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In mathematics, a ball is the
solid figure In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), ...
bounded by a ''
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-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 Euclidean ...
but also for lower and higher dimensions, and for
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
is the same thing as a disk, the area bounded by a circle. In
Euclidean 3-space 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 ...
, a ball is taken to be the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The def ...
bounded by a 2-dimensional sphere. In a
one-dimensional space In physics and mathematics, a sequence of ''n'' numbers can specify a location in ''n''-dimensional space. When , the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where th ...
, a ball is 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 other contexts, such as in
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \operatorname B^n or \operatorname D^n.


In Euclidean space

In Euclidean -space, an (open) -ball of radius and center is the set of all points of distance less than from . A closed -ball of radius is the set of all points of distance less than or equal to away from . In Euclidean -space, every ball is bounded by a
hypersphere In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, ca ...
. The ball is a bounded interval when , is a disk bounded by a circle when , and is bounded by a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
when .


Volume

The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:Equation 5.19.4, ''NIST Digital Library of Mathematical Functions.'

Release 1.0.6 of 2013-05-06.
V_n(R) = \fracR^n, where  is
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
's
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except t ...
(which can be thought of as an extension of the
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) ...
function to fractional arguments). Using explicit formulas for
particular values of the gamma function The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general. Othe ...
at the integers and half integers gives formulas for the volume of a Euclidean ball that do not require an evaluation of the gamma function. These are: \begin V_(R) &= \fracR^\,,\\ pt V_(R) &= \fracR^ = \fracR^\,. \end In the formula for odd-dimensional volumes, the
double factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the ...
is defined for odd integers as .


In general metric spaces

Let be a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
, namely a set with a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
(distance function) . The open (metric) ball of radius centered at a point in , usually denoted by or , is defined by B_r(p) = \, The closed (metric) ball, which may be denoted by or , is defined by B_r = \. Note in particular that a ball (open or closed) always includes itself, since the definition requires . A
unit ball Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (al ...
(open or closed) is a ball of radius 1. A subset of a metric space is bounded if it is contained in some ball. A set is
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
if, given any positive radius, it is covered by finitely many balls of that radius. The open balls of a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
can serve as a base, giving this space a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, the open sets of which are all possible
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...
s of open balls. This topology on a metric space is called the topology induced by the metric . Let denote the closure of the open ball in this topology. While it is always the case that , it is always the case that . For example, in a metric space with the
discrete metric Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, a ...
, one has and , for any .


In normed vector spaces

Any
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
with norm \, \cdot\, is also a metric space with the metric d (x,y)= \, x - y\, . In such spaces, an arbitrary ball B_r(y) of points x around a point y with a distance of less than r may be viewed as a scaled (by r) and translated (by y) copy of a ''unit ball'' B_1(0). Such "centered" balls with y=0 are denoted with B(r). The Euclidean balls discussed earlier are an example of balls in a normed vector space.


-norm

In a
Cartesian space 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 ...
with the -norm , that is \left\, x \right\, _p = \left( , x_1, ^p + , x_2, ^p + \dots + , x_n, ^p \right) ^, an open ball around the origin with radius r is given by the set B(r) = \left\. For , in a 2-dimensional plane \R^2, "balls" according to the -norm (often called the ''
taxicab A taxi, also known as a taxicab or simply a cab, is a type of vehicle for hire with a driver, used by a single passenger or small group of passengers, often for a non-shared ride. A taxicab conveys passengers between locations of their choic ...
'' or ''Manhattan'' metric) are bounded by squares with their ''diagonals'' parallel to the coordinate axes; those according to the -norm, also called the Chebyshev metric, have squares with their ''sides'' parallel to the coordinate axes as their boundaries. The -norm, known as the Euclidean metric, generates the well known disks within circles, and for other values of , the corresponding balls are areas bounded by Lamé curves (hypoellipses or hyperellipses). For , the - balls are within octahedra with axes-aligned ''body diagonals'', the -balls are within cubes with axes-aligned ''edges'', and the boundaries of balls for with are superellipsoids. Obviously, generates the inner of usual spheres.


General convex norm

More generally, given any
centrally symmetric In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
, bounded,
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (Y ...
, and
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polyto ...
subset of , one can define a
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
on where the balls are all translated and uniformly scaled copies of . Note this theorem does not hold if "open" subset is replaced by "closed" subset, because the origin point qualifies but does not define a norm on .


In topological spaces

One may talk about balls in any
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, not necessarily induced by a metric. An (open or closed) -dimensional topological ball of is any subset of which is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
to an (open or closed) Euclidean -ball. Topological -balls are important in
combinatorial topology In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such a ...
, as the building blocks of cell complexes. Any open topological -ball is homeomorphic to the Cartesian space and to the open unit -cube (hypercube) . Any closed topological -ball is homeomorphic to the closed -cube . An -ball is homeomorphic to an -ball if and only if . The homeomorphisms between an open -ball and can be classified in two classes, that can be identified with the two possible topological orientations of . A topological -ball need not be smooth; if it is smooth, it need not be
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
to a Euclidean -ball.


Regions

A number of special regions can be defined for a ball: *'' cap'', bounded by one plane *'' sector'', bounded by a conical boundary with apex at the center of the sphere *'' segment'', bounded by a pair of parallel planes *''
shell Shell may refer to: Architecture and design * Shell (structure), a thin structure ** Concrete shell, a thin shell of concrete, usually with no interior columns or exterior buttresses ** Thin-shell structure Science Biology * Seashell, a hard o ...
'', bounded by two concentric spheres of differing radii *''
wedge A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by conver ...
'', bounded by two planes passing through a sphere center and the surface of the sphere


See also

*
Ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used fo ...
– ordinary meaning *
Disk (mathematics) In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usu ...
*
Formal ball In topology, a formal ball is an extension of the notion of ball to allow unbounded and negative radius. The concept of formal ball was introduced by Weihrauch and Schreiber in 1981 and the negative radius case (the generalized formal ball) by T ...
, an extension to negative radii *
Neighbourhood (mathematics) In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a po ...
*
Sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
, a similar geometric shape * 3-sphere * -sphere, or hypersphere * Alexander horned sphere *
Manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
* Volume of an -ball *
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 ...
– a 3-ball in the metric.


References

* * * {{DEFAULTSORT:Ball (Mathematics) Balls Metric geometry Spheres Topology