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{{Short description|Sphere with radius one, usually centered on the origin of the space}}
[[File:Vector norms.svg|frame|right|Some 1-spheres: {{math|‖'''x'''‖<sub>2</sub>}} is the norm for Euclidean space.]]

In [[Mathematics|mathematics]], a '''unit sphere''' is a [[Sphere|sphere]] of unit [[Radius|radius]]: the [[Locus (mathematics)|set of points]] at [[Euclidean distance]] [[1|1]] from some center point in [[Three-dimensional space|three-dimensional space]]. More generally, the ''unit <math>n</math>-sphere'' is an [[N-sphere|<math>n</math>-sphere]] of unit radius in <math>(n+1)</math>-[[Physics:Dimension|dimension]]al [[Euclidean space]]; the [[Unit circle|unit circle]] is a special case, the unit <math>1</math>-sphere in the [[Euclidean plane|plane]]. An ([[Open set|open]]) '''unit ball''' is the region inside of a unit sphere, the set of points of distance less than 1 from the center.

A sphere or ball with unit radius and center at the [[Origin (mathematics)|origin]] of the space is called ''the'' unit sphere or ''the'' unit ball. Any arbitrary sphere can be transformed to the unit sphere by a combination of [[Translation (geometry)|translation]] and [[Scaling (geometry)|scaling]], so the study of spheres in general can often be reduced to the study of the unit sphere.

The unit sphere is often used as a model for [[Spherical geometry|spherical geometry]] because it has constant [[Sectional curvature|sectional curvature]] of 1, which simplifies calculations. In [[Trigonometry|trigonometry]], circular [[Arc length|arc length]] on the unit circle is called [[Radian|radian]]s and used for measuring [[Astronomy:Angular distance|angular distance]]; in [[Spherical trigonometry|spherical trigonometry]] surface area on the unit sphere is called [[Steradian|steradian]]s and used for measuring [[Solid angle|solid angle]].

In more general contexts, a ''unit sphere'' is the set of points of [[Distance|distance]] 1 from a fixed central point, where different [[Norm (mathematics)|norms]] can be used as general notions of "distance", and an (open) ''unit ball'' is the region inside.

== Unit spheres and balls in Euclidean space ==

In [[Euclidean space]] of <math>n</math> dimensions, the <math>(n-1)</math>-dimensional unit sphere is the set of all points <math>(x_1, \ldots, x_n)</math> which satisfy the equation
: <math> x_1^2 + x_2^2 + \cdots + x_n ^2 = 1.</math>

The open unit <math>n</math>-ball is the set of all points satisfying the [[Inequality (mathematics)|inequality]]
: <math> x_1^2 + x_2^2 + \cdots + x_n ^2 < 1,</math>
and closed unit <math>n</math>-ball is the set of all points satisfying the inequality
: <math> x_1^2 + x_2^2 + \cdots + x_n ^2 \le 1.</math>

=== Volume and area ===
[[File:Graphs of volumes (V) and surface areas (S) of n-balls of radius 1.png|thumb|right|Graphs of volumes ({{mvar|V}}) and surface areas ({{mvar|S}}) of unit {{mvar|n}}-balls.]]

The classical equation of a unit sphere is that of the ellipsoid with a radius of 1 and no alterations to the <math>x</math>-, <math>y</math>-, or <math>z</math>- axes:
: <math>x^2 + y^2 + z^2 = 1</math>

The volume of the unit ball in Euclidean <math>n</math>-space, and the surface area of the unit sphere, appear in many important formulas of [[Mathematical analysis|analysis]].  The volume of the unit <math>n</math>-ball, which we denote <math>V_n,</math> can be expressed by making use of the [[Gamma function|gamma function]].  It is
: <math>V_n = \frac{\pi ^ {n/2}}{\Gamma(1+n/2)} = \begin{cases}
{\pi^{n/2}}/{(n/2)!} & \mathrm{if~}n \ge 0\mathrm{~is~even} \\[6mu]
{2(2\pi)^{(n-1)/2}}/{n!!} & \mathrm{if~}n \ge 0\mathrm{~is~odd,}
\end{cases} </math>
where <math>n!!</math> is the [[Double factorial|double factorial]].

The hypervolume of the <math>(n-1)</math>-dimensional unit sphere (''i.e.'', the "area" of the boundary of the <math>n</math>-dimensional unit ball), which we denote <math>A_{n-1},</math> can be expressed as
: <math>A_{n-1} = n V_n = \frac{n \pi ^ {n/2}}{\Gamma(1+n/2)} = \frac{2 \pi ^ {n/2}}{\Gamma(n/2)} = \begin{cases}
{2 \pi^{n/2}}/{(n/2 - 1)!} & \mathrm{if~}n \ge 1\mathrm{~is~even} \\[6mu]
{2(2\pi)^{(n-1)/2}}/{(n-2)!!} & \mathrm{if~}n \ge 1\mathrm{~is~odd.}
\end{cases}</math>
For example, <math>A_0 = 2</math> is the "area" of the boundary of the unit ball <math>[-1,1] \subset \mathbb{R}</math>, which simply counts the two points. Then <math>A_1 = 2\pi</math> is the "area" of the boundary of the unit disc, which is the circumference of the unit circle. <math>A_2 = 4\pi</math> is the area of the boundary of the unit ball <math>\{ x \in \R^3 : x_1^2 + x_2^2 + x_3^2 \leq 1 \}</math>, which is the surface area of the unit sphere <math>\{ x \in \R^3 : x_1^2 + x_2^2 + x_3^2 = 1 \}</math>.

The surface areas and the volumes for some values of <math>n</math> are as follows:

{| class="wikitable" style="text-align:center"
! <math>n</math>
! colspan=2|<math>A_{n-1}</math> (surface area)
! colspan=2|<math>V_n</math> (volume)
|-
! 0
| || || <math>(1/0!)\pi^0 </math> || 1 
|-
! 1
| <math>1(2^1/1!!)\pi^0 </math> || 2 || <math>(2^1/1!!)\pi^0 </math> || 2 
|-
! 2
| <math>2(1/1!)\pi^1 = 2 \pi </math> || 6.283 || <math>(1/1!)\pi^1 = \pi </math> || 3.141 
|-
! 3
| <math>3(2^2/3!!)\pi^1  = 4 \pi </math> || 12.57 || <math>(2^2/3!!)\pi^1  = (4/3)\pi </math> || 4.189 
|-
! 4
| <math>4(1/2!)\pi^2 = 2 \pi^2 </math> || 19.74 || <math>(1/2!)\pi^2 = (1/2)\pi^2 </math> || 4.935 
|-
! 5
| <math>5(2^3/5!!)\pi^2 = (8/3)\pi^2 </math> || 26.32 || <math>(2^3/5!!)\pi^2 = (8/15)\pi^2 </math> || 5.264 
|-
! 6
| <math>6(1/3!)\pi^3 = \pi^3 </math> || 31.01 || <math>(1/3!)\pi^3 = (1/6)\pi^3 </math> || 5.168 
|-
! 7
| <math>7(2^4/7!!) \pi^3 = (16/15)\pi^3 </math> || 33.07 || <math>(2^4/7!!) \pi^3 = (16/105)\pi^3 </math> || 4.725 
|-
! 8
| <math>8(1/4!)\pi^4 = (1/3)\pi^4 </math> || 32.47 || <math>(1/4!)\pi^4 = (1/24)\pi^4 </math> || 4.059 
|-
! 9
| <math>9(2^5/9!!) \pi^4 = (32/105)\pi^4 </math> || 29.69 || <math>(2^5/9!!) \pi^4 = (32/945)\pi^4 </math> || 3.299 
|-
! 10
| <math>10(1/5!)\pi^5 = (1/12)\pi^5 </math> || 25.50 || <math>(1/5!)\pi^5 = (1/120)\pi^5 </math> || 2.550 
|}
where the decimal expanded values for <math>n \geq 2</math> are rounded to the displayed precision.

==== Recursion ====

The <math>A_n</math> values satisfy the recursion:
: <math>A_0 = 2</math>
: <math>A_1 = 2\pi</math>
: <math>A_n = \frac{2 \pi}{n-1} A_{n-2}</math> for <math>n > 1</math>.

The <math>V_n</math> values satisfy the recursion:
: <math>V_0 = 1</math>
: <math>V_1 = 2</math>
: <math>V_n = \frac{2 \pi}{n} V_{n-2}</math> for <math>n > 1</math>.

==== Non-negative real-valued dimensions ====
{{Main|Hausdorff measure}}

The value <math display=inline>2^{-n} V_n = \pi ^ {n/2} \big/\, 2^n \Gamma\bigl(1+\tfrac12n\bigr)</math> at non-negative real values of <math>n</math> is sometimes used for normalization of Hausdorff measure.<ref>[https://www.math.cuhk.edu.hk/course_builder/1415/math5011/MATH5011_Chapter_3.2014.pdf The Chinese University of Hong Kong, Math 5011, Chapter 3, Lebesgue and Hausdorff Measures]</ref><ref>{{cite journal | last = Manin | first = Yuri I. |  title = The notion of dimension in geometry and algebra | journal = Bulletin of the American Mathematical Society | volume = 43 | issue = 2 | pages = 139–161 | publisher = | location = | date = | language = | url = https://www.ams.org/bull/2006-43-02/S0273-0979-06-01081-0/S0273-0979-06-01081-0.pdf | jstor = | issn = | doi = | id = | mr = | zbl = | jfm = | access-date = 17 December 2021}}</ref>

==== Other radii ====

The surface area of an <math>(n-1)</math>-sphere with radius <math>r</math> is <math>A_{n-1} r^{n-1}</math> and the volume of an <math>n</math>- ball with radius <math>r</math> is <math>V_{n} r^{n}.</math>  For instance, the area is <math>A_2 = 4\pi r^2</math> for the two-dimensional surface of the three-dimensional ball of radius <math>r.</math>  The volume is <math>V_3 = \tfrac43\pi r^3</math> for the three-dimensional ball of radius&nbsp;<math>r</math>.

== Unit balls in normed vector spaces ==

The '''open unit ball''' of a [[Normed vector space|normed vector space]] <math>V</math> with the [[Norm (mathematics)|norm]] <math>\|\cdot\|</math> is given by
: <math> \{ x\in V: \|x\|<1 \}</math>

It is the [[Interior (topology)|topological interior]] of the '''closed unit ball''' of <math>(V, \|\cdot\|)\colon</math>
: <math> \{ x\in V: \|x\|\le 1\}</math>

The latter is the disjoint union of the former and their common border, the '''unit sphere''' of <math>(V, \|\cdot\|)\colon</math>
: <math> \{ x\in V: \|x\| = 1 \}</math>

The "shape" of the ''unit ball'' is entirely dependent on the chosen norm; it may well have "corners", and for example may look like <math>[-1,1]^n</math> in the case of the max-norm in <math>\mathbb{R}^n</math>. One obtains a naturally ''round ball'' as the unit ball pertaining to the usual [[Hilbert space]] norm, based in the finite-dimensional case on the [[Euclidean distance]]; its boundary is what is usually meant by the ''unit sphere''.

Let  <math>x=(x_1,...x_n)\in \R^n.</math> Define the usual <math>\ell_p</math>-norm for <math>p \ge 1</math> as:
: <math>\|x\|_p = \biggl(\sum_{k=1}^n |x_k|^p \biggr)^{1/p}</math>

Then <math>\|x\|_2</math> is the usual [[Hilbert space]] norm.
<math>\|x\|_1</math> is called the Hamming norm, or <math>\ell_1</math>-norm.
The condition <math>p \geq 1</math> is necessary in the definition of the <math>\ell_p</math> norm, as the unit ball in any normed space must be [[Convex set|convex]] as a consequence of the [[Triangle inequality|triangle inequality]].
Let <math>\|x\|_\infty</math> denote the max-norm or <math>\ell_\infty</math>-norm of <math>x</math>.

Note that for the one-dimensional circumferences <math>C_p</math> of the two-dimensional unit balls, we have:
: <math>C_{1} = 4 \sqrt{2}</math> is the minimum value.
: <math>C_{2} = 2 \pi</math>
: <math>C_{\infty} = 8</math> is the maximum value.

== Generalizations ==

=== Metric spaces ===
All three of the above definitions can be straightforwardly generalized to a [[Metric space|metric space]], with respect to a chosen origin. However, topological considerations (interior, closure, border) need not apply in the same way (e.g., in ultrametric spaces, all of the three are simultaneously open and closed sets), and the unit sphere may even be empty in some metric spaces.

=== Quadratic forms ===
If <math>V</math> is a linear space with a real [[Quadratic form|quadratic form]] <math>F : V \to \R,</math> then <math>\{ p \in V : F(p) = 1\}</math> may be called the unit sphere<ref>Takashi Ono (1994) ''Variations on a Theme of Euler: quadratic forms, elliptic curves, and Hopf maps'', chapter 5: Quadratic spherical maps, page 165, Plenum Press, {{isbn|0-306-44789-4}}</ref><ref>F. Reese Harvey (1990) ''Spinors and calibrations'', "Generalized Spheres", page 42, [[Company:Academic Press|Academic Press]], {{isbn|0-12-329650-1}}</ref> or [[Hyperboloid#Relation to the sphere|unit quasi-sphere]] of <math>V.</math> For example, the quadratic form <math>x^2 - y^2</math>, when set equal to one, produces the [[Unit hyperbola|unit hyperbola]], which plays the role of the "unit circle" in the plane of [[Split-complex number|split-complex number]]s. Similarly, the quadratic form <math>x^2</math> yields a pair of lines for the unit sphere in the [[Dual number|dual number]] plane.

== See also ==
* [[Ball (mathematics)|Ball]]
* [[N-sphere|<math>n</math>-sphere]]
* [[Sphere]]
* [[Superellipse]]
* [[Unit circle]]
* [[Unit disk]]
* [[Unit tangent bundle]]
* [[Unit square]]

== Notes and references ==
{{reflist}}
* Mahlon M. Day (1958) ''Normed Linear Spaces'', page 24, [[Physics:Springer-Verlag|Springer-Verlag]].
* {{citation|last1=Deza|first1=E.|first2=M.|last2=Deza|title=Dictionary of Distances|year=2006|publisher=Elsevier|isbn=0-444-52087-2}}. Reviewed in [https://www.scribd.com/doc/2668595/Newsletter-of-the-European-Mathematical-Society-20070664-featuring-Let-Platonism-Die ''Newsletter of the European Mathematical Society'' '''64''' (June 2007)], p.&nbsp;57. This book is organized as a list of distances of many types, each with a brief description.

== External links ==
* {{mathworld | urlname = UnitSphere | title = Unit sphere}}

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