Cube
Regular Hexahedron
Cube
(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-Dynkin CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Symmetry Oh, [4,3], (*432)
References U06, C18, W3
Properties Regular convex zonohedron
Dihedral angle 90°
Cube
4.4.4
(Vertex figure)
Octahedron.png
Octahedron
(dual polyhedron)
Cube
Net

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 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.

Contents

Cartesian coordinates

Orthographic projections
3-cube t0.svg 3-cube t0 B2.svg

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 < x i < 1.

Formulae

For a cube of edge length a,

surface area 6a2
volume a3
face diagonal \sqrt 2a
space diagonal \sqrt 3a
radius of circumscribed sphere \frac{\sqrt 3}{2} a
radius of sphere tangent to edges \frac{a}{\sqrt 2}
radius of inscribed sphere \frac{a}{2}
angles between faces \frac{\pi}{2}

As the volume of a cube is the third power of its sides a×a×a, 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 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 Cuboid Trigonal trapezohedron
Coxeter-Dynkin 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 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Schläfli symbol {4,3} {4}x{} {}x{}x{}
Wythoff symbol 3 | 4 2 4 2 | 2 | 2 2 2
Symmetry Oh
(*432)
D4h
(*422)
D2h
(*222)
D3d
(2*3)
Symmetry order 24 16 8 12
Image
(uniform coloring)
Hexahedron.png
(111)
Tetragonal prism.png
(112)
Uniform polyhedron 222-t012.png
(123)
Trigonal trapezohedron.png
(111), (112), (122), and (222)

Geometric relations

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

A cube has 11 nets (one shown above): that is, there are 11 ways to flatten a hollow cube by cutting 7 edges.[2] 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 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 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

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 \sqrt{2}.

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 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.

Related uniform polyhedra
Cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Truncated cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
cuboctahedron
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncated octahedron
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Octahedron
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Rhombi-cuboctahedron
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
truncated cuboctahedron
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Snub cube
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Stella octangula
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t0.png Uniform polyhedron-43-t01.png Uniform polyhedron-43-t1.png Uniform polyhedron-43-t12.png Uniform polyhedron-43-t2.png Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Stella octangula.png

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]
Hexahedron.svg
Cube
Rhombicdodecahedron.jpg
Rhombic dodecahedron
Rhombictriacontahedron.jpg
Rhombic triacontahedron
Rhombic star tiling.png
Rhombille
Order73 qreg rhombic til.png Uniform dual tiling 433-t01-yellow.png
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

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
Cantellated cubic honeycomb.jpg Cantitruncated cubic honeycomb.jpg Runcitruncated cubic honeycomb.jpg Runcinated alternated cubic honeycomb.jpg

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 4-cube t13.svg 4-cube t03.svg 4-cube t123.svg 4-cube t023.svg

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 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

References

  1. ^ English cube from Old French < Latin cubus < Greek kubos meaning "a cube, a die, vertebra". In turn from PIE *keu(b)-, "to bend, turn".
  2. ^ Weisstein, Eric W., "Cube" from MathWorld.

External links


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Synonyms:

Look at other dictionaries:

  • cube — cube …   Dictionnaire des rimes

  • cubé — cubé …   Dictionnaire des rimes

  • cube — [ kyb ] n. m. • v. 1360; adj. nombres cubes XIIIe; lat. cubus, gr. kubos « dé à jouer » 1 ♦ Géom. Solide à six faces carrées égales, hexaèdre régulier. Le cube est un parallélépipède rectangle dont toutes les arêtes sont égales. Volume d un cube …   Encyclopédie Universelle

  • CUBE — Тип …   Википедия

  • Cube — (engl. Würfel) steht für: Cube (Computerspiel), Computerspiel Rubik s Cube, englische Bezeichnung für den Zauberwürfel von Ernő Rubik Cube (Fahrradhersteller), Fahrradhersteller aus Waldershof Nissan Cube, PKW Modell OLAP Würfel, Darstellungsform …   Deutsch Wikipedia

  • Cube — (k[=u]b), n. [F. cube, L. cubus, fr. Gr. ???? a cube, a cubical die.] 1. (Geom.) A regular solid body, with six equal square sides. [1913 Webster] 2. (Math.) The product obtained by taking a number or quantity three times as a factor; as, 4x4=16 …   The Collaborative International Dictionary of English

  • Cube — Saltar a navegación, búsqueda Cube Título Cube (España) El Cubo (Hispanoamérica) Ficha técnica Dirección Vincenzo Natali Producción Colin Brunton Guión …   Wikipedia Español

  • cube — cube1 [kyo͞ob] n. [Fr < L cubus < Gr kybos, a cube, die, vertebra < IE base * keu(b) , to bend, turn > HIP1, HIVE, L cubare, to lie down] 1. a solid with six equal, square sides: see POLYHEDRON 2. anything having more or less this… …   English World dictionary

  • cube — CUBE. s. m. Corps solide qui a six faces quarrées esgales. Figure posée sur un cube. Il est quelquefois adj. Un quarré cube. pied, toise, cube. racine cube …   Dictionnaire de l'Académie française

  • cube — CUBE. s. m. Corps solide qui a six faces carrées égales. Figure posée sur un cube. [b]f♛/b] Il est quelquefois adjectif. Pied, toise cube. Racine cube …   Dictionnaire de l'Académie Française 1798

  • cube — cube·let; cube; hy·per·cube; mag·i·cube; …   English syllables

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