Cube
Overview

In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a cube is a three-dimensional
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

solid object bounded by six square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

faces, facet
Facet
Facets are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure...

s or sides, with three meeting at each vertex
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

. The cube can also be called a regular
Regular polyhedron
A regular polyhedron is a polyhedron whose faces are congruent 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. it is transitive on its flags...

hexahedron
Hexahedron
A hexahedron is any polyhedron with six faces, although usually implies the cube as a regular hexahedron with all its faces square, and three squares around each vertex....

and is one of the five Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

s. It is a special kind of square prism
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

, of rectangular parallelepiped
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...

and of trigonal trapezohedron
Trigonal trapezohedron
In geometry, the trigonal trapezohedron or deltohedron is the first in an infinite series of face-uniform polyhedra which are dual to the antiprisms. It has six faces which are congruent rhombi....

. The cube is dual
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

to the octahedron
Octahedron
In geometry, 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 vertex....

. It has cubical symmetry (also called octahedral symmetry
Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

). It is special by being a cuboid
Cuboid
In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

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

.

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.
For a cube of edge length ,

As the volume of a cube is the third power of its sides a×a×a, third power
Third Power
Third Power was a hard rock band from Detroit, Michigan, who released one album in 1970.The group was formed in 1969 and became a prominent local bar band before signing to Vanguard Records. Guitarist Drew Abbott and bassist Jem Targal shared singing duties. They released an album, Believe, on the...

s are called cubes, by analogy with squares 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 definitions of a cuboid in mathematical literature...

s (rectangular boxes) with a given surface area
Surface area
Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...

.
Quotations

My parents are these people, I live with them…I'm boring.

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

I don't wanna die, I'm just being realistic. You think they'd go to all the trouble to build this thing if we could just walk out?

Do you think we matter? We don't.

I mean, this is an accident, a forgotten, perpetual public works project. Do you think anybody wants to ask questions? All they want is a clear conscience and a fat paycheck.

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

Just out of curiosity—I mean, don't hit me again, I think—but what are you going to do when you get there?

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

You can't see the big picture from in here, so don't try. Keep your head down, keep it simple. Just look at what's in front of you.

Leaven…you beautiful brain.

Encyclopedia
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a cube is a three-dimensional
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

solid object bounded by six square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

faces, facet
Facet
Facets are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure...

s or sides, with three meeting at each vertex
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

. The cube can also be called a regular
Regular polyhedron
A regular polyhedron is a polyhedron whose faces are congruent 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. it is transitive on its flags...

hexahedron
Hexahedron
A hexahedron is any polyhedron with six faces, although usually implies the cube as a regular hexahedron with all its faces square, and three squares around each vertex....

and is one of the five Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

s. It is a special kind of square prism
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

, of rectangular parallelepiped
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...

and of trigonal trapezohedron
Trigonal trapezohedron
In geometry, the trigonal trapezohedron or deltohedron is the first in an infinite series of face-uniform polyhedra which are dual to the antiprisms. It has six faces which are congruent rhombi....

. The cube is dual
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

to the octahedron
Octahedron
In geometry, 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 vertex....

. It has cubical symmetry (also called octahedral symmetry
Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

). It is special by being a cuboid
Cuboid
In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

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

.

## 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 ,
 surface area volumeVolumeVolume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains.... face diagonalFace diagonalIn geometry, a face diagonal of a polyhedron is a diagonal on one of the faces, in contrast to a space diagonal passing through the interior of the polyhedron.... space diagonalSpace diagonalIn a rectangular box or a magic cube, the four space diagonals are the lines that go from a corner of the box or cube, through the center of the box or cube, to the opposite corner... radius of circumscribed sphereCircumscribed sphereIn geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing... radius of sphere tangent to edges radius of inscribed sphereInscribed sphereIn geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces... angles between facesDihedral angleIn geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...

As the volume of a cube is the third power of its sides a×a×a, third power
Third Power
Third Power was a hard rock band from Detroit, Michigan, who released one album in 1970.The group was formed in 1969 and became a prominent local bar band before signing to Vanguard Records. Guitarist Drew Abbott and bassist Jem Targal shared singing duties. They released an album, Believe, on the...

s are called cubes, by analogy with squares 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 definitions of a cuboid in mathematical literature...

s (rectangular boxes) with a given surface area
Surface area
Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...

. 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
Vertex-transitive
In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same...

coloring the faces. The highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry
Dihedral symmetry in three dimensions
This article deals with three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn .See also point groups in two dimensions.Chiral:...

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, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles....

.
Name Regular hexahedron Square prism
Prism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

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

Trigonal trapezohedron
Trapezohedron
The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites . The faces are symmetrically staggered.The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry...

Coxeter-Dynkin
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

Schläfli symbol {4,3} {4}x{} {}x{}x{}
Wythoff symbol
Wythoff symbol
In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles....

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)

(111)

(112)

(123)

(111), (112), (122), and (222)

## Geometric relations

A cube has 11 nets
Net (polyhedron)
In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron...

(one shown above): that is, there are 11 ways to flatten a hollow cube by cutting 7 edges. To colour the cube so that no two adjacent faces have the same colour, one would need at least 3 colours.

The cube is the cell of the only regular tiling of 3 dimensional Euclidean space
Cubic honeycomb
The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron....

. 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 polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180°. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional...

(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.- Johnson solid :...

s. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron
Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...

is obtained (with pairs of coplanar triangles combined into rhombic faces.)

## Other dimensions

The analogue of a cube in four-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

has a special name—a tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

or hypercube
Hypercube
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...

. 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
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

in dimension 0, a segment in one dimension and a square in two dimensions.

## Related polyhedra

The quotient of the cube by the antipodal
Antipodal point
In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....

map yields a projective polyhedron
Projective polyhedron
In geometry, a projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids....

, the hemicube.

If the original cube has edge length 1, its dual polyhedron
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

(an octahedron
Octahedron
In geometry, 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 vertex....

) has edge length .

The cube is a special case in various classes of general polyhedra:
NameEqual edge-lengths?Equal angles?Right angles?
Cube Yes Yes Yes
Rhombohedron
Rhombohedron
In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombi. It is a special case of a parallelepiped where all edges are the same length....

Yes Yes No
Cuboid No Yes Yes
Parallelepiped No Yes No
quadrilateral
Quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...

ly faced hexahedron
No No No

The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular 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...

; more generally this is referred to as a demicube. 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 polygonal compounds such as the hexagram....

, the stella octangula
Stella octangula
The stellated octahedron, or stella octangula, is the only stellation of the octahedron. It was named by Johannes Kepler in 1609, though it was known to earlier geometers...

. 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
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. As such it is a quasiregular polyhedron,...

. 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
Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....

). The rhombicuboctahedron
Rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles...

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.
 Cube Truncated cubeTruncated cubeIn geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices.... cuboctahedronCuboctahedronIn 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. As such it is a quasiregular polyhedron,... Truncated octahedronTruncated octahedronIn geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron.... OctahedronOctahedronIn geometry, 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 vertex.... Rhombi-cuboctahedronRhombicuboctahedronIn geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles... truncated cuboctahedronTruncated cuboctahedronIn geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges... Snub cubeSnub cubeIn geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each... Stella octangulaStella octangulaThe stellated octahedron, or stella octangula, is the only stellation of the octahedron. It was named by Johannes Kepler in 1609, though it was known to earlier geometers...

All these figures have octahedral symmetry
Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

.

The cube is a part of a sequence of rhombic polyhedra and tilings with [n,3] Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.
Polyhedra Euclidean tiling Hyperbolic tiling
[3,3] [4,3] [5,3] [6,3] [7,3] [8,3]

Cube
Cube
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 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...

Rhombic dodecahedron
Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...

Rhombic triacontahedron
Rhombic triacontahedron
In geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. It is the polyhedral dual of the icosidodecahedron, and it is a zonohedron....

Rhombille

 Compound of three cubesCompound of three cubesThis uniform polyhedron compound is a symmetric arrangement of 3 cubes, considered as square prisms. It can be constructed by superimposing three identical cubes, and then rotating each by 45 degrees about a separate axis .This compound famously appears in the lithograph print Waterfall by M.C.... Compound of five cubesCompound of five cubesThis polyhedral compound is a symmetric arrangement of five cubes. This compound was first described by Edmund Hess in 1876.It is one of five regular compounds, and dual to the compound of five octahedra....

### In uniform honeycombs and polychora

It is an element of 9 of 28 convex uniform honeycomb
Convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...

s:
 Cubic honeycombCubic honeycombThe cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron.... Truncated square prismatic honeycombTruncated square prismatic honeycombThe truncated square prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of octagonal prisms and cubes in a ratio of 1:1.It is constructed from a truncated square tiling extruded into prisms.... Snub square prismatic honeycombSnub square prismatic honeycombThe snub square prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2.It is constructed from a Snub square tiling extruded into prisms.... Elongated triangular prismatic honeycombElongated triangular prismatic honeycombThe elongated triangular prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2.It is constructed from an elongated triangular tiling extruded into prisms.... Gyroelongated triangular prismatic honeycombGyroelongated triangular prismatic honeycombThe gyroelongated triangular prismatic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2.... Cantellated cubic honeycombCantellated cubic honeycombThe cantellated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of small rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3.- References :... Cantitruncated cubic honeycombCantitruncated cubic honeycombThe cantitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3.- Uniform colorings :... Runcitruncated cubic honeycombRuncitruncated cubic honeycombThe runcitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of small rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3.... Runcinated alternated cubic honeycombRuncinated alternated cubic honeycombThe runcinated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of small rhombicuboctahedra, cubes, and tetrahedra in a ratio of 1:1:2.- References :...

It is also an element of five four-dimensional uniform polychora
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....

:
 TesseractTesseractIn geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8... Cantellated 16-cell Runcinated tesseractRuncinated tesseractIn four-dimensional geometry, a runcinated tesseract is a convex uniform polychoron, being a runcination of the regular tesseract.... Cantitruncated 16-cell Runcitruncated 16-cell

## 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
Parallel computing
Parallel computing is a form of computation 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 concurrently . There are several different forms of parallel computing: bit-level,...

in computers.

## See also

• Unit cube
Unit cube
A unit cube, sometimes called a cube of side 1, is a cube 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.- Unit Hypercube :...

• Tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

• Cube (film)
Cube (film)
Cube is a 1997 Canadian science fiction psychological thriller/horror film directed by Vincenzo Natali. The film was a successful product of the Canadian Film Centre's First Feature Project....

• Trapezohedron
Trapezohedron
The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites . The faces are symmetrically staggered.The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry...

• Yoshimoto Cube
Yoshimoto Cube
The Yoshimoto Cube is a polyhedral mechanical puzzle toy invented in 1971 by Japanese man . The cube is made up of eight interconnected cubes and it is capable of folding and unfolding itself in a cyclic fashion. You can keep folding, or unfolding the cube, indefinitely. Once folded, the cube can...

• The Cube (game show)
The Cube (game show)
The Cube is a BAFTA Award–winning British television game show which first aired on ITV on 22 August 2009. Presented by Phillip Schofield, it offers contestants the chance to win a top prize of £250,000 by completing challenges from within a 4x4x4 metre Perspex cube...

• Prince Rupert's cube
Prince Rupert's cube
In geometry, Prince Rupert's cube is the largest cube that can pass through a hole drilled through a unit cube, i.e. through a cube whose sides have length 1. Curiously, it is slightly larger than the unit cube, with a side length of...

• OLAP cube
OLAP cube
An OLAP cube is a data structure that allows fast analysis of data. It can also be defined as the capability of manipulating and analyzing data from multiple perspectives...

## External links

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