Editing Sphere (section)

==Generalizations==
===Ellipsoids===
An [[ellipsoid]] is a sphere that has been stretched or compressed in one or more directions. More exactly, it is the image of a sphere under an [[affine transformation]]. An ellipsoid bears the same relationship to the sphere that an [[ellipse]] does to a circle.

===Dimensionality===
{{Main|n-sphere}}
Spheres can be generalized to spaces of any number of [[dimension]]s. For any [[natural number]] {{mvar|n}}, an ''{{mvar|n}}-sphere,'' often denoted {{math|''S''{{px2}}<sup>''n''</sup>}}, is the set of points in ({{math|''n'' + 1}})-dimensional Euclidean space that are at a fixed distance {{mvar|r}} from a central point of that space, where {{mvar|r}} is, as before, a positive real number. In particular:
*{{math|''S''{{px2}}<sup>0</sup>}}: a 0-sphere consists of two discrete points, {{math|−''r''}} and {{math|''r''}}
*{{math|''S''{{px2}}<sup>1</sup>}}: a 1-sphere is a [[circle]] of radius ''r''
*{{math|''S''{{px2}}<sup>2</sup>}}: a 2-sphere is an ordinary sphere
*{{math|''S''{{px2}}<sup>3</sup>}}: a [[3-sphere]] is a sphere in 4-dimensional Euclidean space.

Spheres for {{math|''n'' > 2}} are sometimes called [[hypersphere]]s.

The {{mvar|n}}-sphere of unit radius centered at the origin is denoted {{math|''S''{{px2}}<sup>''n''</sup>}} and is often referred to as "the" {{mvar|n}}-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.

In [[topology]], the {{mvar|n}}-sphere is an example of a [[compact space|compact]] [[topological manifold]] without [[Boundary (topology)|boundary]]. A topological sphere need not be [[Manifold#Differentiable manifolds|smooth]]; if it is smooth, it need not be [[diffeomorphic]] to the Euclidean sphere (an [[exotic sphere]]).

The sphere is the inverse image of a one-point set under the continuous function {{math|{{norm|''x''}}}}, so it is closed; {{math|''S<sup>n</sup>''}} is also bounded, so it is compact by the [[Heine–Borel theorem]].

===Metric spaces===
{{Main|Metric space}}
More generally, in a [[metric space]] {{math|(''E'',''d'')}}, the sphere of center {{mvar|x}} and radius {{math|''r'' > 0}} is the set of points {{mvar|y}} such that {{math|1=''d''(''x'',''y'') = ''r''}}.

If the center is a distinguished point that is considered to be the origin of {{mvar|E}}, as in a [[norm (mathematics)|normed]] space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a [[unit sphere]].

Unlike a [[ball (mathematics)|ball]], even a large sphere may be an empty set. For example, in {{math|'''Z'''<sup>''n''</sup>}} with [[Euclidean metric]], a sphere of radius {{math|''r''}} is nonempty only if {{math|''r''<sup>2</sup>}} can be written as sum of {{math|''n''}} squares of [[integer]]s.

An [[octahedron]] is a sphere in [[taxicab geometry]], and a [[cube]] is a sphere in geometry using the [[Chebyshev distance]].