Cube
Background Information
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Regular Hexahedron  

(Click here for rotating model) 

Type  Platonic solid 
Elements  F = 6, E = 12 V = 8 (χ = 2) 
Faces by sides  6{4} 
Schläfli symbol  {4,3} 
Wythoff symbol  3  2 4 
Coxeter diagram  
Symmetry  O_{h}, BC_{3}, [4,3], (*432) 
Rotation group  O, [4,3]^{+}, (432) 
References  U_{06}, C_{18}, W_{3} 
Properties  Regular convex zonohedron 
Dihedral angle  90° 
4.4.4 ( Vertex figure) 
Octahedron ( dual polyhedron) 
Net 
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 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.
Orthogonal projections
The cube has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A_{2} and B_{2} Coxeter planes.
Centered by  Face  Vertex 

Coxeter planes  B_{2} 
A_{2} 
Projective symmetry 

Tilted views 
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.
Equation in R^{3}
In analytic geometry, a cube's surface with centre (x_{0}, y_{0}, z_{0}) and edge length of 2a is the locus of all points (x, y, z) such that
Formulae
For a cube of edge length ,
surface area  
volume  
face diagonal  
space diagonal  
radius of circumscribed sphere  
radius of sphere tangent to edges  
radius of inscribed sphere  
angles between faces (in radians) 
As the volume of a cube is the third power of its sides , third powers are called cubes, by analogy with squares and second powers.
A cube has the largest volume among cuboids (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 colour. The dihedral symmetry D_{4h} comes from the cube being a prism, with all four sides being the same colour. 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.
Geometric relations
A cube has eleven nets (one shown above): that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the same colour, one would need at least three colors.
The cube is the cell of the only regular tiling of threedimensional 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 six identical square pyramids. 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 space has 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 antipodal map 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  Yes  Yes  No 
Cuboid  No  Yes  Yes 
Parallelepiped  No  Yes  No 
quadrilaterally 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 ^{1}⁄_{2} 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 six octagonal faces and eight 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 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 polyhedra and tilings with order3 vertex figures.
Polyhedra  Euclidean  Hyperbolic tilings  

{2,3} 
{3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
...  (∞,3} 
The cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.
Symmetry: [4,3], (*432)  [4,3]^{+}, (432)  [1^{+},4,3], (*332)  [4,3^{+}], (3*2)  

{4,3}  t_{0,1}{4,3}  t_{1}{4,3}  t_{1,2}{4,3}  {3,4}  t_{0,2}{4,3}  t_{0,1,2}{4,3}  s{4,3}  h{4,3}  h_{1,2}{4,3} 
Duals to uniform polyhedra  
V4.4.4  V3.8.8  V3.4.3.4  V4.6.6  V3.3.3.3  V3.4.4.4  V4.6.8  V3.3.3.3.4  V3.3.3  V3.3.3.3.3 
The cube is topologically related as a part of sequence of regular tilings, extending into the hyperbolic plane: {4,p}, p=3,4,5...
{4,3} 
{4,4} 
{4,5} 
{4,6} 
{4,7} 
{4,8} 
...  {4,∞} 
With dihedral symmetry, Dih_{4}, the cube is topologically related in a series of uniform polyhedra and tilings 4.2n.2n, extending into the hyperbolic plane:
Symmetry *n42 [n,4] 
Spherical  Euclidean  Hyperbolic...  

*242 [2,4] D_{4h} 
*342 [3,4] O_{h} 
*442 [4,4] P4m 
*542 [5,4] 
*642 [6,4] 
*742 [7,4] 
*842 [8,4]... 
*∞42 [∞,4] 

Truncated figures 
4.4.4 
4.6.6 
4.8.8 
4.10.10 
4.12.12 
4.14.14 
4.16.16 
4.∞.∞ 
Coxeter Schläfli 
t_{1,2}{4,2} 
t_{1,2}{4,3} 
t_{1,2}{4,4} 
t_{1,2}{4,5} 
t_{1,2}{4,6} 
t_{1,2}{4,7} 
t_{1,2}{4,8} 
t_{1,2}{4,∞} 
Uniform dual figures  
nkis figures 
V4.4.4 
V4.6.6 
V4.8.8 
V4.10.10 
V4.12.12 
V4.14.14 
V4.16.16 
V4.∞.∞ 
Coxeter 
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.
Symmetry *n32 [n,3] 
Spherical  Euclidean  Hyperbolic tiling  

*332 [3,3] T_{d} 
*432 [4,3] O_{h} 
*532 [5,3] I_{h} 
*632 [6,3] p6m 
*732 [7,3] 
*832 [8,3] 
*∞32 [∞,3] 

Quasiregular figures configuration 
3.3.3.3 
3.4.3.4 
3.5.3.5 
3.6.3.6 
3.7.3.7 
3.8.3.8 
3.∞.3.∞ 
Coxeter diagram  
Dual (rhombic) figures configuration 
V3.3.3.3 
V3.4.3.4 
V3.5.3.5 
V3.6.3.6 
V3.7.3.7 
V3.8.3.8 
V3.∞.3.∞ 
Coxeter diagram 
The cube is a square prism:
3  4  5  6  7  8  9  10  11  12  ...  ∞ 

As spherical polyhedra  
As a trigonal trapezohedron, the cube is related to the hexagonal dihedral symmetry family.
Symmetry: [6,2], (*622)  [6,2]^{+}, (622)  [1^{+},6,2], (322)  [6,2^{+}], (2*3)  

{6,2}  t_{0,1}{6,2}  t_{1}{6,2}  t_{1,2}{6,2}  t_{2}{6,2}  t_{0,2}{6,2}  t_{0,1,2}{6,2}  s{6,2}  h{6,2}  h_{1,2}{6,2} 
Uniform duals  
V6^{2}  V12^{2}  V6^{2}  V4.4.6  V2^{6}  V4.4.6  V4.4.12  V3.3.3.6  V3^{2}  V3.3.3.3 
Compound of three cubes 
Compound of five cubes 
In uniform honeycombs and polychora
It is an element of 9 of 28 convex uniform honeycombs:
It is also an element of five fourdimensional uniform polychora:
Tesseract 
Cantellated 16cell 
Runcinated 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.
 Unit cube
 Tesseract
 Cube (film)
 Trapezohedron
 Yoshimoto Cube
 The Cube (game show)
 Prince Rupert's cube
 OLAP cube
 Lövheim cube of emotion
 Cube of Heymans
 Necker Cube
 Rubik's Cube