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In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
, an equilateral
cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...
and a right rhombohedron a 3- zonohedron. It is a regular square prism in three orientations, and a
trigonal trapezohedron In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rhomboh ...
in four orientations. The cube is
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces are all squares.


Orthogonal projections

The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes.


Spherical tiling

The cube can also be represented as a
spherical tiling In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most c ...
, and projected onto the plane via a
stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
. This projection is
conformal Conformal may refer to: * Conformal (software), in ASIC Software * Conformal coating in electronics * Conformal cooling channel, in injection or blow moulding * Conformal field theory in physics, such as: ** Boundary conformal field theory ...
, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.


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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
of the vertices are :(±1, ±1, ±1) while the interior consists of all points (''x''0, ''x''1, ''x''2) with −1 < ''x''''i'' < 1 for all ''i''.


Equation in three dimensional space

In
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
, 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 : \max\ = a. A cube can also be considered the limiting case of a 3D superellipsoid as all three exponents approach infinity.


Formulas

For a cube of edge length a: As the volume of a cube is the third power of its sides a \times a \times a, third powers are called ''
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. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
s'', by analogy with squares and second powers. A cube has the largest volume among
cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...
s (rectangular boxes) with a given
surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
. 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: :\frac + \frac = \left(\frac + \frac\right)^2.


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 In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
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, which in 1837
Pierre Wantzel Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge. In a paper from 1837, Wantzel pr ...
proved it to be impossible because the
cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. Fo ...
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 In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
coloring the faces. The highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a solid, with all the six sides being different colors. 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 In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform pol ...
.


Geometric relations

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 In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral 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).


In Theology

Cubes appear in abrahamic religions. The
Kaaba The Kaaba (, ), also spelled Ka'bah or Kabah, sometimes referred to as al-Kaʿbah al-Musharrafah ( ar, ٱلْكَعْبَة ٱلْمُشَرَّفَة, lit=Honored Ka'bah, links=no, translit=al-Kaʿbah al-Musharrafah), is a building at the c ...
in Mecca is one example which is Arabic for “the cube”. They also appear in Judaism as Teffilin and
New Jerusalem In the Book of Ezekiel in the Hebrew Bible, New Jerusalem (, ''YHWH šāmmā'', YHWH sthere") is Ezekiel's prophetic vision of a city centered on the rebuilt Holy Temple, the Third Temple, to be established in Jerusalem, which would be the c ...
in the New Testament is also described as being a Cube.


Other dimensions

The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or
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, ...
. 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 or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
in dimension 0, a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
in one dimension and a square in two dimensions.


Related polyhedra

The quotient of the cube by the antipodal map yields a projective polyhedron, the
hemicube Hemicube can mean: * Hemicube (technology company), a company based in Dubai that develops advanced technology solutions. * Hemicube (computer graphics), a concept in 3D computer graphics rendering *Hemicube (geometry), an abstract regular polytope ...
. If the original cube has edge length 1, its dual polyhedron (an octahedron) has edge length \scriptstyle \sqrt/2. 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. The rectified cube is the cuboctahedron. If smaller corners are cut off we get a polyhedron with six
octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...
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 polyhedral and tilings with order-3 vertex figures. 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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
: , p=3,4,5... With dihedral symmetry, Dih4, the cube is topologically related in a series of uniform polyhedral 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'',3 Coxeter 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 In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rhomboh ...
, 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 In geometry, a convex uniform honeycomb is a uniform polytope, uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex polyhedron, convex uniform polyhedron, uniform polyhedral cells. Twenty-eight such honey ...
s: It is also an element of five four-dimensional
uniform polychora In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There ...
:


Cubical graph

The
skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
of the cube (the vertices and edges) forms a graph with 8 vertices and 12 edges, called the cube graph. It is a special case of the hypercube graph. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid. An extension is the three dimensional ''k''-ARY
Hamming graph Hamming graphs are a special class of graphs named after Richard Hamming and used in several branches of mathematics (graph theory) and computer science. Let be a set of elements and a positive integer. The Hamming graph has vertex set , ...
, which for ''k'' = 2 is the cube graph. Graphs of this sort occur in the theory of parallel processing in computers.


See also

* Pyramid * Tesseract * Trapezohedron


References


External links

*
Cube: Interactive Polyhedron Model


with interactive animation
Cube
(Robert Webb's site) {{Authority control Platonic solids Prismatoid polyhedra Space-filling polyhedra Volume Zonohedra Elementary shapes Cuboids