Cube



A geometric shape with 6 square faces





























Regular hexahedron

Hexahedron.jpg
(Click here for rotating model)
Type
Platonic solid
Elements
F = 6, E = 12
V = 8 (χ = 2)
Faces by sides64
Conway notationC
Schläfli symbols4,3
t2,4 or 4×
tr2,2 or ××
Face configurationV3.3.3.3
Wythoff symbol3 | 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, convexzonohedron
Dihedral angle90°

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 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





  • 1 Orthogonal projections


  • 2 Spherical tiling


  • 3 Cartesian coordinates


  • 4 Equation in R3


  • 5 Formulas

    • 5.1 Point in space



  • 6 Doubling the cube


  • 7 Uniform colorings and symmetry


  • 8 Geometric relations


  • 9 Other dimensions


  • 10 Related polyhedra

    • 10.1 In uniform honeycombs and polychora



  • 11 Cubical graph


  • 12 See also


  • 13 References


  • 14 External links




Orthogonal projections


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 B2Coxeter 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


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


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 for all i.



Equation in R3


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


max,=a.displaystyle max=a.max=a.


Formulas


For a cube of edge length adisplaystyle aa:



















surface area

6a2displaystyle 6a^2,6a^2,

volume

a3displaystyle a^3,a^3,

face diagonal

2adisplaystyle sqrt 2asqrt 2a

space diagonal

3atextstyle sqrt 3atextstyle sqrt 3a
radius of circumscribed sphere

32adisplaystyle frac sqrt 32afrac sqrt 32a
radius of sphere tangent to edges

a2displaystyle frac asqrt 2frac asqrt 2
radius of inscribed sphere

a2displaystyle frac a2frac a2

angles between faces (in radians)

π2displaystyle frac pi 2frac pi 2

As the volume of a cube is the third power of its sides a×a×adisplaystyle atimes atimes aatimes atimes 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).



Point in space


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]


∑i=18di48+16R49=(∑i=18di28+2R23)2.displaystyle frac sum _i=1^8d_i^48+frac 16R^49=left(frac sum _i=1^8d_i^28+frac 2R^23right)^2.displaystyle frac sum _i=1^8d_i^48+frac 16R^49=left(frac sum _i=1^8d_i^28+frac 2R^23right)^2.


Doubling the cube


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





Octahedral symmetry tree


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

rr4,2
s22,4
3
tr2,2
×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




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


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 line segment in one dimension and a square in two dimensions.



Related polyhedra




The dual of a cube is an octahedron, seen here with vertices at the center of the cube's square faces.




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 2/2displaystyle scriptstyle sqrt 2/2scriptstyle sqrt 2/2.


The cube is a special case in various classes of general polyhedra:


























NameEqual edge-lengths?Equal angles?Right angles?
CubeYesYes
Yes
RhombohedronYesYesNo
CuboidNoYesYes
ParallelepipedNoYesNo

quadrilaterally faced hexahedron
NoNoNo

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:


















































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


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




















Cubical graph
3-cube column graph.svg
Named afterQ3
Vertices8
Edges12
Radius3
Diameter3
Girth4
Automorphisms48
Chromatic number2
Properties
Hamiltonian, regular, symmetric, distance-regular, distance-transitive, 3-vertex-connected, 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 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


  • Tesseract

  • Trapezohedron

Miscellaneous cubes


  • Cube (film)

  • Diamond cubic

  • Lövheim cube of emotion

  • Cube of Heymans

  • Necker Cube

  • OLAP cube

  • Prince Rupert's cube

  • Rubik's Cube

  • The Cube (game show)

  • Unit cube

  • Yoshimoto Cube

  • Kaaba


References




  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..mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em


  4. ^ Weisstein, Eric W. "Cubical graph". MathWorld.




External links


  • 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


Family

An

Bn

I2(p) / Dn

E6 / E7 / E8 / F4 / G2

Hn

Regular polygon

Triangle

Square

p-gon

Hexagon

Pentagon

Uniform polyhedron

Tetrahedron

Octahedron • Cube

Demicube


Dodecahedron • Icosahedron

Uniform 4-polytope

5-cell

16-cell • Tesseract

Demitesseract

24-cell

120-cell • 600-cell

Uniform 5-polytope

5-simplex

5-orthoplex • 5-cube

5-demicube



Uniform 6-polytope

6-simplex

6-orthoplex • 6-cube

6-demicube

122 • 221


Uniform 7-polytope

7-simplex

7-orthoplex • 7-cube

7-demicube

132 • 231 • 321


Uniform 8-polytope

8-simplex

8-orthoplex • 8-cube

8-demicube

142 • 241 • 421


Uniform 9-polytope

9-simplex

9-orthoplex • 9-cube

9-demicube



Uniform 10-polytope

10-simplex

10-orthoplex • 10-cube

10-demicube


Uniform n-polytope

n-simplex

n-orthoplex • n-cube

n-demicube

1k2 • 2k1 • k21

n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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