(Wikipedia) - Cube This article is about the geometric shape. For other uses, see Cube (disambiguation). Regular Hexahedron

Spherical tiling

Cartesian coordinates

Geometric relationsThe 11 nets of the cube.These familiar six-sided dice are cube-shaped.

Regular and uniform compounds of cubes

In uniform honeycombs and polychora

Combinatorial cubes

(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} |

{4}×{}, {}×{}×{} | |

Wythoff symbol | 3 | 2 4 |

Coxeter diagram | |

Symmetry | Oh, BC3, , (*432) |

Rotation group | O, +, (432) |

References | U06, C18, W3 |

Properties | Regular convex zonohedron |

Dihedral angle | 90° |

4.4.4 (Vertex figure) | Octahedron (dual polyhedron) |

Net |

In geometry, a cube is a three-dimensional 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.

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.

Contents- 1 Orthogonal projections
- 2 Spherical tiling
- 3 Cartesian coordinates
- 4 Equation in R3
- 5 Formulae
- 6 Uniform colorings and symmetry
- 7 Geometric relations
- 8 Other dimensions
- 9 Related polyhedra
- 9.1 In uniform honeycombs and polychora

- 10 Combinatorial cubes
- 11 See also
- 12 References
- 13 External links

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 A2 and B2 Coxeter planes.

Orthogonal projections Centered by Face Vertex Coxeter planes Projective symmetry Tilted viewsB2 | A2 |

The Cube 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 projectionsquare-centered | |

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 (x0, x1, x2) with −1 < xi < 1.

Equation in R3In analytic geometry, a cube''s surface with center (x0, y0, z0) and edge length of 2a is the locus of all points (x, y, z) such that

A direct formula for the surface without using limits is:

FormulaeFor 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 symmetryOctahedral symmetry treeThe 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 vertex-transitive coloring the faces. The highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h 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 Wythoff symbol Symmetry Symmetry order Image (uniform coloring){4,3} | {4}×{ } rr{4,2} | s2{2,4} | { }3 tr{2,2} | { }×2{ } | |

3 | 4 2 | 4 2 | 2 | 2 2 2 | | |||

Oh (*432) | D4h (*422) | D2d (2*2) | D2h (*222) | D3d (2*3) | |

24 | 16 | 8 | 8 | 12 | |

(111) | (112) | (112) | (123) | (112) | (111), (112) |

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 color, one would need at least three colors.

The cube is the cell of 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 (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 dimensionsThe analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or 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 in dimension 0, a segment in one dimension and a square in two dimensions.

Related polyhedraThe dual of a cube is an octahedron.The hemicube is the 2-to-1 quotient of the cube.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 edge-lengths? 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 order-3 vertex figures.

Spherical Polyhedra 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.

Uniform octahedral polyhedra Symmetry: , (*432) + (432) = (*332) (3*2) Duals to uniform polyhedra{4,3} | t{4,3} | r{4,3} r{31,1} | t{3,4} t{31,1} | {3,4} {31,1} | rr{4,3} s2{3,4} | tr{4,3} | sr{4,3} | h{4,3} {3,3} | h2{4,3} t{3,3} | s{3,4} s{31,1} |

= | = | = | = or | = or | = | |||||

V43 | V3.82 | V(3.4)2 | V4.62 | V34 | V3.43 | V4.6.8 | V34.4 | V33 | V3.62 | V35 |

The cube is topologically related as a part of sequence of regular tilings, extending into the hyperbolic plane: {4,p}, p=3,4,5...

Finite Euclidean Compact hyperbolic Paracompact{4,3} | {4,4} | {4,5} | {4,6} | {4,7} | {4,8}... | {4,∞} |

With dihedral symmetry, Dih4, the cube is topologically related in a series of uniform polyhedra and tilings 4.2n.2n, extending into the hyperbolic plane:

Dimensional family of truncated polyhedra and tilings: 4.2n.2n Symmetry *n42 Spherical Euclidean Compact hyperbolic Paracompact *242 D4h *342 Oh *442 P4m *542 *642 *742 *842 ... *∞42 Truncated figures Coxeter Schläflit{2,4}t{3,4}t{4,4}t{5,4}t{6,4}t{7,4}t{8,4}t{4,∞} Uniform dual figures n-kis figures Coxeter4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ |

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 Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.

Dimensional family of quasiregular polyhedra and tilings: 3.n.3.n Symmetry *n32 Spherical Euclidean Compact hyperbolic Paracompact Noncompact *332 Td *432 Oh *532 Ih *632 p6m *732 *832 ... *∞32 Quasiregular figures configuration Coxeter diagram Dual (rhombic) figures configuration Coxeter diagram3.3.3.3 | 3.4.3.4 | 3.5.3.5 | 3.6.3.6 | 3.7.3.7 | 3.8.3.8 | 3.∞.3.∞ | 3.∞.3.∞ |

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.∞ | |

The cube is a square prism:

Family of uniform prisms Symmetry 3 4 5 6 7 8 9 10 11 12 Image As spherical polyhedra ImageAs a trigonal trapezohedron, the cube is related to the hexagonal dihedral symmetry family.

Uniform hexagonal dihedral spherical polyhedra Symmetry: , (*622) +, (622) , (322) , (2*3) Uniform duals{6,2} | t{6,2} | r{6,2} | 2t{6,2}=t{2,6} | 2r{6,2}={2,6} | rr{6,2} | tr{6,2} | sr{6,2} | h{6,2} | s{2,6} |

V62 | V122 | V62 | V4.4.6 | V26 | V4.4.6 | V4.4.12 | V3.3.3.6 | V32 | V3.3.3.3 |

Compound of three cubes | Compound of five cubes |

It is an element of 9 of 28 convex uniform honeycombs:

Cubic honeycomb | Truncated square prismatic honeycomb | Snub square prismatic honeycomb | Elongated triangular prismatic honeycomb | Gyroelongated triangular prismatic honeycomb |

Cantellated cubic honeycomb | Cantitruncated cubic honeycomb | Runcitruncated cubic honeycomb | Runcinated alternated cubic honeycomb | |

It is also an element of five four-dimensional uniform polychora:

Tesseract | Cantellated 16-cell | Runcinated tesseract | Cantitruncated 16-cell | Runcitruncated 16-cell |

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 k-ary Hamming graph, which for k = 2 is the cube graph. Graphs of this sort occur in the theory of parallel processing in computers.

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