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Cube

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Title: Cube  
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Subject: Combination puzzle, Convex uniform honeycomb, Tesseract, Octahedron, Platonic solid
Collection: Cubes, Elementary Shapes, Platonic Solids, Prismatoid Polyhedra, Space-Filling Polyhedra, Volume, Zonohedra
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Cube

Regular Hexahedron

(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, [4,3], (*432)
Rotation group O, [4,3]+, (432)
References U06, C18, W3
Properties Regular convex zonohedron
Dihedral angle 90°

4.4.4
(Vertex figure)

Octahedron
(dual polyhedron)

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 and 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.

Contents

  • Orthogonal projections 1
  • Spherical tiling 2
  • Cartesian coordinates 3
  • Equation in R3 4
  • Formulae 5
  • Doubling the cube 6
  • Uniform colorings and symmetry 7
  • Geometric relations 8
  • Other dimensions 9
  • Related polyhedra 10
    • In uniform honeycombs and polychora 10.1
  • Cubical graph 11
  • See also 12
  • References 13
  • External links 14

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 B2 Coxeter planes.
Orthogonal projections
Centered by Face Vertex
Coxeter planes B2
A2
Projective
symmetry
[4] [6]
Tilted views

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.

square-centered
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.

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\{ |x-x_0|,|y-y_0|,|z-z_0| \} = a.

Formulae

For a cube of edge length a,
surface area 6 a^2\,
volume a^3\,
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 (in radians) \frac{\pi}{2}

As the volume of a cube is the third power of its sides a \times a \times 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).

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

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
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)

(111)

(112)

(112)

(123)

(112)

(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.[2] 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 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 \scriptstyle \sqrt{2}/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 13 of that of the cube. The remaining space consists of four equal irregular tetrahedra with a volume of 16 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.
*n32 symmetry mutation of regular tilings: n3 or {n',3}
Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic
{2,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {12i,3} {9i,3} {6i,3} {3i,3}

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...
*n42 symmetry mutation of regular tilings: 4n
Spherical 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:
*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
Config. 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 [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.
Symmetry mutations of dual quasiregular tilings: V(3.n)2
Spherical Euclidean Hyperbolic
*n32 *332 *432 *532 *632 *732 *832... *∞32
Tiling
Conf. V(3.3)2 V(3.4)2 V(3.5)2 V(3.6)2 V(3.7)2 V(3.8)2 V(3.∞)2
The cube is a square prism:
Family of uniform prisms
Polyhedron
Tiling
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 ...∞.4.4
As a trigonal trapezohedron, the cube is related to the hexagonal dihedral symmetry family.
Uniform hexagonal dihedral spherical polyhedra
Symmetry: [6,2], (*622) [6,2]+, (622) [6,2+], (2*3)
{6,2} t{6,2} r{6,2} t{2,6} {2,6} rr{6,2} tr{6,2} sr{6,2} s{2,6}
Duals to uniforms
V62 V122 V62 V4.4.6 V26 V4.4.6 V4.4.12 V3.3.3.6 V3.3.3.3
Regular and uniform compounds of cubes

Compound of three cubes

Compound of five cubes

In uniform honeycombs and polychora

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

Cubical 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.[3] 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

Miscellaneous cubes

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. ^ Weisstein, Eric W., "Cube", MathWorld.
  3. ^ 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)
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