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A cube is a three-dimensional
Three-dimensional space

Three-dimensional space is a geometric model of the physical universe in which we live. The three dimensions are commonly called length, width, and depth , although any three mutually perpendicular directions can serve as the three dimensions....
 solid object bounded by six square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular
Regular polyhedron

A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
 hexahedron
Hexahedron

A hexahedron is a polyhedron with six faces. A Regular polyhedron hexahedron, with all its faces Square , is a cube.There are many kinds of hexahedra, some topologically similar to the cube and some not....
 and is one of the five Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s. It is a special kind of square prism
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
, of rectangular parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
 and of 3-sided trapezohedron
Trapezohedron

The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kite ....
. The cube is dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 to the octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
. It has cubical symmetry (also called octahedral symmetry
Octahedral symmetry

A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
). A cube is the three-dimensional case of the more general concept of a hypercube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
, which exists in any dimension.
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.

a cube of edge length ,

As the volume of a cube is the third power of its sides a×a×a, third powers are called cubes, by analogy with square
Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
s and second powers.

A cube has the largest volume among cuboid
Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
s (rectangular boxes) with a given surface area
Surface area

Surface area is how much exposed area an object has. It is expressed in square units. If an object has flat Face , its surface area can be calculated by adding together the areas of its faces....
.






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Quotations


Every day I mop up after your bleeding heart. The only reason you even exist is because I keep you!

I know your type. No kids, no man to fuck you. So you go around outraged, sticking your nose up other people's assholes, sniffing their business!

I looked through the walls. I dreamed him, at his desk, designing everything. He can't let you solve the puzzle, see, because there is a purpose.

I mean, nobody wants to see the big picture. Life's too complicated.

I'm just a guy. I work in an office building, doing office building stuff.

Let's rule out aliens for now and concentrate on what we know.






Encyclopedia


A cube is a three-dimensional
Three-dimensional space

Three-dimensional space is a geometric model of the physical universe in which we live. The three dimensions are commonly called length, width, and depth , although any three mutually perpendicular directions can serve as the three dimensions....
 solid object bounded by six square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular
Regular polyhedron

A regular polyhedron is a polyhedron whose faces are Congruence regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e....
 hexahedron
Hexahedron

A hexahedron is a polyhedron with six faces. A Regular polyhedron hexahedron, with all its faces Square , is a cube.There are many kinds of hexahedra, some topologically similar to the cube and some not....
 and is one of the five Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s. It is a special kind of square prism
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
, of rectangular parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
 and of 3-sided trapezohedron
Trapezohedron

The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kite ....
. The cube is dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 to the octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
. It has cubical symmetry (also called octahedral symmetry
Octahedral symmetry

A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
). A cube is the three-dimensional case of the more general concept of a hypercube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
, which exists in any dimension.

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.

Formulae

For a cube of edge length ,
surface area 
volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
 
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 square
Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
s and second powers.

A cube has the largest volume among cuboid
Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
s (rectangular boxes) with a given surface area
Surface area

Surface area is how much exposed area an object has. It is expressed in square units. If an object has flat Face , its surface area can be calculated by adding together the areas of its faces....
. 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
Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
 coloring the faces. The highest octahedral symmetry
Octahedral symmetry

A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
 Oh has all the faces the same color. The dihedral symmetry
Dihedral symmetry in three dimensions

This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ....
 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
Wythoff symbol

In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a Wythoff construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane....
.

(3 colors)
| 2 2 2
D2h
Tetragonal Prism

(2 colors)
4 2 | 2
D4h
Hexahedron

(1 color)
3 | 4 2
Oh


Geometric relations

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
Zonohedron

A zonohedron is a convex set polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180?....
 (every face has point symmetry).

The cube can be cut into 6 identical square pyramid
Square pyramid

In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
s. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron
Rhombic dodecahedron

The rhombic dodecahedron is a convex set polyhedron with 12 rhombus faces. It is an Archimedean solid solid, or a Catalan solid. Its dual is the cuboctahedron....
 is obtained.

Other dimensions

The analogue of a cube in four-dimensional Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
 has a special name — a tesseract
Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
 or (rarely) hypercube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
.

The analogue of the cube in n-dimensional Euclidean space is called a hypercube
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a Square and a cube . It is a Closed set, Compact space, Convex set figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other and of the same length....
 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
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
 in dimension 0, a segment in one dimension and a square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 in two dimensions.

Related polyhedra

The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron
Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
. These two together form a regular compound
Polyhedral compound

A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of star polygon#Star figuress such as the hexagram....
, the stella octangula
Stella octangula

The stella octangula, also known as the stellated octahedron, Star Tetrahedron, eight-pointed star, or 2D geometric model as the Star of David....
. The intersection of the two forms a regular octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
. 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 ? 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
Rectification (geometry)

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....
 cube is the cuboctahedron
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
. If smaller corners are cut off we get a polyhedron with 6 octagon
Octagon

In geometry, an octagon is a polygon that has 8 sides. A regular octagon is represented by the Schl?fli symbol ....
al faces and 8 triangular ones. In particular we can get regular octagons (truncated cube
Truncated cube

The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....
). The rhombicuboctahedron
Rhombicuboctahedron

The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangle and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each....
 is obtained by cutting off both corners and edges to the correct amount.

A cube can be inscribed in a dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 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
Polyhedral compound

A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of star polygon#Star figuress such as the hexagram....
 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
Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
.

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

A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
).

The cube is a special case in various classes of general polyhedra:
NameEqual edge-lengths?Equal angles?Right angles?
CubeYesYesYes
Rhombohedron
Rhombohedron

In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombus. It is a special case of a parallelepiped where all edges are the same length....
YesYesNo
Cuboid
Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
NoYesYes
Parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
NoYesNo
quadrilaterally
Quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on....
-faced hexahedron
Hexahedron

A hexahedron is a polyhedron with six faces. A Regular polyhedron hexahedron, with all its faces Square , is a cube.There are many kinds of hexahedra, some topologically similar to the cube and some not....
NoNoNo


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
Hypercube graph

In the mathematics field of graph theory, the hypercube graph Qn is a regular graph with 2n vertex , which correspond to the subsets of a Set with n elements....
.

An extension is the 3-dimensional k-ary Hamming graph
Hamming graph

Hamming graphs are a special class of graph used in several branches of mathematics and computer science. Let S be a set of q elements and d a positive integer....
, which for k = 2 is the cube graph. Graphs of this sort occur in the theory of parallel processing
Parallel computing

Parallel computing is a form of computing in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved Concurrency ....
 in computers.

See also

  • Kaaba
    Kaaba

    The Kaaba "Cube" is a cuboidal building in Mecca, Saudi Arabia, and is the Most holy place#Islam in Islam. The building is more than two thousand years old, and according to Islamic tradition the first building at the site was built by Abraham ....
    , a large masonry structure roughly the shape of a cube
  • Unit cube
    Unit cube

    A unit cube is a cube all of whose sides are 1 unit long. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units....
  • Square
    Square

    Square may mean:...
  • Tesseract
    Tesseract

    In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
  • Cube (film)
    Cube (film)

    Cube is a 1997 in film Canada psychological thriller/horror /science fiction film film director by Vincenzo Natali. The film was a very successful product of the Canadian Film Centre's First Feature Project....


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