Cube

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. Template:Reg polyhedra db In geometry, a cube 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.

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

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

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.

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

$\underset{n\to \mathrm{\infty }}{lim}[(x-x_0{)}^n+(y-y_0{)}^n+(z-z_0{)}^n-a^n]=0.$

Formulae

For a cube of edge length $a$,

surface area	$6a^2$
volume	$a^3$
face diagonal	$\sqrt{2}a$
space diagonal	$\sqrt{3}a$
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).

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 O_h has all the faces the same color. The dihedral symmetry D_4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D_2h 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 diagram	Template:CDD	Template:CDD	Template:CDD	Template:CDD
Schläfli symbol	{4,3}	{4}×{}	{}×{}×{}
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)	File:Hexahedron.png (111)	File:Tetragonal prism.png (112)	File:Uniform polyhedron 222-t012.png (123)	File:Trigonal trapezohedron.png (111), (112), (122), and (222)

Geometric relations

File:Planificacao cubo.gif

The 11 nets of the cube.

File:Stone Dice 17.JPG

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. 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

File:Dual Cube-Octahedron.svg

The dual of a cube is an octahedron.

File:Hemicube2.PNG

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 Template:Frac of that of the cube. The remaining space consists of four equal irregular tetrahedra with a volume of Template:Frac 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. Template:Order-3 tiling table

The cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron. Template:Octahedral truncations

The cube is topologically related as a part of sequence of regular tilings, extending into the hyperbolic plane: {4,p}, p=3,4,5... Template:Regular square tiling table

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: Template:Truncated figure3 table

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. Template:Quasiregular figure table

The cube is a square prism: Template:UniformPrisms

As a trigonal trapezohedron, the cube is related to the hexagonal dihedral symmetry family. Template:Hexagonal dihedral truncations

Regular and uniform compounds of cubes

File:UC08-3 cubes.png Compound of three cubes

File: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 Template:CDD Template:CDD	Truncated square prismatic honeycomb Template:CDD	Snub square prismatic honeycomb Template:CDD	Elongated triangular prismatic honeycomb	Gyroelongated triangular prismatic honeycomb
File:Partial cubic honeycomb.png	File:Truncated square prismatic honeycomb.png	File:Snub square prismatic honeycomb.png	File:Elongated triangular prismatic honeycomb.png	File:Gyroelongated triangular prismatic honeycomb.png
Cantellated cubic honeycomb Template:CDD	Cantitruncated cubic honeycomb Template:CDD	Runcitruncated cubic honeycomb Template:CDD	Runcinated alternated cubic honeycomb Template:CDD
File:HC A5-A3-P2.png	File:HC A6-A4-P2.png	File:HC A5-A2-P2-Pr8.png	File:HC A5-P2-P1.png

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

Tesseract Template:CDD	Cantellated 16-cell Template:CDD	Runcinated tesseract Template:CDD	Cantitruncated 16-cell Template:CDD	Runcitruncated 16-cell Template:CDD
File:4-cube t0.svg	File:4-cube t13.svg	File:4-cube t03.svg	File:4-cube t123.svg	File: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

• Tesseract
• Trapezohedron

Miscellaneous cubes

• Cube (film)
• 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

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.
  
  Here is my web site - cottagehillchurch.com
• Cube: Interactive Polyhedron Model*
• Volume of a cube, with interactive animation
• Cube (Robert Webb's site)