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In geometry, 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 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.
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
while the interior consists of all points (x_{0}, x_{1}, x_{2}) with −1 < x_{i} < 1.
In analytic geometry, a cube's surface with center (x_{0}, y_{0}, z_{0}) and edge length of 2a is the locus of all points (x, y, z) such that
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).
The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123.
A cube has eleven nets (one shown above): that is, there are eleven ways to flatten a hollow cube by cutting seven edges.^{[2]} 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.)
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.
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 \scriptstyle \sqrt{2}.
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 ^{1}⁄_{3} 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 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.
All these figures have octahedral symmetry.
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.
Miscellaneous cubes
Titanium, India, De Beers, Indiana, New York City
Dungeons & Dragons, American Civil War, Bible, Latin, Asia
Mathematics, Area, 0 (number), Euclidean space, Lebesgue measure
Leonardo da Vinci, Tetrahedron, Dodecahedron, Octahedron, Icosahedron
Tesseract, 24-cell, 16-cell, Dodecahedron, Icosahedron