, a cube is a threedimensional
solid object bounded by six square
faces, facet
s or sides, with three meeting at each 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
, of rectangular parallelepiped
and of trigonal trapezohedron
. The cube is dual
to the octahedron
. It has cubical symmetry (also called octahedral symmetry
). It is special by being a cuboid
and a rhombohedron
.
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_{0}, x_{1}, x_{2}) with −1 < x_{ i} < 1.
For a cube of edge length ,
As the volume of a cube is the third power of its sides a×a×a, third power
s are called cubes, by analogy with squares and second powers.
A cube has the largest volume among cuboid
s (rectangular boxes) with a given surface area
.
My parents are these people, I live with them…I'm boring.
I'm just a guy. I work in an office building, doing office building stuff.
I don't wanna die, I'm just being realistic. You think they'd go to all the trouble to build this thing if we could just walk out?
Do you think we matter? We don't.
I mean, this is an accident, a forgotten, perpetual public works project. Do you think anybody wants to ask questions? All they want is a clear conscience and a fat paycheck.
I mean, nobody wants to see the big picture. Life's too complicated.
Just out of curiosity—I mean, don't hit me again, I think—but what are you going to do when you get there?
Let's rule out aliens for now and concentrate on what we know.
You can't see the big picture from in here, so don't try. Keep your head down, keep it simple. Just look at what's in front of you.
Leaven…you beautiful brain.
, a cube is a threedimensional
solid object bounded by six square
faces, facet
s or sides, with three meeting at each 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
, of rectangular parallelepiped
and of trigonal trapezohedron
. The cube is dual
to the octahedron
. It has cubical symmetry (also called octahedral symmetry
). It is special by being a cuboid
and a rhombohedron
.
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_{0}, x_{1}, x_{2}) with −1 < x_{ i} < 1.
Formulae
For a cube of edge length ,surface area  
volume Volume Volume is the quantity of threedimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains.... 

face diagonal Face diagonal In geometry, a face diagonal of a polyhedron is a diagonal on one of the faces, in contrast to a space diagonal passing through the interior of the polyhedron.... 

space diagonal Space diagonal In a rectangular box or a magic cube, the four space diagonals are the lines that go from a corner of the box or cube, through the center of the box or cube, to the opposite corner... 

radius of circumscribed sphere Circumscribed sphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing... 

radius of sphere tangent to edges  
radius of inscribed sphere Inscribed sphere In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces... 

angles between faces Dihedral angle In geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection... 
As the volume of a cube is the third power of its sides a×a×a, third power
s are called cubes, by analogy with squares 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).
Uniform colorings and symmetry
The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123.The cube has three classes of symmetry, which can be represented by vertextransitive
coloring the faces. The highest octahedral symmetry O_{h} has all the faces the same color. The dihedral symmetry
D_{4h} comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D_{2h} 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
.
Name  Regular hexahedron  Square prism Prism (geometry) In geometry, a prism is a polyhedron with an nsided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All crosssections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a... 
Cuboid Cuboid In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature... 
Trigonal trapezohedron Trapezohedron The ngonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an ngonal antiprism. Its 2n faces are congruent kites . The faces are symmetrically staggered.The ngon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry... 

CoxeterDynkin CoxeterDynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... 

Schläfli symbol  {4,3}  {4}x{}  {}x{}x{}  
Wythoff symbol Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, LongeutHiggens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.... 
3  4 2  4 2  2   2 2 2  
Symmetry  O_{h} (*432) 
D_{4h} (*422) 
D_{2h} (*222) 
D_{3d} (2*3) 
Symmetry order  24  16  8  12 
Image (uniform coloring) 
(111) 
(112) 
(123) 
(111), (112), (122), and (222) 
Geometric relations
A cube has 11 nets(one shown above): that is, there are 11 ways to flatten a hollow cube by cutting 7 edges. To colour the cube so that no two adjacent faces have the same colour, one would need at least 3 colours.
The cube is the cell of the only regular tiling of 3 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
(every face has point symmetry).
The cube can be cut into 6 identical square pyramid
s. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron
is obtained (with pairs of coplanar triangles combined into rhombic faces.)
Other dimensions
The analogue of a cube in fourdimensional Euclidean spacehas a special name—a tesseract
or hypercube
. More properly, a hypercube (or ndimensional cube or simply ncube) is the analogue of the cube in ndimensional Euclidean space and a tesseract is the order4 hypercube. A hypercube is also called a measure polytope.
There are analogues of the cube in lower dimensions too: a point
in dimension 0, a segment in one dimension and a square in two dimensions.
Related polyhedra
The quotient of the cube by the antipodalmap yields a projective polyhedron
, the hemicube.
If the original cube has edge length 1, its dual polyhedron
(an octahedron
) has edge length .
The cube is a special case in various classes of general polyhedra:
Name  Equal edgelengths?  Equal angles?  Right angles? 

Cube  Yes  Yes  Yes 
Rhombohedron Rhombohedron In geometry, a rhombohedron is a threedimensional figure like a cube, except that its faces are not squares but rhombi. It is a special case of a parallelepiped where all edges are the same length.... 
Yes  Yes  No 
Cuboid  No  Yes  Yes 
Parallelepiped  No  Yes  No 
quadrilateral Quadrilateral In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on... ly faced hexahedron 
No  No  No 
The vertices of a cube can be grouped into two groups of four, each forming a regular 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 ⅓ of that of the cube. The remaining space consists of four equal irregular tetrahedra with a volume of 1/6 of that of the cube, each.
The rectified
cube is the cuboctahedron
. If smaller corners are cut off we get a polyhedron with 6 octagonal 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 compound of five cubes.
If two opposite corners of a cube are truncated at the depth of the 3 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.
Cube 
Truncated cube Truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices.... 
cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,... 
Truncated octahedron Truncated octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron.... 
Octahedron Octahedron In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.... 
Rhombicuboctahedron Rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles... 
truncated cuboctahedron Truncated cuboctahedron In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges... 
Snub cube Snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each... 
Stella octangula Stella octangula The stellated octahedron, or stella octangula, is the only stellation of the octahedron. It was named by Johannes Kepler in 1609, though it was known to earlier geometers... 
All these figures have 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.
Polyhedra  Euclidean tiling  Hyperbolic tiling  

[3,3]  [4,3]  [5,3]  [6,3]  [7,3]  [8,3] 
Cube Cube In geometry, a cube is a threedimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... 
Rhombic dodecahedron Rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.Properties:... 
Rhombic triacontahedron Rhombic triacontahedron In geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. It is the polyhedral dual of the icosidodecahedron, and it is a zonohedron.... 
Rhombille 
Compound of three cubes Compound of three cubes This uniform polyhedron compound is a symmetric arrangement of 3 cubes, considered as square prisms. It can be constructed by superimposing three identical cubes, and then rotating each by 45 degrees about a separate axis .This compound famously appears in the lithograph print Waterfall by M.C.... 
Compound of five cubes Compound of five cubes This polyhedral compound is a symmetric arrangement of five cubes. This compound was first described by Edmund Hess in 1876.It is one of five regular compounds, and dual to the compound of five octahedra.... 
In uniform honeycombs and polychora
It is an element of 9 of 28 convex uniform honeycombs:
Cubic honeycomb Cubic honeycomb The cubic honeycomb is the only regular spacefilling tessellation in Euclidean 3space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron.... 
Truncated square prismatic honeycomb Truncated square prismatic honeycomb The truncated square prismatic honeycomb is a spacefilling tessellation in Euclidean 3space. It is composed of octagonal prisms and cubes in a ratio of 1:1.It is constructed from a truncated square tiling extruded into prisms.... 
Snub square prismatic honeycomb Snub square prismatic honeycomb The snub square prismatic honeycomb is a spacefilling tessellation in Euclidean 3space. It is composed of cubes and triangular prisms in a ratio of 1:2.It is constructed from a Snub square tiling extruded into prisms.... 
Elongated triangular prismatic honeycomb Elongated triangular prismatic honeycomb The elongated triangular prismatic honeycomb is a spacefilling tessellation in Euclidean 3space. It is composed of cubes and triangular prisms in a ratio of 1:2.It is constructed from an elongated triangular tiling extruded into prisms.... 
Gyroelongated triangular prismatic honeycomb Gyroelongated triangular prismatic honeycomb The gyroelongated triangular prismatic honeycomb is a uniform spacefilling tessellation in Euclidean 3space. It is composed of cubes and triangular prisms in a ratio of 1:2.... 
Cantellated cubic honeycomb Cantellated cubic honeycomb The cantellated cubic honeycomb is a uniform spacefilling tessellation in Euclidean 3space. It is composed of small rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3. References :... 
Cantitruncated cubic honeycomb Cantitruncated cubic honeycomb The cantitruncated cubic honeycomb is a uniform spacefilling tessellation in Euclidean 3space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3. Uniform colorings :... 
Runcitruncated cubic honeycomb Runcitruncated cubic honeycomb The runcitruncated cubic honeycomb is a uniform spacefilling tessellation in Euclidean 3space. It is composed of small rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3.... 
Runcinated alternated cubic honeycomb Runcinated alternated cubic honeycomb The runcinated alternated cubic honeycomb is a uniform spacefilling tessellation in Euclidean 3space. It is composed of small rhombicuboctahedra, cubes, and tetrahedra in a ratio of 1:1:2. References :... 

It is also an element of five fourdimensional uniform polychora
:
Tesseract Tesseract In geometry, the tesseract, also called an 8cell or regular octachoron or cubic prism, is the fourdimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8... 
Cantellated 16cell 
Runcinated tesseract Runcinated tesseract In fourdimensional geometry, a runcinated tesseract is a convex uniform polychoron, being a runcination of the regular tesseract.... 
Cantitruncated 16cell 
Runcitruncated 16cell 
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 three dimensional kary Hamming graph, which for k = 2 is the cube graph.
Graphs of this sort occur in the theory of parallel processing
in computers.
See also
 Unit cubeUnit cubeA unit cube, sometimes called a cube of side 1, is a cube whose sides are 1 unit long. The volume of a 3dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units. Unit Hypercube :...
 TesseractTesseractIn geometry, the tesseract, also called an 8cell or regular octachoron or cubic prism, is the fourdimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
 Cube (film)Cube (film)Cube is a 1997 Canadian science fiction psychological thriller/horror film directed by Vincenzo Natali. The film was a successful product of the Canadian Film Centre's First Feature Project....
 TrapezohedronTrapezohedronThe ngonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an ngonal antiprism. Its 2n faces are congruent kites . The faces are symmetrically staggered.The ngon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry...
 Yoshimoto CubeYoshimoto CubeThe Yoshimoto Cube is a polyhedral mechanical puzzle toy invented in 1971 by Japanese man . The cube is made up of eight interconnected cubes and it is capable of folding and unfolding itself in a cyclic fashion. You can keep folding, or unfolding the cube, indefinitely. Once folded, the cube can...
 The Cube (game show)The Cube (game show)The Cube is a BAFTA Award–winning British television game show which first aired on ITV on 22 August 2009. Presented by Phillip Schofield, it offers contestants the chance to win a top prize of £250,000 by completing challenges from within a 4x4x4 metre Perspex cube...
 Prince Rupert's cubePrince Rupert's cubeIn geometry, Prince Rupert's cube is the largest cube that can pass through a hole drilled through a unit cube, i.e. through a cube whose sides have length 1. Curiously, it is slightly larger than the unit cube, with a side length of...
 OLAP cubeOLAP cubeAn OLAP cube is a data structure that allows fast analysis of data. It can also be defined as the capability of manipulating and analyzing data from multiple perspectives...