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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a cube is a
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
solid object bounded by six
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
faces,
facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
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 positio ...
. Viewed from a corner it is a
hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...
and its
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
is usually depicted as a
cross A cross is a geometrical figure consisting of two intersecting lines or bars, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of the Latin letter X, is termed a sa ...
. The cube is the only regular
hexahedron A hexahedron (plural: hexahedra or hexahedrons) or sexahedron (plural: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex. Ther ...
and is one of the five
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s. 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 In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be us ...
a 3-
zonohedron In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
. It is a regular square
prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...
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 to the
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
. It has cubical or
octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
. The cube is the only convex polyhedron whose faces are all
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
s.


Orthogonal projections

The ''cube'' has four special
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
s, centered, on a vertex, edges, face and normal to its
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
. The first and third correspond to the A2 and B2
Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there ar ...
s.


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, 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 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 Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...
of all points (''x'', ''y'', ''z'') such that : \max\ = a. A cube can also be considered the limiting case of a 3D
superellipsoid In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same exponent ''r'', and whose vertical sections through the center are superellipses with the same exponent '' ...
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
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
s 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. 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 Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
, or the ''Delian problem'', was the problem posed by
ancient Greek mathematicians Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
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, 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 In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is cons ...
.


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 In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...
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 In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...
(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 In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedro ...
is obtained (with pairs of coplanar triangles combined into rhombic faces).


In Theology

Cubes appear in
abrahamic religions The Abrahamic religions are a group of religions centered around worship of the God of Abraham. Abraham, a Hebrew patriarch, is extensively mentioned throughout Abrahamic religious scriptures such as the Bible and the Quran. Jewish tradition ...
. 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 Tefillin (; Israeli Hebrew: / ; Ashkenazic pronunciation: ), or phylacteries, are a set of small black leather boxes with leather straps containing scrolls of parchment inscribed with verses from the Torah. Tefillin are worn by adult Jews durin ...
and New Jerusalem in the New Testament is also described as being a Cube.


Other dimensions

The analogue of a cube in four-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
has a special name—a
tesseract In geometry, a tesseract is the four-dimensional analogue 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 six square faces, the hypersurface of the tesseract consists of eig ...
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 In geometry, a (globally) 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. Projec ...
, 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 In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...
(an
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
) 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 In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
; more generally this is referred to as a
demicube In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex ( ...
. These two together form a regular
compound Compound may refer to: Architecture and built environments * Compound (enclosure), a cluster of buildings having a shared purpose, usually inside a fence or wall ** Compound (fortification), a version of the above fortified with defensive struc ...
, the
stella octangula The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depict ...
. 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 A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
. 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 In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...
). The
rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at ea ...
is obtained by cutting off both corners and edges to the correct amount. A cube can be inscribed in a
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
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 figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s. 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 In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...
, 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 A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
. The cube is a part of a sequence of rhombic polyhedra and tilings with 'n'',3
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 reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...
symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. The cube is a
square prism 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 ...
: 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 tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. Twenty-eight such honeycombs are known: * the familiar cubic honeycomb and 7 tr ...
s: It is also an element of five four-dimensional uniform polychora:


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 Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
with 8 vertices and 12 edges, called the cube graph. It is a special case of the
hypercube graph In graph theory, the hypercube graph is the graph formed from the vertices and edges of an -dimensional hypercube. For instance, the cube graph is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. has vertices, e ...
. It is one of 5
Platonic graph In the mathematical field of graph theory, a Platonic graph is a graph that has one of the Platonic solids as its skeleton. There are 5 Platonic graphs, and all of them are regular, polyhedral (and therefore by necessity also 3-vertex-conne ...
s, each a skeleton of its
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
. 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 A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...
*
Tesseract In geometry, a tesseract is the four-dimensional analogue 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 six square faces, the hypersurface of the tesseract consists of eig ...
*
Trapezohedron In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...


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