cube

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cube

To calculate the volume of a cube, multiply the length of an edge of the cube by itself twice. The volume of this cube is 125 cubic feet. (Academy Artworks)

1. Mathematics. A regular solid having six congruent square faces.
2. 1. Something having the general shape of a cube: a cube of sugar.
   2. A cubicle, used for work or study.
3. Mathematics. The third power of a number or quantity.
4. cubes Slang. Cubic inches. Used especially of an internal combustion engine.

1. Mathematics. To raise (a quantity or number) to the third power.
2. To determine the cubic contents of.
3. To form or cut into cubes; dice.
4. To tenderize (meat) by breaking the fibers with superficial cuts in a pattern of squares.

[Latin cubus, from Greek kubos. N., sense 2b, short for CUBICLE.]

Any of several tropical American woody plants of the genus Lonchocarpus in the pea family, whose roots are used locally as a fish poison and commercially as a source of rotenone.

[American Spanish.]

A parallelepiped whose six faces are all squares. The cube is one of the five regular solids known to the ancient Greeks. A cube has eight vertices and twelve edges. Each vertex is on three edges and three faces, each edge is on two vertices and two faces, and each plane is on four vertices and four edges. See also Polyhedron; Regular polytopes.

A multidimensional database that holds data more like a 3D spreadsheet rather than a relational database. The cube allows different views of the data to be quickly displayed. See OLAP and multidimensional views.

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1. To cut food (such as meat or cheese) into 1⁄2-inch cubes. Cubes of food are larger than diced or mirepoix. 2. A term also used to describe tenderizing meat with an instrument that leaves cube-shaped imprints on the surface (see cube steak).

1. [short for ‘cubicle’] A module in the open-plan offices used at many programming shops. “I've got the manuals in my cube.”

2. A NeXT machine (which resembles a matte-black cube).

Feed for animals compressed into cubes. May be entirely concentrate but many nowadays are a complete ration including hammer-milled fiber. Are more expensive but reduce wastage and can be eaten more quickly than a milled grain mixture. See also pelleted feed.

This article is about the geometric shape. For other meanings of the word "cube", see cube (disambiguation).

Regular Hexahedron
(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 2 4 | 2 2 2 2 |
Coxeter-Dynkin
Symmetry	Oh or (*432)
References	U06, C18, W3
Properties	Regular convex zonohedron
Dihedral angle	90°
4.4.4 (Vertex figure)	Octahedron (dual polyhedron)
Net

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). A cube is the three-dimensional case of the more general concept of a hypercube.

Contents 1 Cartesian coordinates 2 Formulae 3 Symmetry 4 Geometric relations 5 Other dimensions 6 Related polyhedra 7 Combinatorial cubes 8 See also 9 References 10 External links

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 a,

surface area	6a2
volume	a3
radius of circumscribed sphere
radius of sphere tangent to edges
radius of inscribed sphere

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).

Symmetry

The cube has 3 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.

(3 colors) | 2 2 2 D2h

(2 colors) 4 2 | 2 D4h

(1 color) 3 | 4 2 Oh

Geometric relations

These familiar six-sided dice are cube-shaped.

The cube is unique among the Platonic solids for being able to tile space regularly. 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.

Other dimensions

The analogue of a cube in four-dimensional Euclidean space has a special name — a tesseract or (rarely) hypercube.

The analogue of the cube in n-dimensional Euclidean space is called a hypercube or n-dimensional cube or simply n-cube. It 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 vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. 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.

The tetrahedra in the cube (stella octangula)	The rectified cube (cuboctahedron)	Truncated cube	Rhombicuboctahedron
Compound of three cubes	An alternately truncated cube

All but the last of the figures shown have the same symmetries as the cube (see octahedral symmetry).

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

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 3-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

• Kaaba, a large masonry structure roughly the shape of a cube
• Unit cube
• Square
• Tesseract
• Cube (film)
• Trapezohedron

References

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

External links

Listen to this article (info/dl)

This audio file was created from a revision dated 2006-07-07, and does not reflect subsequent edits to the article. (Audio help)

More spoken articles

• Eric W. Weisstein, Cube at MathWorld.
• Cube: Interactive Polyhedron Model
• K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
• The Uniform Polyhedra
• Virtual Reality Polyhedra
• Volume of a cube, with interactive animation

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Dansk (Danish)
n. - terning, sukkerknald, kubus, kubiktal
v. tr. - uddrage kubikroden af, opløfte til tredje potens, kubere

idioms:

• cube root    kubikrod

Nederlands (Dutch)
kubus, blokje, derdemacht, tot de derdemacht verheffen/ verheven, in blokjes snijden, inhoud berekenen van

Français (French)
n. - cube
v. tr. - (Math) mettre au cube, couper en cube

idioms:

• cube root    racine cubique

Deutsch (German)
n. - Würfel, dritte Potenz
v. - zur dritten Potenz erheben, in kleine Würfel schneiden

idioms:

• cube root    Kubikwurzel

Ελληνική (Greek)
n. - κύβος, στερεό σώμα σε σχήμα κύβου, ζάρι, (μαθημ.) κύβος, τρίτη δύναμη
v. - (μαθημ.) υψώνω στον κύβο, κόβω (τρόφιμα κ.λπ.) σε μικρούς κύβους

idioms:

• cube root    (μαθημ.) κυβική ρίζα

Italiano (Italian)
cubo

idioms:

• cube root    radice cubica
• ice cube    cubetto di ghiaccio

Português (Portuguese)
n. - cubo (m)
v. - cubar

idioms:

• cube root    raiz (f) cúbica (Mat.)
• ice cube    pedra (f) de gelo

Русский (Russian)
куб

idioms:

• cube root    кубический корень
• ice cube    кубик льда

Español (Spanish)
n. - cubo, terrón
v. tr. - cubicar, dar forma de cubo

idioms:

• cube root    raíz cúbica

Svenska (Swedish)
n. - kub, tärning, kub (matem.)
v. - upphöja till tre, dra kubikroten ur, beräkna kubikinnehållet hos, skära i tärningar

中文（简体）(Chinese (Simplified))
立方体, 立方, 使成立方形, 量...的体积, 将...切成小方块, 使自乘二次

idioms:

• cube root    立方根

中文（繁體）(Chinese (Traditional))
n. - 立方體, 立方
v. tr. - 使成立方形, 量...的體積, 將...切成小方塊, 使自乘二次

idioms:

• cube root    立方根

한국어 (Korean)
n. - 정육면체, 세제곱, 주사위
v. tr. - 입방체로 하다, 벽돌을 깔다, 세제곱하다, 부피를 구하다

日本語 (Japanese)
n. - 立方体, 立方体のもの, 三乗, 立方
v. - 三乗する, さいの目に切る, 体積を求める

idioms:

• cube root    立方根

العربيه (Arabic)
‏(الاسم) مكعب (فعل) كعب‏

עברית (Hebrew)
n. - ‮קוביה, חזקה שלישית‬
v. tr. - ‮עיקב, העלה בחזקה השלישית, חתך לקוביות‬

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