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
ball is a region in a space comprising all points within a fixed distance, called the
radius
In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
, from a given point; that is, it is the region enclosed 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 c ...
or
hypersphere. An -ball is a ball in an -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 ...
. The volume of a -ball is the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
of this ball, which generalizes to any dimension the usual volume of a ball in
3-dimensional space. The volume of a -ball of radius is
where
is the volume of the
unit -ball, the -ball of radius .
The
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
can be expressed via a two-dimension
recurrence relation.
Closed-form expressions involve the
gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter r ...
,
factorial, or
double factorial function.
The volume can also be expressed in terms of
, the
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of the
unit -sphere.
Formulas
The first volumes are as follows:
Two-dimension recurrence relation
As is proved
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
using a vector-calculus
double integral in
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
, the volume of an -ball of radius can be expressed recursively in terms of the volume of an -ball, via the interleaved
recurrence relation:
:
This allows computation of in approximately steps.
Closed form
The -dimensional volume of a Euclidean ball of
radius
In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
in -dimensional Euclidean space is:
:
where is
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 excep ...
. The gamma function is offset from but otherwise extends the
factorial function to non-
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
arguments. It satisfies if is a positive integer and if is a non-negative integer.
Alternative forms
The volume can also be expressed in terms of an -ball using the one-dimension recurrence relation:
:
Inverting the above, the radius of an -ball of volume can be expressed recursively in terms of the radius of an - or -ball:
:
Using explicit formulas for
particular values of the gamma function at the integers and
half-integer
In mathematics, a half-integer is a number of the form
:n + \tfrac,
where n is an whole number. For example,
:, , , 8.5
are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
s gives formulas for the volume of a Euclidean ball in terms of
factorials. These are:
:
The volume can also be expressed in terms of
double factorials. For an odd integer , the double factorial is defined by
:
The volume of an odd-dimensional ball is
:
There are multiple conventions for double factorials of even integers. Under the convention in which the double factorial satisfies
:
the volume of an -dimensional ball is, regardless of whether is even or odd,
:
Instead of expressing the volume of the ball in terms of its radius , the formulas can be
inverted to express the radius as a function of the volume:
:
Approximation for high dimensions
Stirling's formula for the gamma function can be used to approximate the volume when the number of dimensions is high.
:
:
In particular, for any fixed value of the volume tends to a limiting value of 0 as goes to infinity. For example, the volume is increasing for , achieves its maximum when , and is decreasing for .
Relation with surface area
Let denote the hypervolume of the
-sphere of radius . The -sphere is the -dimensional boundary (surface) of the -dimensional ball of radius , and the sphere's hypervolume and the ball's hypervolume are related by:
:
Thus, inherits formulas and recursion relationships from , such as
:
There are also formulas in terms of factorials and double factorials.
Proofs
There are many proofs of the above formulas.
The volume is proportional to the th power of the radius
An important step in several proofs about volumes of -balls, and a generally useful fact besides, is that the volume of the -ball of radius is proportional to :
:
The proportionality constant is the volume of the unit ball.
This is a special case of a general fact about volumes in -dimensional space: If
is a body (
measurable set) in that space and is the body obtained by stretching in all directions by the factor then the volume of equals times the volume of . This is a direct consequence of the change of variables formula:
:
where and the substitution was made.
Another proof of the above relation, which avoids multi-dimensional
integration, uses
induction: The base case is , where the proportionality is obvious. For the inductive step, assume that proportionality is true in dimension . Note that the intersection of an ''n''-ball with a hyperplane is an -ball. When the volume of the -ball is written as an integral of volumes of -balls:
:
it is possible by the inductive hypothesis to remove a factor of from the radius of the -ball to get:
:
Making the change of variables leads to:
:
which demonstrates the proportionality relation in dimension . By induction, the proportionality relation is true in all dimensions.
The two-dimension recursion formula
A proof of the recursion formula relating the volume of the -ball and an -ball can be given using the proportionality formula above and integration in
cylindrical coordinates. Fix a plane through the center of the ball. Let denote the distance between a point in the plane and the center of the sphere, and let denote the azimuth. Intersecting the -ball with the -dimensional plane defined by fixing a radius and an azimuth gives an -ball of radius . The volume of the ball can therefore be written as an iterated integral of the volumes of the -balls over the possible radii and azimuths:
:
The azimuthal coordinate can be immediately integrated out. Applying the proportionality relation shows that the volume equals
:
The integral can be evaluated by making the substitution to get
:
which is the two-dimension recursion formula.
The same technique can be used to give an inductive proof of the volume formula. The base cases of the induction are the 0-ball and the 1-ball, which can be checked directly using the facts and . The inductive step is similar to the above, but instead of applying proportionality to the volumes of the -balls, the inductive hypothesis is applied instead.
The one-dimension recursion formula
The proportionality relation can also be used to prove the recursion formula relating the volumes of an -ball and an -ball. As in the proof of the proportionality formula, the volume of an -ball can be written as an integral over the volumes of -balls. Instead of making a substitution, however, the proportionality relation can be applied to the volumes of the -balls in the
integrand:
:
The integrand is an
even function
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power se ...
, so by symmetry the interval of integration can be restricted to . On the interval , it is possible to apply the substitution . This transforms the expression into
:
The integral is a value of a well-known
special function called the
beta function , and the volume in terms of the beta function is
:
The beta function can be expressed in terms of the gamma function in much the same way that factorials are related to
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s. Applying this relationship gives
:
Using the value gives the one-dimension recursion formula:
:
As with the two-dimension recursive formula, the same technique can be used to give an inductive proof of the volume formula.
Direct integration in spherical coordinates
The volume of the ''n''-ball
can be computed by integrating the volume element in
spherical coordinates. The spherical coordinate system has a radial coordinate and angular coordinates , where the domain of each except is , and the domain of is . The spherical volume element is:
:
and the volume is the integral of this quantity over between 0 and and all possible angles:
:
Each of the factors in the integrand depends on only a single variable, and therefore the iterated integral can be written as a product of integrals:
:
The integral over the radius is . The intervals of integration on the angular coordinates can, by the symmetry of the sine about , be changed to :
:
Each of the remaining integrals is now a particular value of the beta function:
:
The beta functions can be rewritten in terms of gamma functions:
:
This product telescopes. Combining this with the values and and the functional equation leads to
:
Gaussian integrals
The volume formula can be proven directly using
Gaussian integrals. Consider the function:
:
This function is both rotationally invariant and a product of functions of one variable each. Using the fact that it is a product and the formula for the Gaussian integral gives:
:
where is the -dimensional volume element. Using rotational invariance, the same integral can be computed in spherical coordinates:
:
where is an -sphere of radius (being the surface of an -ball of radius ) and is the area element (equivalently, the -dimensional volume element). The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If is the surface area of an -sphere of radius , then:
:
Applying this to the above integral gives the expression
:
Substituting :
:
The integral on the right is the gamma function evaluated at .
Combining the two results shows that
:
To derive the volume of an -ball of radius from this formula, integrate the surface area of a sphere of radius for and apply the functional equation :
:
Geometric proof
The relations
and
and thus the volumes of ''n''-balls and areas of ''n''-spheres can also be derived geometrically. As noted above, because a ball of radius
is obtained from a unit ball
by rescaling all directions in
times,
is proportional to
, which implies
. Also,
because a ball is a union of concentric spheres and increasing radius by ''ε'' corresponds to a shell of thickness ''ε''. Thus,
; equivalently,
.
follows from existence of a volume-preserving
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between the unit sphere
and
:
:
(
is an ''n''-tuple;
; we are ignoring sets of measure 0). Volume is preserved because at each point, the difference from
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
is a stretching in the ''xy'' plane (in
times in the direction of constant
) that exactly matches the compression in the direction of the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of
on
(the relevant angles being equal). For
, a similar argument was originally made by
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
in ''
On the Sphere and Cylinder
''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of t ...
''.
Balls in norms
There are also explicit expressions for the volumes of balls in
norms. The norm of the vector in is
:
and an ball is the set of all vectors whose norm is less than or equal to a fixed number called the radius of the ball. The case is the standard Euclidean distance function, but other values of occur in diverse contexts such as
information theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...
,
coding theory, and
dimensional regularization
__NOTOC__
In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini as well as – independently and more comprehensively – by 't Hooft and Veltman for regularizing integrals in the evaluation of ...
.
The volume of an ball of radius is
:
These volumes satisfy recurrence relations similar to those for :
:
and
:
which can be written more concisely using a
generalized binomial coefficient,
:
For , one recovers the recurrence for the volume of a Euclidean ball because .
For example, in the cases (
taxicab norm) and (
max norm), the volumes are:
:
These agree with elementary calculations of the volumes of
cross-polytope
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahe ...
s and
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 ...
s.
Relation with surface area
For most values of , the surface area
of an sphere of radius (the boundary of an -ball of radius ) cannot be calculated by
differentiating the volume of an ball with respect to its radius. While the volume can be expressed as an integral over the surface areas using the
coarea formula, the coarea formula contains a correction factor that accounts for how the -norm varies from point to point. For and , this factor is one. However, if then the correction factor is : the surface area of an sphere of radius in is times the derivative of the volume of an ball. This can be seen most simply by applying the
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
to the vector field to get
:
For other values of , the constant is a complicated integral.
Generalizations
The volume formula can be generalized even further. For positive real numbers , define the ball with limit to be
:
The volume of this ball has been known since the time of Dirichlet:
:
Comparison to norm
Using the
harmonic mean and defining