Cube

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This article is about the geometric shape. For other uses, see Cube (disambiguation).
Regular Hexahedron
Hexahedron.jpg
(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 {}×{}×{}
Wythoff symbol 3 | 2 4
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Symmetry Oh, B3, [4,3], (*432)
Rotation group O, [4,3]+, (432)
References U06, C18, W3
Properties Regular convex zonohedron
Dihedral angle 90°
Cube vertfig.png
4.4.4
(Vertex figure)
Octahedron (vector).svg
Octahedron
(dual polyhedron)
Hexahedron flat color.svg
Net
Net of cube

In geometry, a cube[1] 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. It has 12 edges, 6 faces 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.

Orthogonal projections[edit]

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 B2
2-cube.svg
A2
3-cube t0.svg
Projective
symmetry
[4] [6]
Tilted views Cube t0 e.png Cube t0 fb.png

Spherical tiling[edit]

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.

Uniform tiling 432-t0.png Cube stereographic projection.svg
Orthographic projection Stereographic projection

Cartesian coordinates[edit]

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 R3[edit]

In 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

Formulas[edit]

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 space[edit]

For a cube whose circumscribing sphere has radius R, and for a given point in its 3-dimensional space with distances di from the cube's eight vertices, we have:[2]

Doubling the cube[edit]

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 symmetry[edit]

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 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
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.png CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 2.pngCDel node f1.pngCDel 2x.pngCDel node f1.png CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 6.pngCDel node.png
Schläfli
symbol
{4,3} {4}×{ }
rr{4,2}
s2{2,4} { }3
tr{2,2}
{ }×2{ }
Wythoff
symbol
3 | 4 2 4 2 | 2 2 2 2 |
Symmetry Oh
[4,3]
(*432)
D4h
[4,2]
(*422)
D2d
[4,2+]
(2*2)
D2h
[2,2]
(*222)
D3d
[6,2+]
(2*3)
Symmetry
order
24 16 8 8 12
Image
(uniform
coloring)
Hexahedron.png
(111)
Tetragonal prism.png
(112)
Cube rotorotational symmetry.png
(112)
Uniform polyhedron 222-t012.png
(123)
Cube rhombic symmetry.png
(112)
Trigonal trapezohedron.png
(111), (112)

Geometric relations[edit]

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

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 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 dimensions[edit]

The 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 polyhedra[edit]

The 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.

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 hyperbolic plane: {4,p}, p=3,4,5...

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:

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.

The cube is a square prism:

Family of uniform prisms
Polyhedron Triangular prism.png Tetragonal prism.png Pentagonal prism.png Hexagonal prism.png Prism 7.png Octagonal prism.png Prism 9.png Decagonal prism.png Hendecagonal prism.png Dodecagonal prism.png
Coxeter CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 9.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 10.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 11.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 12.pngCDel node.pngCDel 2.pngCDel node 1.png
Tiling Spherical triangular prism.png Spherical square prism.png Spherical pentagonal prism.png Spherical hexagonal prism.png Spherical heptagonal prism.png Spherical octagonal prism.png Spherical decagonal prism.png
Config. 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.

Regular and uniform compounds of cubes
UC08-3 cubes.png
Compound of three cubes
Compound of five cubes.png
Compound of five cubes

In uniform honeycombs and polychora[edit]

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

Cubic honeycomb
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
Truncated square prismatic honeycomb
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
Snub square prismatic honeycomb
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
Elongated triangular prismatic honeycomb Gyroelongated triangular prismatic honeycomb
Partial cubic honeycomb.png Truncated square prismatic honeycomb.png Snub square prismatic honeycomb.png Elongated triangular prismatic honeycomb.png Gyroelongated triangular prismatic honeycomb.png
Cantellated cubic honeycomb
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Cantitruncated cubic honeycomb
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Runcitruncated cubic honeycomb
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Runcinated alternated cubic honeycomb
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node 1.png
HC A5-A3-P2.png HC A6-A4-P2.png HC A5-A2-P2-Pr8.png HC A5-P2-P1.png

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

Tesseract
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cantellated 16-cell
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcinated tesseract
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Cantitruncated 16-cell
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Runcitruncated 16-cell
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
4-cube t0.svg 24-cell t1 B4.svg 4-cube t03.svg 4-cube t123.svg 4-cube t023.svg

Cubical graph[edit]

Cubical graph
3-cube column graph.svg
Named after Q3
Vertices 8
Edges 12
Radius 3
Diameter 3
Girth 4
Automorphisms 48
Chromatic number 2
Properties Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, planar graph

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 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.

See also[edit]

Miscellaneous cubes

References[edit]

  1. ^ 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".
  2. ^ Park, Poo-Sung. "Regular polytope distances", Forum Geometricorum 16, 2016, 227-232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf
  3. ^ Weisstein, Eric W. "Cube". MathWorld. 
  4. ^ Weisstein, Eric W. "Cubical graph". MathWorld. 

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds