Cube
Regular hexahedron  

(Click here for rotating model)  
Type  Platonic solid 
Elements  F = 6, E = 12 V = 8 (χ = 2) 
Faces by sides  6{4} 
Conway notation  C 
Schläfli symbols  {4,3} 
t{2,4} or {4}×{} tr{2,2} or {}×{}×{}  
Face configuration  V3.3.3.3 
Wythoff symbol  3  2 4 
Coxeter diagram  
Symmetry  O_{h}, B_{3}, [4,3], (*432) 
Rotation group  O, [4,3]^{+}, (432) 
References  U_{06}, C_{18}, W_{3} 
Properties  regular, convexzonohedron 
Dihedral angle  90° 
4.4.4 (Vertex figure) 
Octahedron (dual polyhedron) 
Net 
In geometry, a cube^{[1]} is a threedimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices.
The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.
The cube is dual to the octahedron. It has cubical or octahedral symmetry
The cube is the only convex polyhedron whose faces are all squares.
Contents
Orthogonal projectionsEdit
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 
[4]  [6] 
Tilted views 
Spherical tilingEdit
The cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Orthographic projection  Stereographic projection 

Cartesian coordinatesEdit
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 all i.
Equation in R^{3}Edit
In analytic geometry, a cube's surface with center (x_{0}, y_{0}, z_{0}) and edge length of 2a is the locus of all points (x, y, z) such that
FormulasEdit
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).
Point in spaceEdit
For a cube whose circumscribing sphere has radius R, and for a given point in its 3dimensional space with distances d_{i} from the cube's eight vertices, we have:^{[2]}
Doubling the cubeEdit
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, and in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number.
Uniform colorings and symmetryEdit
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 
Rectangular cuboid 
Rhombic prism 
Trigonal trapezohedron  

Coxeter diagram 

Schläfli symbol 
{4,3}  {4}×{ } rr{4,2} 
s_{2}{2,4}  { }^{3} tr{2,2} 
{ }×2{ }  
Wythoff symbol 
3  4 2  4 2  2  2 2 2   
Symmetry  O_{h} [4,3] (*432) 
D_{4h} [4,2] (*422) 
D_{2d} [4,2^{+}] (2*2) 
D_{2h} [2,2] (*222) 
D_{3d} [6,2^{+}] (2*3)  
Symmetry order 
24  16  8  8  12  
Image (uniform coloring) 
(111) 
(112) 
(112) 
(123) 
(112) 
(111), (112) 
Geometric relationsEdit
A cube has eleven nets (one shown above): that is, there are eleven ways to flatten a hollow cube by cutting seven edges.^{[3]} 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 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 dimensionsEdit
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 line segment in one dimension and a square in two dimensions.
Related polyhedraEdit
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 1/3 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.
*n32 symmetry mutation of regular tilings: {n,3}  

Spherical  Euclidean  Compact hyperb.  Paraco.  Noncompact hyperbolic  
{2,3}  {3,3}  {4,3}  {5,3}  {6,3}  {7,3}  {8,3}  {∞,3}  {12i,3}  {9i,3}  {6i,3}  {3i,3} 
The cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.
Uniform octahedral polyhedra  

Symmetry: [4,3], (*432)  [4,3]^{+} (432) 
[1^{+},4,3] = [3,3] (*332) 
[3^{+},4] (3*2)  
{4,3}  t{4,3}  r{4,3} r{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3}  h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{3,4} s{3^{1,1}} 
= 
= 
= 
= or 
= or 
=  




 
Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
The cube is topologically related as a part of sequence of regular tilings, extending into the hyperbolic plane: {4,p}, p=3,4,5...
*n42 symmetry mutation of regular tilings: {4,n}  

Spherical  Euclidean  Compact hyperbolic  Paracompact  
{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:
*n42 symmetry mutation of truncated tilings: 4.2n.2n  

Symmetry *n42 [n,4] 
Spherical  Euclidean  Compact hyperbolic  Paracomp.  
*242 [2,4] 
*342 [3,4] 
*442 [4,4] 
*542 [5,4] 
*642 [6,4] 
*742 [7,4] 
*842 [8,4]... 
*∞42 [∞,4]  
Truncated figures 

Config.  4.4.4  4.6.6  4.8.8  4.10.10  4.12.12  4.14.14  4.16.16  4.∞.∞  
nkis figures 

Config.  V4.4.4  V4.6.6  V4.8.8  V4.10.10  V4.12.12  V4.14.14  V4.16.16  V4.∞.∞ 
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 mutations of dual quasiregular tilings: V(3.n)^{2}  

*n32  Spherical  Euclidean  Hyperbolic  
*332  *432  *532  *632  *732  *832...  *∞32  
Tiling  
Conf.  V(3.3)^{2}  V(3.4)^{2}  V(3.5)^{2}  V(3.6)^{2}  V(3.7)^{2}  V(3.8)^{2}  V(3.∞)^{2} 
The cube is a square prism:
Family of uniform prisms  

Polyhedron  
Coxeter  
Tiling  
Config.  2.4.4  3.4.4  4.4.4  5.4.4  6.4.4  7.4.4  8.4.4  9.4.4  10.4.4  11.4.4  12.4.4 
As a trigonal trapezohedron, the cube is related to the hexagonal dihedral symmetry family.
Uniform hexagonal dihedral spherical polyhedra  

Symmetry: [6,2], (*622)  [6,2]^{+}, (622)  [6,2^{+}], (2*3)  
{6,2}  t{6,2}  r{6,2}  t{2,6}  {2,6}  rr{6,2}  tr{6,2}  sr{6,2}  s{2,6}  
Duals to uniforms  
V6^{2}  V12^{2}  V6^{2}  V4.4.6  V2^{6}  V4.4.6  V4.4.12  V3.3.3.6  V3.3.3.3 
Compound of three cubes 
Compound of five cubes 
In uniform honeycombs and polychoraEdit
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 
Cubical graphEdit
Cubical graph  

Named after  Q_{3} 
Vertices  8 
Edges  12 
Radius  3 
Diameter  3 
Girth  4 
Automorphisms  48 
Chromatic number  2 
Properties  Hamiltonian, regular, symmetric, distanceregular, distancetransitive, 3vertexconnected, planar graph 
Table of graphs and parameters 
The skeleton of the cube (the vertices and edges) form a graph, with 8 vertices, and 12 edges. It is a special case of the hypercube graph.^{[4]} It is one of 5 Platonic graphs, each a skeleton of its Platonic solid.
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 alsoEdit
Miscellaneous cubes
ReferencesEdit
 ^ English cube from Old French < Latin cubus < Greek κύβος (kubos) meaning "a cube, a die, vertebra". In turn from PIE *keu(b), "to bend, turn".
 ^ Park, PooSung. "Regular polytope distances", Forum Geometricorum 16, 2016, 227232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
 ^ Weisstein, Eric W. "Cube". MathWorld.
 ^ Weisstein, Eric W. "Cubical graph". MathWorld.
External linksEdit
 Weisstein, Eric W. "Cube". MathWorld.
 Cube: Interactive Polyhedron Model*
 Volume of a cube, with interactive animation
 Cube (Robert Webb's site)
Fundamental convex regular and uniform polytopes in dimensions 2–10
 

A_{n}  B_{n}  I_{2}(p) / D_{n}  E_{6} / E_{7} / E_{8} / F_{4} / G_{2}  H_{n}  
Triangle  Square  pgon  Hexagon  Pentagon  
Tetrahedron  Octahedron • Cube  Demicube  Dodecahedron • Icosahedron  
5cell  16cell • Tesseract  Demitesseract  24cell  120cell • 600cell  
5simplex  5orthoplex • 5cube  5demicube  
6simplex  6orthoplex • 6cube  6demicube  1_{22} • 2_{21}  
7simplex  7orthoplex • 7cube  7demicube  1_{32} • 2_{31} • 3_{21}  
8simplex  8orthoplex • 8cube  8demicube  1_{42} • 2_{41} • 4_{21}  
9simplex  9orthoplex • 9cube  9demicube  
10simplex  10orthoplex • 10cube  10demicube  
nsimplex  northoplex • ncube  ndemicube  1_{k2} • 2_{k1} • k_{21}  npentagonal polytope  
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds 