Cube: Difference between revisions
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==Combinatorial cubes== |
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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]]. |
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An extension is the three dimensional ''k''-ary [[Hamming graph]], which for ''k'' = 2 is the cube graph. |
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Graphs of this sort occur in the theory of [[parallel computing|parallel processing]] in computers. |
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==See also== |
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* [[Unit cube]] |
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* [[Tesseract]] |
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* [[Cube (film)]] |
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* [[Trapezohedron]] |
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* [[Yoshimoto Cube]] |
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* [[The Cube (game show)]] |
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* [[Prince Rupert's cube]] |
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* [[OLAP cube]] |
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==References== |
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{{reflist}} |
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==External links== |
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* {{mathworld | urlname = Cube| title = Cube}} |
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* {{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3o4x - cube}} |
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* [http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=yMtAqU4z1OXaI6DjTGFCpy12MxHeA0unxJpd4YVkhHgSd&name=Cube#applet Editable printable net of a cube with interactive 3D view] |
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* [http://polyhedra.org/poly/show/1/cube Cube: Interactive Polyhedron Model] |
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* [http://www.kjmaclean.com/Geometry/GeometryHome.html K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra] |
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* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra] |
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* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] |
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* [http://www.mathopenref.com/cubevolume.html Volume of a cube], with interactive animation |
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{{Convex polyhedron navigator|state=collapsed}} |
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{{Polytopes|state=collapsed}} |
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[[Category:Platonic solids]] |
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[[Category:Prismatoid polyhedra]] |
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[[Category:Space-filling polyhedra]] |
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[[Category:Volume]] |
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[[Category:Zonohedra]] |
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[[Category:Cubes]] |
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[[af:Kubus]] |
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[[ar:مكعب]] |
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[[bs:Kocka]] |
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[[et:Kuup]] |
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[[kn:ಕ್ಯೂಬ್ (ಘನಾಕೃತಿ)]] |
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[[ta:அறுமுக கட்டகம்]] |
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Revision as of 19:49, 9 November 2011
Regular hexahedron | |
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(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} {}×{}×{} = {}3 | |
Face configuration | V3.3.3.3 |
Wythoff symbol | 3 | 2 4 |
Coxeter diagram | |
Symmetry | Oh, B3, [4,3], (*432) |
Rotation group | O, [4,3]+, (432) |
References | U06, C18, W3 |
Properties | regular, convexzonohedron, Hanner polytope |
Dihedral angle | 90° |
4.4.4 (Vertex figure) |
Octahedron (dual polyhedron) |
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.
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 (x0, x1, x2) with −1 < x i < 1.
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 |
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 | ||||
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) |
(111) |
(112) |
(123) |
(111), (112), (122), and (222) |
Geometric relations
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 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 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.
Cube |
Truncated cube |
cuboctahedron |
Truncated octahedron |
Octahedron |
Rhombi-cuboctahedron |
truncated cuboctahedron |
Snub cube |
Stella octangula |
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] |
Cube |
Rhombic dodecahedron |
Rhombic triacontahedron |
Rhombille |
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 four-dimensional uniform polychora:
Tesseract |
Cantellated 16-cell |
Runcinated tesseract |
Cantitruncated 16-cell |
Runcitruncated 16-cell |