, a cube is a three-dimensional
solid object bounded by six square
or sides, with three meeting at each vertex
The cube is the only regular hexahedron
and is one of the five Platonic solid
s. It has 6 faces, 12 edges, 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
The cube is the only convex polyhedron whose faces are all square
The ''cube'' has four special orthogonal projection
s, centered, on a vertex, edges, face and normal to its vertex figure
. The first and third correspond to the A2
and B2 Coxeter plane
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.
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 (''x''0
) with −1 < ''x''''i''
< 1 for all ''i''.
In analytic geometry
, a cube's surface with center (''x''0
) and edge length of ''2a'' is the locus
of all points (''x'', ''y'', ''z'') such that
A cube can also be considered the limiting case of a 3D superellipsoid
as all three exponents approach infinity.
For a cube of edge length
As the volume of a cube is the third power of its sides
, third power
s are called ''cube
s'', by analogy with square
s and second powers.
A cube has the largest volume among cuboid
s (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).
Point in space
For a cube whose circumscribing sphere has radius ''R'', and for a given point in its 3-dimensional space with distances ''di
'' from the cube's eight vertices, we have:
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 four 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
comes from the cube being a prism, with all four sides being the same color. The prismatic subsets D2d
has the same coloring as the previous one and D2h
has alternating colors for its sides for a total of three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol
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 pyramid
s. 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).
The analogue of a cube in four-dimensional Euclidean space
has a special name—a tesseract
. 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 line segment
in one dimension and a square in two dimensions.
thumb|200px|right|The dual of a cube is an octahedron
, seen here with vertices at the center of the cube's square faces.
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
) has edge length
The cube is a special case in various classes of general polyhedra:
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 of that of the cube. The remaining space consists of four equal irregular tetrahedra with a volume of of that of the cube, each.
cube is the cuboctahedron
. If smaller corners are cut off we get a polyhedron with six octagon
al 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 figure
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
: , p=3,4,5...
With dihedral symmetry
, the cube is topologically related in a series of uniform polyhedra and tilings 4.2n.2n, extending into the hyperbolic plane:
All these figures have octahedral symmetry
The cube is a part of a sequence of rhombic polyhedra and tilings with 'n'',3Coxeter group
symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares.
The cube is a square prism
As a trigonal trapezohedron
, the cube is related to the hexagonal dihedral symmetry family.
In uniform honeycombs and polychora
It is an element of 9 of 28 convex uniform honeycomb
It is also an element of five four-dimensional uniform polychora
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
It is one of 5 Platonic graph
s, 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
*Cube: Interactive Polyhedron Model
with interactive animationCube
(Robert Webb's site)