2}{3}\right)^2. </math>

==Doubling the cube==
[[Doubling the cube]], or the ''Delian problem'', was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube. They were unable to solve this problem, which in 1837 [[Biography:Pierre Wantzel|Pierre Wantzel]] proved it to be impossible because the [[Cube root|cube root]] of 2 is not a [[Constructible number|constructible number]].

==Uniform colorings and symmetry==
[[File:Octahedral subgroup tree.png|thumb|[[Octahedral symmetry|Octahedral symmetry]] tree]]
The cube has three uniform colorings, named by the unique colors of the square faces around each vertex: 111, 112, 123.

The cube has four 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 solid, with all the six sides being different colors. The prismatic subsets D<sub>2d</sub> has the same coloring as the previous one and D<sub>2h</sub> has alternating colors for its sides for a total of three colors, paired by opposite sides. Each symmetry form has a different [[Wythoff symbol]].
{|class="wikitable"
|- align=center
!Name
!Regular<BR>hexahedron
!Square prism
!Rectangular<BR>trapezoprism
!Rectangular<BR>cuboid
![[Rhombus|Rhombic]]<BR>prism
!Trigonal<BR>[[Trapezohedron|trapezohedron]]
|- align=center
!Coxeter<BR>diagram
|{{CDD|node_1|4|node|3|node}}
|{{CDD|node_1|4|node|2|node_1}}
|{{CDD|node_1|4|node_h|2x|node_h}}
|{{CDD|node_1|2|node_1|2|node_1}}
|{{CDD|node_1|2|node_f1|2x|node_f1}}
|{{CDD|node_fh|2x|node_fh|6|node}}
|- align=center
![[Schläfli symbol|Schläfli<BR>symbol]]
|{4,3}
|{4}×{ }<BR>rr{4,2}
|s<sub>2</sub>{2,4}
|{ }<sup>3</sup><BR>tr{2,2}
|{ }×2{ }
|
|- align=center
![[Wythoff symbol|Wythoff<BR>symbol]]
|3 &#124; 4 2
|4 2 &#124; 2
|
|2 2 2 &#124;
|
|
|- align=center
![[List of spherical symmetry groups|Symmetry]]
|O<sub>h</sub><br>[4,3]<BR>(*432)
|D<sub>4h</sub><br>[4,2]<BR>(*422)
|D<sub>2d</sub><br>[4,2<sup>+</sup>]<BR>(2*2)
|colspan=2|D<sub>2h</sub><br>[2,2]<BR>(*222)
|D<sub>3d</sub><br>[6,2<sup>+</sup>]<BR>(2*3)
|- align=center
!Symmetry<BR>order
|24
|16
|8
|colspan=2|8
|12
|- align=center
!Image<br>(uniform<BR>coloring)
|[[Image:Hexahedron.png|80px]]<br>(111)
|[[Image:Tetragonal prism.png|80px]]<br>(112)
|[[File:Cube rotorotational symmetry.png|80px]]<BR>(112)
|[[Image:Uniform polyhedron 222-t012.png|80px]]<br>(123)
|[[File:Cube_rhombic_symmetry.png|80px]]<BR>(112)
|[[File:Trigonal trapezohedron.png|80px]]<br>(111), (112)
|}

==Geometric relations==
[[File:The 11 cubic nets.svg|thumb|upright=1.2|right|The 11 nets of the cube]]
[[File:Cubo desarrollo.gif|thumb|upright|Net of a cube folding into 3 dimensions]]
A cube has eleven [[Net (polyhedron)|nets]]: that is, there are eleven ways to flatten a hollow cube by cutting seven edges.<ref>{{cite book
 | last = Uehara | first = Ryuhei
 | contribution = Figure 1.1
 | doi = 10.1007/978-981-15-4470-5
 | isbn = 978-981-15-4469-9
 | location = Singapore
 | mr = 4215620
 | page = 4
 | publisher = Springer
 | title = Introduction to Computational Origami: The World of New Computational Geometry
 | year = 2020| s2cid = 220150682
 }}</ref> To color the cube so that no two adjacent faces have the same color, one would need at least three colors.

The cube is the cell of [[Cubic honeycomb|the only regular tiling of three-dimensional Euclidean space]]. 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|zonohedron]] (every face has point symmetry).

The cube can be cut into six identical [[Square pyramid|square pyramid]]s. If these square pyramids are then attached to the faces of a second cube, a [[Rhombic dodecahedron|rhombic dodecahedron]] is obtained (with pairs of coplanar triangles combined into rhombic faces).

==In theology==
Cubes appear in [[Religion:Abrahamic religions|Abrahamic religions]]. The [[Religion:Kaaba|Kaaba]] (Arabic for 'cube') in Mecca is one example. Cubes also appear in Judaism as tefillin, and the [[Place:New Jerusalem|New Jerusalem]] is described in the New Testament as a cube.<ref>{{cite web | url=https://www.deilataylor.com/symbolism-of-the-cube/ | title=Symbolism of the Cube • Eve Out of the Garden | date=30 October 2020 }}</ref>

==Other dimensions==
The analogue of a cube in four-dimensional [[Euclidean space]] has a special name—a [[Tesseract|tesseract]] or [[Hypercube|hypercube]]. More properly, a hypercube (or ''n''-dimensional cube or simply ''n''-cube) is the analogue of the cube in ''n''-dimensional Euclidean space and a tesseract is the order-4 hypercube. A hypercube is also called a ''measure polytope''.

There are analogues of the cube in lower dimensions too: a [[Point (geometry)|point]] in dimension 0, a [[Line segment|line segment]] in one dimension and a square in two dimensions.

==Related polyhedra==
[[Image:Dual Cube-Octahedron.svg|thumb|upright=.9|right|The dual of a cube is an [[Octahedron|octahedron]], seen here with vertices at the center of the cube's square faces.]]
[[File:Hemicube.svg|upright=.9|thumb|The hemicube is the 2-to-1 quotient of the cube.]]
The quotient of the cube by the [[Antipodal point|antipodal]] map yields a projective polyhedron, the hemicube.

If the original cube has edge length 1, its [[Dual polyhedron|dual polyhedron]] (an [[Octahedron|octahedron]]) has edge length <math>\scriptstyle \sqrt{2}/2</math>.

The cube is a special case in various classes of general polyhedra:
{|class=wikitable
!Name!!Equal edge-lengths?!!Equal angles?!!Right angles?
|-
|'''Cube'''||'''Yes'''||'''Yes'''||'''Yes'''
|-
|[[Rhombohedron]]||Yes||Yes||No
|-
|[[Cuboid]]||No||Yes||Yes
|-
|[[Parallelepiped]]||No||Yes||No
|-
|[[Quadrilateral|quadrilateral]]ly faced hexahedron||No||No||No
|}

The vertices of a cube can be grouped into two groups of four, each forming a regular [[Tetrahedron|tetrahedron]]; more generally this is referred to as a demicube. These two together form a regular 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 {{sfrac|1|3}} of that of the cube. The remaining space consists of four equal irregular tetrahedra with a volume of {{sfrac|1|6}} of that of the cube, each.

The [[Rectification (geometry)|rectified]] cube is the [[Cuboctahedron|cuboctahedron]]. If smaller corners are cut off we get a polyhedron with six [[Octagon|octagon]]al faces and eight triangular ones. In particular we can get regular octagons ([[Truncated cube|truncated cube]]). The [[Rhombicuboctahedron|rhombicuboctahedron]] is obtained by cutting off both corners and edges to the correct amount.

A cube can be inscribed in a [[Dodecahedron|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 compound of five cubes.

If two opposite corners of a cube are truncated at the depth of the three vertices directly connected to them, an irregular octahedron is obtained. Eight of these irregular octahedra can be attached to the triangular faces of a regular octahedron to obtain the cuboctahedron.

The cube is topologically related to a series of spherical polyhedral and tilings with order-3 [[Vertex figure|vertex figure]]s.

The cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.

The cube is topologically related as a part of sequence of regular tilings, extending into the [[List of regular polytopes and compounds#Hyperbolic tilings|hyperbolic plane]]: {4,p}, p=3,4,5...
{{Regular square tiling table}}

With dihedral symmetry, Dih<sub>4</sub>, the cube is topologically related in a series of uniform polyhedral and tilings 4.2n.2n, extending into the hyperbolic plane:

All these figures have [[Octahedral symmetry|octahedral symmetry]].

The cube is a part of a sequence of rhombic polyhedra and tilings with [''n'',3] [[Coxeter group]] symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.

The cube is a [[Prism (geometry)|square prism]]:

As a [[Trapezohedron|trigonal trapezohedron]], the cube is related to the hexagonal dihedral symmetry family.

{|class=wikitable
|+ Regular and uniform compounds of cubes
|- align=center valign=top
|[[Image:UC08-3 cubes.png|100px]]<br>[[Compound of three cubes]]
|[[Image:Compound of five cubes.png|100px]]<br>[[Compound of five cubes]]
|}

===In uniform honeycombs and polychora===
It is an element of 9 of 28 [[Convex uniform honeycomb|convex uniform honeycomb]]s: 
{|class=wikitable width=500
|- align=center valign=top
|[[Cubic honeycomb]]<br>{{CDD|node_1|4|node|3|node|4|node}}<br>{{CDD|node_1|4|node|4|node|2|node_1|infin|node}}
|Truncated square prismatic honeycomb<br>{{CDD|node_1|4|node_1|4|node|2|node_1|infin|node}}
|[[Snub square prismatic honeycomb]]<br>{{CDD|node_h|4|node_h|4|node_h|2|node_1|infin|node}}
|Elongated triangular prismatic honeycomb
|Gyroelongated triangular prismatic honeycomb
|- align=center
|[[File:Partial cubic honeycomb.png|100px]]
|[[File:Truncated square prismatic honeycomb.png|100px]]
|[[File:Snub square prismatic honeycomb.png|100px]]
|[[File:Elongated triangular prismatic honeycomb.png|100px]]
|[[File:Gyroelongated triangular prismatic honeycomb.png|100px]]
|- align=center
|Cantellated cubic honeycomb<br>{{CDD|node|4|node_1|3|node|4|node_1}}
|Cantitruncated cubic honeycomb<br>{{CDD|node|4|node_1|3|node_1|4|node_1}}
|Runcitruncated cubic honeycomb<br>{{CDD|node_1|4|node|3|node_1|4|node_1}}
|Runcinated alternated cubic honeycomb<br>{{CDD|nodes_10ru|split2|node|4|node_1}}
|- align=center
|[[File:HC A5-A3-P2.png|100px]]
|[[File:HC A6-A4-P2.png|100px]]
|[[File:HC A5-A2-P2-Pr8.png|100px]]
|[[File:HC A5-P2-P1.png|100px]]
|}

It is also an element of five four-dimensional uniform polychora: 
{|class=wikitable width=500
|- align=center
|[[Tesseract]]<br>{{CDD|node_1|4|node|3|node|3|node}}
|Cantellated 16-cell<br>{{CDD|node|4|node_1|3|node|3|node_1}}
|Runcinated tesseract<br>{{CDD|node_1|4|node|3|node|3|node_1}}
|Cantitruncated 16-cell<br>{{CDD|node|4|node_1|3|node_1|3|node_1}}
|Runcitruncated 16-cell<br>{{CDD|node_1|4|node|3|node_1|3|node_1}}
|- align=center
|[[File:4-cube t0.svg|100px]]
|[[File:24-cell t1 B4.svg|100px]]
|[[File:4-cube t03.svg|100px]]
|[[File:4-cube t123.svg|100px]]
|[[File:4-cube t023.svg|100px]]
|}

==Cubical graph==
{{Infobox graph
 | name = Cubical graph
 | image = [[File:3-cube column graph.svg|160px]]
 | image_caption = 
 | namesake = [[Hypercube graph|Q<sub>3</sub>]]
 | vertices = 8
 | edges = 12
 | automorphisms = 48
 | radius = 3
 | diameter = 3
 | girth = 4
 | chromatic_number = 2
 | chromatic_index = 
 | fractional_chromatic_index = 
 | properties = Hamiltonian, [[Regular graph|regular]], [[Symmetric graph|symmetric]], [[Distance-regular graph|distance-regular]], [[Distance-transitive graph|distance-transitive]], [[K-vertex-connected graph|3-vertex-connected]], [[Bipartite graph|bipartite]], [[Planar graph|planar graph]]
}}
The [[N-skeleton|skeleton]] of the cube (the vertices and edges) forms a [[Graph (discrete mathematics)|graph]] with 8 vertices and 12 edges, called the '''cube graph'''. It is a special case of the [[Hypercube graph|hypercube graph]].<ref>{{cite journal
 | last2 = Hayes | first2 = John P.
 | last3 = Wu | first3 = Horng-Jyh
 | doi = 10.1016/0898-1221(88)90213-1
 | issue = 4
 | journal = Computers & Mathematics with Applications
 | mr = 949280
 | pages = 277–289
 | title = A survey of the theory of hypercube graphs
 | volume = 15
 | year = 1988| url = http://deepblue.lib.umich.edu/bitstream/handle/2027.42/27522/0000566.pdf?sequence=1
 | hdl = 2027.42/27522
 | hdl-access = free
 }}</ref> It is one of 5 [[Platonic graph]]s, each a skeleton of its [[Platonic solid]].

An extension is the three 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.

{{Clear}}

==See also==
* [[Pyramid (geometry)|Pyramid]]
* [[Tesseract]]
* [[Trapezohedron]]
* Prince Rupert's cube

==Notes==
{{notelist}}

==References==
{{reflist}}

===Works cited===
* {{Cite book |last=Coxeter |first=H.S.M. |year=1973 |title=Regular Polytopes |publisher=Dover |place=New York |edition=3rd |title-link=Regular Polytopes (book) |pages=[https://archive.org/details/regularpolytopes00coxe_869/page/n140 122]–123}}

==External links==
*{{mathworld |urlname=Cube |title=Cube}}
*[https://web.archive.org/web/20071009235233/http://polyhedra.org/poly/show/1/cube Cube: Interactive Polyhedron Model]*
*[http://www.mathopenref.com/cubevolume.html Volume of a cube], with interactive animation
*[http://www.software3d.com/Cube.php Cube] (Robert Webb's site)
{{Convex polyhedron navigator|state=collapsed}}
{{Polytopes|state=collapsed}}

[[Category:Platonic solids]]
[[Category:Prismatoid polyhedra]]
[[Category:Space-filling polyhedra]]
[[Category:Volume]]
[[Category:Zonohedra]]
[[Category:Elementary shapes]]
[[Category:Cubes| ]]

{{Sourceattribution|Cube}}