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 inform ...
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
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
and its
net 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 s ...
.
The cube is the only
regular hexahedron 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 e ...
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 Euclid ...
, 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 c ...
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 rh ...
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
squares.
Orthogonal projections
The ''cube'' has four special
orthogonal projections, 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
. The first and third correspond to the A
2 and B
2 Coxeter planes.
Spherical tiling
The cube can also be represented as a
spherical tiling, 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 (the ''projection plane'') perpendicular to the diameter thro ...
. 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
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 ...
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 enginee ...
, 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
:
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
:
As the volume of a cube is the third power of its sides
, 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 ...
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 c ...
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 ...
. 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 ''d
i'' 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
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
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 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. F ...
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 fa ...
coloring the faces. The highest octahedral symmetry O
h has all the faces the same color. The
dihedral symmetry D
4h comes from the cube being a solid, with all the six sides being different colors. The prismatic subsets D
2d has the same coloring as the previous one and D
2h has alternating colors for its sides for a total of three colors, paired by opposite sides. Each symmetry form has a different
Wythoff symbol.
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 pyrami ...
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 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 duri ...
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 ...
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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
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 e ...
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, p ...
. 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 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 i ...
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.
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 oth ...
(an
octahedron) 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
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 ...
; 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, w ...
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 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
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=3,4,5...
With
dihedral symmetry, Dih
4, 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 rh ...
, 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 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
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 e ...
.
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 se ...
, 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, quadrilate ...
*
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 e ...
*
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