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n-sphere
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an n-sphere or hypersphere is an n {\displaystyle n} -<a href="/facts/Dimension/0ktmUyac">dimensional</a> generalization of the 1 {\displaystyle 1} -dimensional <a href="/facts/Circle/9fy5ggKn">circle</a> and 2 {\displaystyle 2} -dimensional <a href="/facts/Sphere/4Ivb1cTj">sphere</a> to any non-negative <a href="/facts/Integer/PUwwrawx">integer</a> n {\displaystyle n} . </p><p>The circle is considered <a href="/facts/2-dimensional/f4b397W8">2-dimensional</a> and the sphere <a href="/facts/3-dimensional/KAqKWUbS">3-dimensional</a> because a point within them has one and two degrees of freedom respectively. However, the typical <a href="/facts/Embedding/HKeVKc42">embedding</a> of the 2-dimensional circle is in 3-dimensional space, the 3-dimensional sphere is usually depicted embedded in <a href="/facts/4-dimensional_space/vFQd0M7T">4-dimensional space</a>, and a general n {\displaystyle n} -sphere is embedded in an n + 1 {\displaystyle n+1} -dimensional space. The term <i>hyper</i>sphere is commonly used to distinguish spheres of dimension n ≥ 3 {\displaystyle n\geq 3} which are thus embedded in a space of dimension n + 1 ≥ 4 {\displaystyle n+1\geq 4} , which means that they cannot be easily visualized. The n {\displaystyle n} -sphere is the setting for n {\displaystyle n} -dimensional <a href="/facts/Spherical_geometry/WGLQb1Ys">spherical geometry</a>. </p><p>Considered extrinsically, as a <a href="/facts/Hypersurface/cTvtLVsV">hypersurface</a> embedded in ( n + 1 ) {\displaystyle (n+1)} -dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, an n {\displaystyle n} -sphere is the <a href="/facts/Locus_(mathematics)/t4swdGJq">locus</a> of <a href="/facts/Point_(geometry)/6YPAvX2a">points</a> at equal <a href="/facts/Euclidean_distance/9qDoQKQe">distance</a> (the <i><a href="/facts/Radius/9Gr9deYD">radius</a></i>) from a given <i><a href="/facts/Center_(geometry)/oKPIwkY3">center</a></i> point. Its <a href="/facts/Interior_(topology)/5sxGnQfY">interior</a>, consisting of all points closer to the center than the radius, is an ( n + 1 ) {\displaystyle (n+1)} -dimensional <a href="/facts/Ball_(mathematics)/f9vpMVMp">ball</a>. In particular: </p> <ul><li>The 0 {\displaystyle 0} -sphere is the pair of points at the ends of a <a href="/facts/Line_segment/V8E0M4qk">line segment</a> ( 1 {\displaystyle 1} -ball).</li> <li>The 1 {\displaystyle 1} -sphere is a <a href="/facts/Circle/9fy5ggKn">circle</a>, the <a href="/facts/Circumference/r1rntU0f">circumference</a> of a <a href="/facts/Disk_(mathematics)/DgMVr97H">disk</a> ( 2 {\displaystyle 2} -ball) in the two-dimensional <a href="/facts/Euclidean_plane/tAUtRFcn">plane</a>.</li> <li>The 2 {\displaystyle 2} -sphere, often simply called a sphere, is the <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> of a 3 {\displaystyle 3} -ball in <a href="/facts/Three-dimensional_space/KAqKWUbS">three-dimensional space</a>.</li> <li>The <a href="/facts/3-sphere/u67nImpY">3-sphere</a> is the boundary of a 4 {\displaystyle 4} -ball in <a href="/facts/Four-dimensional_space/vFQd0M7T">four-dimensional space</a>.</li> <li>The ( n − 1 ) {\displaystyle (n-1)} -sphere is the boundary of an n {\displaystyle n} -ball.</li></ul> <p>Given a <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">Cartesian coordinate system</a>, the <i><a href="/facts/Unit_n-sphere/5UeoTcug">unit n {\displaystyle n} -sphere</a></i> of radius 1 {\displaystyle 1} can be defined as: </p> S n = { x ∈ R n + 1 : ‖ x ‖ = 1 } . {\displaystyle S^{n}=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=1\right\}.} <p>Considered intrinsically, when n ≥ 1 {\displaystyle n\geq 1} , the n {\displaystyle n} -sphere is a <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> of positive <a href="/facts/Constant_curvature/VuJg6YlV">constant curvature</a>, and is <a href="/facts/Orientable_manifold/1SEQM2Pt">orientable</a>. The <a href="/facts/Geodesics/fbWR6l80">geodesics</a> of the n {\displaystyle n} -sphere are called <a href="/facts/Great_circle/x4fq6eNm">great circles</a>. </p><p>The <a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a> maps the n {\displaystyle n} -sphere onto n {\displaystyle n} -space with a single adjoined <a href="/facts/Point_at_infinity/GAuS0wHQ">point at infinity</a>; under the <a href="/facts/Metric_tensor/tbqek1gJ">metric</a> thereby defined, R n ∪ { ∞ } {\displaystyle \mathbb {R} ^{n}\cup \{\infty \}} is a model for the n {\displaystyle n} -sphere. </p><p>In the more general setting of <a href="/facts/Topology/2EqW47cU">topology</a>, any <a href="/facts/Topological_space/v2ut7CbK">topological space</a> that is <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> to the unit n {\displaystyle n} -sphere is called an n {\displaystyle n} -<i>sphere</i>. Under inverse stereographic projection, the n {\displaystyle n} -sphere is the <a href="/facts/One-point_compactification/d3Xc0QKu">one-point compactification</a> of n {\displaystyle n} -space. The n {\displaystyle n} -spheres admit several other topological descriptions: for example, they can be constructed by gluing two n {\displaystyle n} -dimensional spaces together, by identifying the boundary of an <a href="/facts/Hypercube/q2mRPbIz"> n {\displaystyle n} -cube</a> with a point, or (inductively) by forming the <a href="/facts/Suspension_(topology)/pIWr58dM">suspension</a> of an ( n − 1 ) {\displaystyle (n-1)} -sphere. When n ≥ 2 {\displaystyle n\geq 2} it is <a href="/facts/Simply_connected/xFlta63J">simply connected</a>; the 1 {\displaystyle 1} -sphere (circle) is not simply connected; the 0 {\displaystyle 0} -sphere is not even connected, consisting of two discrete points. </p>