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Unit sphere - Wikipedia
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class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <p>In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematics">mathematics</a>, a <b>unit sphere</b> is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sphere?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sphere">sphere</a> of unit <a href="https://en-m-wikipedia-org.translate.goog/wiki/Radius?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Radius">radius</a>: the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Locus_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Locus (mathematics)">set of points</a> at <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_distance?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean distance">Euclidean distance</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="1">1</a> from some <a href="https://en-m-wikipedia-org.translate.goog/wiki/Center_(geometry)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Center (geometry)">center point</a> in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Three-dimensional_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Three-dimensional space">three-dimensional space</a>. More generally, the <i>unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-sphere</i> is an <a href="https://en-m-wikipedia-org.translate.goog/wiki/N-sphere?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="N-sphere"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-sphere</a> of unit radius in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n+1)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (n+1)} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30a29cfd35628469f9dbffea4804f5b422f3037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n+1)}"></span>-<a href="https://en-m-wikipedia-org.translate.goog/wiki/Dimension?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dimension">dimensional</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean space">Euclidean space</a>; the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Unit_circle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unit circle">unit circle</a> is a special case, the unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 1} </annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>-sphere in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_plane?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean plane">plane</a>. An (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Open_set?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Open set">open</a>) <b>unit ball</b> is the region inside of a unit sphere, the set of points of distance less than 1 from the center.</p> <figure class="mw-halign-right" typeof="mw:File/Frame"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Vector_norms.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Vector_norms.svg/140px-Vector_norms.svg.png" decoding="async" width="140" height="460" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Vector_norms.svg/210px-Vector_norms.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Vector_norms.svg/280px-Vector_norms.svg.png 2x" data-file-width="140" data-file-height="460"></a> <figcaption> Some 1-spheres: <span class="texhtml">‖<b>x</b>‖<sub>2</sub></span> is the norm for Euclidean space. </figcaption> </figure> <p>A sphere or ball with unit radius and center at the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Origin_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Origin (mathematics)">origin</a> of the space is called <i>the</i> unit sphere or <i>the</i> unit ball. Any arbitrary sphere can be transformed to the unit sphere by a combination of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Translation_(geometry)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Translation (geometry)">translation</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Scaling_(geometry)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Scaling (geometry)">scaling</a>, so the study of spheres in general can often be reduced to the study of the unit sphere.</p> <p>The unit sphere is often used as a model for <a href="https://en-m-wikipedia-org.translate.goog/wiki/Spherical_geometry?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spherical geometry">spherical geometry</a> because it has constant <a href="https://en-m-wikipedia-org.translate.goog/wiki/Sectional_curvature?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sectional curvature">sectional curvature</a> of 1, which simplifies calculations. In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Trigonometry?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Trigonometry">trigonometry</a>, circular <a href="https://en-m-wikipedia-org.translate.goog/wiki/Arc_length?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Arc length">arc length</a> on the unit circle is called <a href="https://en-m-wikipedia-org.translate.goog/wiki/Radian?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Radian">radians</a> and used for measuring <a href="https://en-m-wikipedia-org.translate.goog/wiki/Angular_distance?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Angular distance">angular distance</a>; in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Spherical_trigonometry?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spherical trigonometry">spherical trigonometry</a> surface area on the unit sphere is called <a href="https://en-m-wikipedia-org.translate.goog/wiki/Steradian?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Steradian">steradians</a> and used for measuring <a href="https://en-m-wikipedia-org.translate.goog/wiki/Solid_angle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Solid angle">solid angle</a>.</p> <p>In more general contexts, a <i>unit sphere</i> is the set of points of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Distance?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Distance">distance</a> 1 from a fixed central point, where different <a href="https://en-m-wikipedia-org.translate.goog/wiki/Norm_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Norm (mathematics)">norms</a> can be used as general notions of "distance", and an (open) <i>unit ball</i> is the region inside.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Unit_spheres_and_balls_in_Euclidean_space"><span class="tocnumber">1</span> <span class="toctext">Unit spheres and balls in Euclidean space</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Volume_and_area"><span class="tocnumber">1.1</span> <span class="toctext">Volume and area</span></a> <ul> <li class="toclevel-3 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Recursion"><span class="tocnumber">1.1.1</span> <span class="toctext">Recursion</span></a></li> <li class="toclevel-3 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Non-negative_real-valued_dimensions"><span class="tocnumber">1.1.2</span> <span class="toctext">Non-negative real-valued dimensions</span></a></li> <li class="toclevel-3 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Other_radii"><span class="tocnumber">1.1.3</span> <span class="toctext">Other radii</span></a></li> </ul></li> </ul></li> <li class="toclevel-1 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Unit_balls_in_normed_vector_spaces"><span class="tocnumber">2</span> <span class="toctext">Unit balls in normed vector spaces</span></a></li> <li class="toclevel-1 tocsection-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Generalizations"><span class="tocnumber">3</span> <span class="toctext">Generalizations</span></a> <ul> <li class="toclevel-2 tocsection-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Metric_spaces"><span class="tocnumber">3.1</span> <span class="toctext">Metric spaces</span></a></li> <li class="toclevel-2 tocsection-9"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Quadratic_forms"><span class="tocnumber">3.2</span> <span class="toctext">Quadratic forms</span></a></li> </ul></li> <li class="toclevel-1 tocsection-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notes_and_references"><span class="tocnumber">5</span> <span class="toctext">Notes and references</span></a></li> <li class="toclevel-1 tocsection-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#External_links"><span class="tocnumber">6</span> <span class="toctext">External links</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Unit_spheres_and_balls_in_Euclidean_space">Unit spheres and balls in Euclidean space</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Unit spheres and balls in Euclidean space" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean space">Euclidean space</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> dimensions, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (n-1)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}"> </noscript><span class="lazy-image-placeholder" style="width: 7.207ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" data-alt="{\displaystyle (n-1)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-dimensional unit sphere is the set of all points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> …<!-- … --> </mo> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x_{1},\ldots ,x_{n})} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7935f7983d8a5ae59fea84efe65415235fa7c47b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\ldots ,x_{n})}"> </noscript><span class="lazy-image-placeholder" style="width: 11.92ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7935f7983d8a5ae59fea84efe65415235fa7c47b" data-alt="{\displaystyle (x_{1},\ldots ,x_{n})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> which satisfy the equation</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <mo> ⋯<!-- ⋯ --> </mo> <mo> + </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> = </mo> <mn> 1. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8394763b6f55d5c34a7231d365907a1445d0847e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.468ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1.}"> </noscript><span class="lazy-image-placeholder" style="width: 23.468ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8394763b6f55d5c34a7231d365907a1445d0847e" data-alt="{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The open unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-ball is the set of all points satisfying the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Inequality_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Inequality (mathematics)">inequality</a></p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <mo> ⋯<!-- ⋯ --> </mo> <mo> + </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> < </mo> <mn> 1 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73f519261799847164ee15f2c0a16265f4a6294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.468ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1,}"> </noscript><span class="lazy-image-placeholder" style="width: 23.468ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73f519261799847164ee15f2c0a16265f4a6294" data-alt="{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>and closed unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-ball is the set of all points satisfying the inequality</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\leq 1.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <mo> ⋯<!-- ⋯ --> </mo> <mo> + </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> ≤<!-- ≤ --> </mo> <mn> 1. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\leq 1.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92b7796924e56063c7f88625253fdeecd0a35e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.468ex; height:3.176ex;" alt="{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\leq 1.}"> </noscript><span class="lazy-image-placeholder" style="width: 23.468ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92b7796924e56063c7f88625253fdeecd0a35e1" data-alt="{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\leq 1.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Volume_and_area">Volume and area</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Volume and area" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style> <div role="note" class="hatnote navigation-not-searchable"> See also: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Volume_of_an_n-ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Volume of an n-ball">Volume of an n-ball</a> </div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Graphs_of_volumes_(V)_and_surface_areas_(S)_of_n-balls_of_radius_1.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Graphs_of_volumes_%28V%29_and_surface_areas_%28S%29_of_n-balls_of_radius_1.png/220px-Graphs_of_volumes_%28V%29_and_surface_areas_%28S%29_of_n-balls_of_radius_1.png" decoding="async" width="220" height="266" class="mw-file-element" data-file-width="756" data-file-height="914"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 266px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Graphs_of_volumes_%28V%29_and_surface_areas_%28S%29_of_n-balls_of_radius_1.png/220px-Graphs_of_volumes_%28V%29_and_surface_areas_%28S%29_of_n-balls_of_radius_1.png" data-width="220" data-height="266" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/Graphs_of_volumes_%28V%29_and_surface_areas_%28S%29_of_n-balls_of_radius_1.png/330px-Graphs_of_volumes_%28V%29_and_surface_areas_%28S%29_of_n-balls_of_radius_1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/Graphs_of_volumes_%28V%29_and_surface_areas_%28S%29_of_n-balls_of_radius_1.png/440px-Graphs_of_volumes_%28V%29_and_surface_areas_%28S%29_of_n-balls_of_radius_1.png 2x" data-class="mw-file-element"> </span></a> <figcaption> Graphs of volumes (<span class="texhtml mvar" style="font-style:italic;">V</span>) and surface areas (<span class="texhtml mvar" style="font-style:italic;">S</span>) of unit <span class="texhtml mvar" style="font-style:italic;">n</span>-balls </figcaption> </figure> <p>The classical equation of a unit sphere is that of the ellipsoid with a radius of 1 and no alterations to the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> </noscript><span class="lazy-image-placeholder" style="width: 1.155ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" data-alt="{\displaystyle y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-, or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> </noscript><span class="lazy-image-placeholder" style="width: 1.088ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" data-alt="{\displaystyle z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>- axes:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}+z^{2}=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x^{2}+y^{2}+z^{2}=1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b5c3e3525a1604f9425edb80835d236fe7af82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.685ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}=1}"> </noscript><span class="lazy-image-placeholder" style="width: 16.685ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b5c3e3525a1604f9425edb80835d236fe7af82" data-alt="{\displaystyle x^{2}+y^{2}+z^{2}=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The volume of the unit ball in Euclidean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-space, and the surface area of the unit sphere, appear in many important formulas of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematical_analysis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematical analysis">analysis</a>. The volume of the unit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-ball, which we denote <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{n},} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399bc67c862217cde2df3a84fa59cede0c50f7dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.221ex; height:2.509ex;" alt="{\displaystyle V_{n},}"> </noscript><span class="lazy-image-placeholder" style="width: 3.221ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/399bc67c862217cde2df3a84fa59cede0c50f7dd" data-alt="{\displaystyle V_{n},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> can be expressed by making use of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gamma_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gamma function">gamma function</a>. It is</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}={\frac {\pi ^{n/2}}{\Gamma (1+n/2)}}={\begin{cases}{\pi ^{n/2}}/{(n/2)!}&\mathrm {if~} n\geq 0\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{n!!}&\mathrm {if~} n\geq 0\mathrm {~is~odd,} \end{cases}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> <mrow> <mi mathvariant="normal"> Γ<!-- Γ --> </mi> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> + </mo> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> { </mo> <mtable columnalign="left left" rowspacing="0.533em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> f </mi> <mtext> </mtext> </mrow> <mi> n </mi> <mo> ≥<!-- ≥ --> </mo> <mn> 0 </mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> s </mi> <mtext> </mtext> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> v </mi> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> n </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> f </mi> <mtext> </mtext> </mrow> <mi> n </mi> <mo> ≥<!-- ≥ --> </mo> <mn> 0 </mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> s </mi> <mtext> </mtext> <mi mathvariant="normal"> o </mi> <mi mathvariant="normal"> d </mi> <mi mathvariant="normal"> d </mi> <mo> , </mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{n}={\frac {\pi ^{n/2}}{\Gamma (1+n/2)}}={\begin{cases}{\pi ^{n/2}}/{(n/2)!}&\mathrm {if~} n\geq 0\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{n!!}&\mathrm {if~} n\geq 0\mathrm {~is~odd,} \end{cases}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf208735d51b452fb423a2e121ada1e6c269bf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.155ex; height:7.509ex;" alt="{\displaystyle V_{n}={\frac {\pi ^{n/2}}{\Gamma (1+n/2)}}={\begin{cases}{\pi ^{n/2}}/{(n/2)!}&\mathrm {if~} n\geq 0\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{n!!}&\mathrm {if~} n\geq 0\mathrm {~is~odd,} \end{cases}}}"> </noscript><span class="lazy-image-placeholder" style="width: 56.155ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf208735d51b452fb423a2e121ada1e6c269bf9" data-alt="{\displaystyle V_{n}={\frac {\pi ^{n/2}}{\Gamma (1+n/2)}}={\begin{cases}{\pi ^{n/2}}/{(n/2)!}&\mathrm {if~} n\geq 0\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{n!!}&\mathrm {if~} n\geq 0\mathrm {~is~odd,} \end{cases}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ! </mo> <mo> ! </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n!!} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511717d541dba5357928e8d8631f1b4d4f8d5b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.688ex; height:2.176ex;" alt="{\displaystyle n!!}"> </noscript><span class="lazy-image-placeholder" style="width: 2.688ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511717d541dba5357928e8d8631f1b4d4f8d5b31" data-alt="{\displaystyle n!!}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Double_factorial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Double factorial">double factorial</a>.</p> <p>The hypervolume of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (n-1)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}"> </noscript><span class="lazy-image-placeholder" style="width: 7.207ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" data-alt="{\displaystyle (n-1)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-dimensional unit sphere (<i>i.e.</i>, the "area" of the boundary of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-dimensional unit ball), which we denote <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n-1},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{n-1},} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e33c569176315734b3d64dbe8db9857a7eab228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.709ex; height:2.509ex;" alt="{\displaystyle A_{n-1},}"> </noscript><span class="lazy-image-placeholder" style="width: 5.709ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e33c569176315734b3d64dbe8db9857a7eab228" data-alt="{\displaystyle A_{n-1},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> can be expressed as</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n-1}=nV_{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\geq 1\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{(n-2)!!}&\mathrm {if~} n\geq 1\mathrm {~is~odd.} \end{cases}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mi> n </mi> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> n </mi> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal"> Γ<!-- Γ --> </mi> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> + </mo> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal"> Γ<!-- Γ --> </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> { </mo> <mtable columnalign="left left" rowspacing="0.533em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo> −<!-- − --> </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> f </mi> <mtext> </mtext> </mrow> <mi> n </mi> <mo> ≥<!-- ≥ --> </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> s </mi> <mtext> </mtext> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> v </mi> <mi mathvariant="normal"> e </mi> <mi mathvariant="normal"> n </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> <mo> ! </mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> f </mi> <mtext> </mtext> </mrow> <mi> n </mi> <mo> ≥<!-- ≥ --> </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> s </mi> <mtext> </mtext> <mi mathvariant="normal"> o </mi> <mi mathvariant="normal"> d </mi> <mi mathvariant="normal"> d </mi> <mo> . </mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{n-1}=nV_{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\geq 1\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{(n-2)!!}&\mathrm {if~} n\geq 1\mathrm {~is~odd.} \end{cases}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef608cca0068ac18865a9203858bacc8780fe7ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:82.439ex; height:7.509ex;" alt="{\displaystyle A_{n-1}=nV_{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\geq 1\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{(n-2)!!}&\mathrm {if~} n\geq 1\mathrm {~is~odd.} \end{cases}}}"> </noscript><span class="lazy-image-placeholder" style="width: 82.439ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef608cca0068ac18865a9203858bacc8780fe7ac" data-alt="{\displaystyle A_{n-1}=nV_{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\geq 1\mathrm {~is~even} \\[6mu]{2(2\pi )^{(n-1)/2}}/{(n-2)!!}&\mathrm {if~} n\geq 1\mathrm {~is~odd.} \end{cases}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}=2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{0}=2} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab6024095f5fe12c19f8ca41458072885667409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.058ex; height:2.509ex;" alt="{\displaystyle A_{0}=2}"> </noscript><span class="lazy-image-placeholder" style="width: 7.058ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab6024095f5fe12c19f8ca41458072885667409" data-alt="{\displaystyle A_{0}=2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the "area" of the boundary of the unit ball <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]\subset \mathbb {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mo> −<!-- − --> </mo> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> <mo stretchy="false"> ] </mo> <mo> ⊂<!-- ⊂ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [-1,1]\subset \mathbb {R} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2016a25b4496666df761c56045fc221d3fbd2c08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.237ex; height:2.843ex;" alt="{\displaystyle [-1,1]\subset \mathbb {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 11.237ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2016a25b4496666df761c56045fc221d3fbd2c08" data-alt="{\displaystyle [-1,1]\subset \mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, which simply counts the two points. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}=2\pi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{1}=2\pi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cb892d36f56187e992d89272aa2a5261815e09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.39ex; height:2.509ex;" alt="{\displaystyle A_{1}=2\pi }"> </noscript><span class="lazy-image-placeholder" style="width: 8.39ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cb892d36f56187e992d89272aa2a5261815e09" data-alt="{\displaystyle A_{1}=2\pi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the "area" of the boundary of the unit disc, which is the circumference of the unit circle. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}=4\pi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mn> 4 </mn> <mi> π<!-- π --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{2}=4\pi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b8af242317819b4c41019dc4f48b87cf7ca590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.39ex; height:2.509ex;" alt="{\displaystyle A_{2}=4\pi }"> </noscript><span class="lazy-image-placeholder" style="width: 8.39ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b8af242317819b4c41019dc4f48b87cf7ca590" data-alt="{\displaystyle A_{2}=4\pi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the area of the boundary of the unit ball <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leq 1\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <mi> x </mi> <mo> ∈<!-- ∈ --> </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> : </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> ≤<!-- ≤ --> </mo> <mn> 1 </mn> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leq 1\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66304e768495a8bbb56ce45bcee32bd854429723" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.258ex; height:3.343ex;" alt="{\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leq 1\}}"> </noscript><span class="lazy-image-placeholder" style="width: 28.258ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66304e768495a8bbb56ce45bcee32bd854429723" data-alt="{\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leq 1\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, which is the surface area of the unit sphere <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <mi> x </mi> <mo> ∈<!-- ∈ --> </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> : </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> = </mo> <mn> 1 </mn> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d759699417fdbda6a3e1aabcc6fdf71a41add95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.258ex; height:3.343ex;" alt="{\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}}"> </noscript><span class="lazy-image-placeholder" style="width: 28.258ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d759699417fdbda6a3e1aabcc6fdf71a41add95" data-alt="{\displaystyle \{x\in \mathbb {R} ^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <p>The surface areas and the volumes for some values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> are as follows:</p> <table class="wikitable" style="text-align:center"> <tbody> <tr> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></th> <th colspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n-1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{n-1}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b576e75f0336d126580faaad6039e2e84f6f3ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.062ex; height:2.509ex;" alt="{\displaystyle A_{n-1}}"> </noscript><span class="lazy-image-placeholder" style="width: 5.062ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b576e75f0336d126580faaad6039e2e84f6f3ee2" data-alt="{\displaystyle A_{n-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (surface area)</th> <th colspan="2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc5a637019ce3415183f06995aeeca93547767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.509ex;" alt="{\displaystyle V_{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.574ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc5a637019ce3415183f06995aeeca93547767" data-alt="{\displaystyle V_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (volume)</th> </tr> <tr> <th>0</th> <td></td> <td></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1/0!)\pi ^{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 0 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (1/0!)\pi ^{0}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753341d8e4b62398fabafa0c7df2208cc75edfdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.332ex; height:3.176ex;" alt="{\displaystyle (1/0!)\pi ^{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.332ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753341d8e4b62398fabafa0c7df2208cc75edfdd" data-alt="{\displaystyle (1/0!)\pi ^{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>1</td> </tr> <tr> <th>1</th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1(2^{1}/1!!)\pi ^{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 1 </mn> <mo stretchy="false"> ( </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 1 </mn> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 1(2^{1}/1!!)\pi ^{0}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c17143124cfa481d3295821a507e8a2a65b7d76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.195ex; height:3.176ex;" alt="{\displaystyle 1(2^{1}/1!!)\pi ^{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.195ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c17143124cfa481d3295821a507e8a2a65b7d76" data-alt="{\displaystyle 1(2^{1}/1!!)\pi ^{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>2</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2^{1}/1!!)\pi ^{0}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 1 </mn> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2^{1}/1!!)\pi ^{0}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32cc6dc082078a92ec7d972a952231fd8956ab00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.033ex; height:3.176ex;" alt="{\displaystyle (2^{1}/1!!)\pi ^{0}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.033ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32cc6dc082078a92ec7d972a952231fd8956ab00" data-alt="{\displaystyle (2^{1}/1!!)\pi ^{0}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>2</td> </tr> <tr> <th>2</th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(1/1!)\pi ^{1}=2\pi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 1 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 2(1/1!)\pi ^{1}=2\pi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7812cfeb8842212cdb5ba0698191340dd789f5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.087ex; height:3.176ex;" alt="{\displaystyle 2(1/1!)\pi ^{1}=2\pi }"> </noscript><span class="lazy-image-placeholder" style="width: 15.087ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7812cfeb8842212cdb5ba0698191340dd789f5a2" data-alt="{\displaystyle 2(1/1!)\pi ^{1}=2\pi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>6.283</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1/1!)\pi ^{1}=\pi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 1 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <mi> π<!-- π --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (1/1!)\pi ^{1}=\pi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9990032793168fde5e5ac5f991391351f4e68f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.762ex; height:3.176ex;" alt="{\displaystyle (1/1!)\pi ^{1}=\pi }"> </noscript><span class="lazy-image-placeholder" style="width: 12.762ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9990032793168fde5e5ac5f991391351f4e68f" data-alt="{\displaystyle (1/1!)\pi ^{1}=\pi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>3.141</td> </tr> <tr> <th>3</th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3(2^{2}/3!!)\pi ^{1}=4\pi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 3 </mn> <mo stretchy="false"> ( </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 3 </mn> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <mn> 4 </mn> <mi> π<!-- π --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 3(2^{2}/3!!)\pi ^{1}=4\pi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8e2ad09144361ec1574d1bfdb7df8cf0e08717" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.788ex; height:3.176ex;" alt="{\displaystyle 3(2^{2}/3!!)\pi ^{1}=4\pi }"> </noscript><span class="lazy-image-placeholder" style="width: 16.788ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8e2ad09144361ec1574d1bfdb7df8cf0e08717" data-alt="{\displaystyle 3(2^{2}/3!!)\pi ^{1}=4\pi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>12.57</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2^{2}/3!!)\pi ^{1}=(4/3)\pi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 3 </mn> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 4 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 3 </mn> <mo stretchy="false"> ) </mo> <mi> π<!-- π --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2^{2}/3!!)\pi ^{1}=(4/3)\pi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/580a26a0a195280909564306669f76124eafff07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.76ex; height:3.176ex;" alt="{\displaystyle (2^{2}/3!!)\pi ^{1}=(4/3)\pi }"> </noscript><span class="lazy-image-placeholder" style="width: 19.76ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/580a26a0a195280909564306669f76124eafff07" data-alt="{\displaystyle (2^{2}/3!!)\pi ^{1}=(4/3)\pi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>4.189</td> </tr> <tr> <th>4</th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4(1/2!)\pi ^{2}=2\pi ^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 4 </mn> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mn> 2 </mn> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 4(1/2!)\pi ^{2}=2\pi ^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4539444a8d24279ecfe298a40de99be32de3ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.144ex; height:3.176ex;" alt="{\displaystyle 4(1/2!)\pi ^{2}=2\pi ^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 16.144ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d4539444a8d24279ecfe298a40de99be32de3ab" data-alt="{\displaystyle 4(1/2!)\pi ^{2}=2\pi ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>19.74</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1/2!)\pi ^{2}=(1/2)\pi ^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (1/2!)\pi ^{2}=(1/2)\pi ^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e7ddf66f36528770817066c85282ca771ca720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.116ex; height:3.176ex;" alt="{\displaystyle (1/2!)\pi ^{2}=(1/2)\pi ^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 19.116ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e7ddf66f36528770817066c85282ca771ca720" data-alt="{\displaystyle (1/2!)\pi ^{2}=(1/2)\pi ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>4.935</td> </tr> <tr> <th>5</th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5(2^{3}/5!!)\pi ^{2}=(8/3)\pi ^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 5 </mn> <mo stretchy="false"> ( </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 5 </mn> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 8 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 3 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 5(2^{3}/5!!)\pi ^{2}=(8/3)\pi ^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6434fe6441351dc205ef35f9a21cb01acbb8feb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.979ex; height:3.176ex;" alt="{\displaystyle 5(2^{3}/5!!)\pi ^{2}=(8/3)\pi ^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 21.979ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6434fe6441351dc205ef35f9a21cb01acbb8feb" data-alt="{\displaystyle 5(2^{3}/5!!)\pi ^{2}=(8/3)\pi ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>26.32</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2^{3}/5!!)\pi ^{2}=(8/15)\pi ^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 5 </mn> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 8 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 15 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2^{3}/5!!)\pi ^{2}=(8/15)\pi ^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6546c1225f08fdf36ea9813ab793b539f02fd416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.979ex; height:3.176ex;" alt="{\displaystyle (2^{3}/5!!)\pi ^{2}=(8/15)\pi ^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 21.979ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6546c1225f08fdf36ea9813ab793b539f02fd416" data-alt="{\displaystyle (2^{3}/5!!)\pi ^{2}=(8/15)\pi ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>5.264</td> </tr> <tr> <th>6</th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6(1/3!)\pi ^{3}=\pi ^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 6 </mn> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 3 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 6(1/3!)\pi ^{3}=\pi ^{3}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5b2da1a649efc1ffed0347d0f928cb6b282780" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.981ex; height:3.176ex;" alt="{\displaystyle 6(1/3!)\pi ^{3}=\pi ^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.981ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5b2da1a649efc1ffed0347d0f928cb6b282780" data-alt="{\displaystyle 6(1/3!)\pi ^{3}=\pi ^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>31.01</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1/3!)\pi ^{3}=(1/6)\pi ^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 3 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 6 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (1/3!)\pi ^{3}=(1/6)\pi ^{3}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70a1407c3a61b67e17fa611951f975d28c49f20c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.116ex; height:3.176ex;" alt="{\displaystyle (1/3!)\pi ^{3}=(1/6)\pi ^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 19.116ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70a1407c3a61b67e17fa611951f975d28c49f20c" data-alt="{\displaystyle (1/3!)\pi ^{3}=(1/6)\pi ^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>5.168</td> </tr> <tr> <th>7</th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7(2^{4}/7!!)\pi ^{3}=(16/15)\pi ^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 7 </mn> <mo stretchy="false"> ( </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 7 </mn> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 16 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 15 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 7(2^{4}/7!!)\pi ^{3}=(16/15)\pi ^{3}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c846f836d8ffc14dddba434b6300a68267c98d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.304ex; height:3.176ex;" alt="{\displaystyle 7(2^{4}/7!!)\pi ^{3}=(16/15)\pi ^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 24.304ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c846f836d8ffc14dddba434b6300a68267c98d1" data-alt="{\displaystyle 7(2^{4}/7!!)\pi ^{3}=(16/15)\pi ^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>33.07</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2^{4}/7!!)\pi ^{3}=(16/105)\pi ^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 7 </mn> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 16 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 105 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2^{4}/7!!)\pi ^{3}=(16/105)\pi ^{3}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11909965248fa4d4e14fc140273a29a9f5789811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.304ex; height:3.176ex;" alt="{\displaystyle (2^{4}/7!!)\pi ^{3}=(16/105)\pi ^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 24.304ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11909965248fa4d4e14fc140273a29a9f5789811" data-alt="{\displaystyle (2^{4}/7!!)\pi ^{3}=(16/105)\pi ^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>4.725</td> </tr> <tr> <th>8</th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8(1/4!)\pi ^{4}=(1/3)\pi ^{4}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 8 </mn> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 4 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 3 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 8(1/4!)\pi ^{4}=(1/3)\pi ^{4}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b37bd10864430a769882d5a82574b768b2cfe97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.278ex; height:3.176ex;" alt="{\displaystyle 8(1/4!)\pi ^{4}=(1/3)\pi ^{4}}"> </noscript><span class="lazy-image-placeholder" style="width: 20.278ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b37bd10864430a769882d5a82574b768b2cfe97" data-alt="{\displaystyle 8(1/4!)\pi ^{4}=(1/3)\pi ^{4}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>32.47</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1/4!)\pi ^{4}=(1/24)\pi ^{4}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 4 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 24 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (1/4!)\pi ^{4}=(1/24)\pi ^{4}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f09d3ff1c2fd72388cf93ced78f311045ef5c9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.278ex; height:3.176ex;" alt="{\displaystyle (1/4!)\pi ^{4}=(1/24)\pi ^{4}}"> </noscript><span class="lazy-image-placeholder" style="width: 20.278ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f09d3ff1c2fd72388cf93ced78f311045ef5c9a" data-alt="{\displaystyle (1/4!)\pi ^{4}=(1/24)\pi ^{4}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>4.059</td> </tr> <tr> <th>9</th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9(2^{5}/9!!)\pi ^{4}=(32/105)\pi ^{4}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 9 </mn> <mo stretchy="false"> ( </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 9 </mn> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 32 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 105 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 9(2^{5}/9!!)\pi ^{4}=(32/105)\pi ^{4}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a22dbf3c326bd323eecf83d2532868b8f432909d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.466ex; height:3.176ex;" alt="{\displaystyle 9(2^{5}/9!!)\pi ^{4}=(32/105)\pi ^{4}}"> </noscript><span class="lazy-image-placeholder" style="width: 25.466ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a22dbf3c326bd323eecf83d2532868b8f432909d" data-alt="{\displaystyle 9(2^{5}/9!!)\pi ^{4}=(32/105)\pi ^{4}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>29.69</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2^{5}/9!!)\pi ^{4}=(32/945)\pi ^{4}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 9 </mn> <mo> ! </mo> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 32 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 945 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (2^{5}/9!!)\pi ^{4}=(32/945)\pi ^{4}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b313054e58d4dc7f9253f6a7810133aa4e2bf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.304ex; height:3.176ex;" alt="{\displaystyle (2^{5}/9!!)\pi ^{4}=(32/945)\pi ^{4}}"> </noscript><span class="lazy-image-placeholder" style="width: 24.304ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b313054e58d4dc7f9253f6a7810133aa4e2bf7" data-alt="{\displaystyle (2^{5}/9!!)\pi ^{4}=(32/945)\pi ^{4}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>3.299</td> </tr> <tr> <th>10</th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10(1/5!)\pi ^{5}=(1/12)\pi ^{5}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 10 </mn> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 5 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 12 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 10(1/5!)\pi ^{5}=(1/12)\pi ^{5}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1869aa572d8a7af2c7b452e1b0f584b72e39d28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.603ex; height:3.176ex;" alt="{\displaystyle 10(1/5!)\pi ^{5}=(1/12)\pi ^{5}}"> </noscript><span class="lazy-image-placeholder" style="width: 22.603ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1869aa572d8a7af2c7b452e1b0f584b72e39d28" data-alt="{\displaystyle 10(1/5!)\pi ^{5}=(1/12)\pi ^{5}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>25.50</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1/5!)\pi ^{5}=(1/120)\pi ^{5}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 5 </mn> <mo> ! </mo> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 120 </mn> <mo stretchy="false"> ) </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (1/5!)\pi ^{5}=(1/120)\pi ^{5}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa61da03540b53ffe13ebbe3a0873afcbe124d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.44ex; height:3.176ex;" alt="{\displaystyle (1/5!)\pi ^{5}=(1/120)\pi ^{5}}"> </noscript><span class="lazy-image-placeholder" style="width: 21.44ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa61da03540b53ffe13ebbe3a0873afcbe124d6d" data-alt="{\displaystyle (1/5!)\pi ^{5}=(1/120)\pi ^{5}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>2.550</td> </tr> </tbody> </table> <p>where the decimal expanded values for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ≥<!-- ≥ --> </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n\geq 2} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"> </noscript><span class="lazy-image-placeholder" style="width: 5.656ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" data-alt="{\displaystyle n\geq 2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> are rounded to the displayed precision.</p> <div class="mw-heading mw-heading4"> <h4 id="Recursion">Recursion</h4><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Recursion" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.962ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" data-alt="{\displaystyle A_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> values satisfy the recursion:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{0}=2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{0}=2} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab6024095f5fe12c19f8ca41458072885667409" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.058ex; height:2.509ex;" alt="{\displaystyle A_{0}=2}"> </noscript><span class="lazy-image-placeholder" style="width: 7.058ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab6024095f5fe12c19f8ca41458072885667409" data-alt="{\displaystyle A_{0}=2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}=2\pi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{1}=2\pi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cb892d36f56187e992d89272aa2a5261815e09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.39ex; height:2.509ex;" alt="{\displaystyle A_{1}=2\pi }"> </noscript><span class="lazy-image-placeholder" style="width: 8.39ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35cb892d36f56187e992d89272aa2a5261815e09" data-alt="{\displaystyle A_{1}=2\pi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}={\frac {2\pi }{n-1}}A_{n-2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> π<!-- π --> </mi> </mrow> <mrow> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{n}={\frac {2\pi }{n-1}}A_{n-2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5c7a4f70985f20b387a06f17715580553180402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.356ex; height:5.343ex;" alt="{\displaystyle A_{n}={\frac {2\pi }{n-1}}A_{n-2}}"> </noscript><span class="lazy-image-placeholder" style="width: 17.356ex;height: 5.343ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5c7a4f70985f20b387a06f17715580553180402" data-alt="{\displaystyle A_{n}={\frac {2\pi }{n-1}}A_{n-2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> > </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n>1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.656ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" data-alt="{\displaystyle n>1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </dd> </dl> <p>The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc5a637019ce3415183f06995aeeca93547767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.509ex;" alt="{\displaystyle V_{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.574ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebc5a637019ce3415183f06995aeeca93547767" data-alt="{\displaystyle V_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> values satisfy the recursion:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{0}=1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f3abbd454c5271df714b0871ea552c0de4398c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.67ex; height:2.509ex;" alt="{\displaystyle V_{0}=1}"> </noscript><span class="lazy-image-placeholder" style="width: 6.67ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f3abbd454c5271df714b0871ea552c0de4398c" data-alt="{\displaystyle V_{0}=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}=2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{1}=2} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2aa53273bfe974ffe4a5beff301129f5645cdb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.67ex; height:2.509ex;" alt="{\displaystyle V_{1}=2}"> </noscript><span class="lazy-image-placeholder" style="width: 6.67ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2aa53273bfe974ffe4a5beff301129f5645cdb7" data-alt="{\displaystyle V_{1}=2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}={\frac {2\pi }{n}}V_{n-2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 2 </mn> <mi> π<!-- π --> </mi> </mrow> <mi> n </mi> </mfrac> </mrow> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{n}={\frac {2\pi }{n}}V_{n-2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19b9ff790ef7fb5bcbed5442b847e701c274d0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.677ex; height:5.176ex;" alt="{\displaystyle V_{n}={\frac {2\pi }{n}}V_{n-2}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.677ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19b9ff790ef7fb5bcbed5442b847e701c274d0f4" data-alt="{\displaystyle V_{n}={\frac {2\pi }{n}}V_{n-2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> > </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n>1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.656ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64" data-alt="{\displaystyle n>1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </dd> </dl> <div class="mw-heading mw-heading4"> <h4 id="Non-negative_real-valued_dimensions">Non-negative real-valued dimensions</h4><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Non-negative real-valued dimensions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> Main article: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hausdorff_measure?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hausdorff measure">Hausdorff measure</a> </div> <p>The value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2^{-n}V_{n}=\pi ^{n/2}{\big /}\,2^{n}\Gamma {\bigl (}1+{\tfrac {1}{2}}n{\bigr )}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> −<!-- − --> </mo> <mi> n </mi> </mrow> </msup> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <msup> <mi> π<!-- π --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em"> / </mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mi mathvariant="normal"> Γ<!-- Γ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mn> 1 </mn> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mstyle> </mrow> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\textstyle 2^{-n}V_{n}=\pi ^{n/2}{\big /}\,2^{n}\Gamma {\bigl (}1+{\tfrac {1}{2}}n{\bigr )}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47b0fb41ef6753b6e2f9bcc6d16c16fb55749b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.278ex; height:3.509ex;" alt="{\textstyle 2^{-n}V_{n}=\pi ^{n/2}{\big /}\,2^{n}\Gamma {\bigl (}1+{\tfrac {1}{2}}n{\bigr )}}"> </noscript><span class="lazy-image-placeholder" style="width: 28.278ex;height: 3.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47b0fb41ef6753b6e2f9bcc6d16c16fb55749b9" data-alt="{\textstyle 2^{-n}V_{n}=\pi ^{n/2}{\big /}\,2^{n}\Gamma {\bigl (}1+{\tfrac {1}{2}}n{\bigr )}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> at non-negative real values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is sometimes used for normalization of Hausdorff measure.<sup id="cite_ref-1" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></p> <div class="mw-heading mw-heading4"> <h4 id="Other_radii">Other radii</h4><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Other radii" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> Main article: <a href="https://en-m-wikipedia-org.translate.goog/wiki/N-sphere?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Volume_and_area" title="N-sphere">N-sphere § Volume and area</a> </div> <p>The surface area of an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (n-1)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}"> </noscript><span class="lazy-image-placeholder" style="width: 7.207ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e" data-alt="{\displaystyle (n-1)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-sphere with radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n-1}r^{n-1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msub> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> <mo> −<!-- − --> </mo> <mn> 1 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{n-1}r^{n-1}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093809d8e0ef1a9ae6d9e3cc9068e305a11f2768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.43ex; height:3.009ex;" alt="{\displaystyle A_{n-1}r^{n-1}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.43ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093809d8e0ef1a9ae6d9e3cc9068e305a11f2768" data-alt="{\displaystyle A_{n-1}r^{n-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and the volume of an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>- ball with radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{n}r^{n}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{n}r^{n}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8d9105ce48043febf98edf91fd7e19411b03db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.488ex; height:2.676ex;" alt="{\displaystyle V_{n}r^{n}.}"> </noscript><span class="lazy-image-placeholder" style="width: 5.488ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8d9105ce48043febf98edf91fd7e19411b03db" data-alt="{\displaystyle V_{n}r^{n}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> For instance, the area is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}=4\pi r^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mn> 4 </mn> <mi> π<!-- π --> </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A_{2}=4\pi r^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5ce1b9fbf0d96cc36ac2584242033c468505db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.493ex; height:3.009ex;" alt="{\displaystyle A_{2}=4\pi r^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.493ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5ce1b9fbf0d96cc36ac2584242033c468505db" data-alt="{\displaystyle A_{2}=4\pi r^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for the two-dimensional surface of the three-dimensional ball of radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10110093812676dd04a92ce4c8b75940c366330a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.695ex; height:1.676ex;" alt="{\displaystyle r.}"> </noscript><span class="lazy-image-placeholder" style="width: 1.695ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10110093812676dd04a92ce4c8b75940c366330a" data-alt="{\displaystyle r.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> The volume is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{3}={\tfrac {4}{3}}\pi r^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> </mstyle> </mrow> <mi> π<!-- π --> </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V_{3}={\tfrac {4}{3}}\pi r^{3}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5caf8973012c14bf0b7989eb628802568fd788ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.601ex; height:3.676ex;" alt="{\displaystyle V_{3}={\tfrac {4}{3}}\pi r^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.601ex;height: 3.676ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5caf8973012c14bf0b7989eb628802568fd788ff" data-alt="{\displaystyle V_{3}={\tfrac {4}{3}}\pi r^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for the three-dimensional ball of radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Unit_balls_in_normed_vector_spaces">Unit balls in normed vector spaces</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Unit balls in normed vector spaces" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>The <b>open unit ball</b> of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Normed_vector_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Normed vector space">normed vector space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> </noscript><span class="lazy-image-placeholder" style="width: 1.787ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" data-alt="{\displaystyle V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> with the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Norm_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Norm (mathematics)">norm</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mo> ⋅<!-- ⋅ --> </mo> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|\cdot \|} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}"> </noscript><span class="lazy-image-placeholder" style="width: 4.004ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1" data-alt="{\displaystyle \|\cdot \|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is given by</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in V:\|x\|<1\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <mi> x </mi> <mo> ∈<!-- ∈ --> </mo> <mi> V </mi> <mo> : </mo> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mi> x </mi> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mo> < </mo> <mn> 1 </mn> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{x\in V:\|x\|<1\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374dce5ac10f958523793e605ccbbeb89e117ff4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.135ex; height:2.843ex;" alt="{\displaystyle \{x\in V:\|x\|<1\}}"> </noscript><span class="lazy-image-placeholder" style="width: 18.135ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374dce5ac10f958523793e605ccbbeb89e117ff4" data-alt="{\displaystyle \{x\in V:\|x\|<1\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>It is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Interior_(topology)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Interior (topology)">topological interior</a> of the <b>closed unit ball</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V,\|\cdot \|)\colon }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> V </mi> <mo> , </mo> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mo> ⋅<!-- ⋅ --> </mo> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mo stretchy="false"> ) </mo> <mo> :<!-- : --> </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (V,\|\cdot \|)\colon } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34284f7e2dddc8337b2613f749758aed0adbefcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.281ex; height:2.843ex;" alt="{\displaystyle (V,\|\cdot \|)\colon }"> </noscript><span class="lazy-image-placeholder" style="width: 9.281ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34284f7e2dddc8337b2613f749758aed0adbefcb" data-alt="{\displaystyle (V,\|\cdot \|)\colon }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in V:\|x\|\leq 1\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <mi> x </mi> <mo> ∈<!-- ∈ --> </mo> <mi> V </mi> <mo> : </mo> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mi> x </mi> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mo> ≤<!-- ≤ --> </mo> <mn> 1 </mn> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{x\in V:\|x\|\leq 1\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52442e128feeb5dfdf756da43b48812f4c747dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.135ex; height:2.843ex;" alt="{\displaystyle \{x\in V:\|x\|\leq 1\}}"> </noscript><span class="lazy-image-placeholder" style="width: 18.135ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52442e128feeb5dfdf756da43b48812f4c747dff" data-alt="{\displaystyle \{x\in V:\|x\|\leq 1\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The latter is the disjoint union of the former and their common border, the <b>unit sphere</b> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (V,\|\cdot \|)\colon }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> V </mi> <mo> , </mo> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mo> ⋅<!-- ⋅ --> </mo> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mo stretchy="false"> ) </mo> <mo> :<!-- : --> </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (V,\|\cdot \|)\colon } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34284f7e2dddc8337b2613f749758aed0adbefcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.281ex; height:2.843ex;" alt="{\displaystyle (V,\|\cdot \|)\colon }"> </noscript><span class="lazy-image-placeholder" style="width: 9.281ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34284f7e2dddc8337b2613f749758aed0adbefcb" data-alt="{\displaystyle (V,\|\cdot \|)\colon }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\in V:\|x\|=1\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <mi> x </mi> <mo> ∈<!-- ∈ --> </mo> <mi> V </mi> <mo> : </mo> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mi> x </mi> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mo> = </mo> <mn> 1 </mn> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{x\in V:\|x\|=1\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f9c8e38c6832a42f9b52a6b2d74271eb338ff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.135ex; height:2.843ex;" alt="{\displaystyle \{x\in V:\|x\|=1\}}"> </noscript><span class="lazy-image-placeholder" style="width: 18.135ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f9c8e38c6832a42f9b52a6b2d74271eb338ff8" data-alt="{\displaystyle \{x\in V:\|x\|=1\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>The "shape" of the <i>unit ball</i> is entirely dependent on the chosen norm; it may well have "corners", and for example may look like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mo> −<!-- − --> </mo> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [-1,1]^{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a008254b1bf6d63ac3b13548c4c31180bcd43de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.679ex; height:2.843ex;" alt="{\displaystyle [-1,1]^{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 7.679ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a008254b1bf6d63ac3b13548c4c31180bcd43de" data-alt="{\displaystyle [-1,1]^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> in the case of the max-norm in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{n}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.897ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" data-alt="{\displaystyle \mathbb {R} ^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. One obtains a naturally <i>round ball</i> as the unit ball pertaining to the usual <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hilbert_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hilbert space">Hilbert space</a> norm, based in the finite-dimensional case on the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_distance?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean distance">Euclidean distance</a>; its boundary is what is usually meant by the <i>unit sphere</i>.</p> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(x_{1},...x_{n})\in \mathbb {R} ^{n}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> ∈<!-- ∈ --> </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=(x_{1},...x_{n})\in \mathbb {R} ^{n}.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86661b34262250238b6fdc87225ad70f9253195b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.689ex; height:2.843ex;" alt="{\displaystyle x=(x_{1},...x_{n})\in \mathbb {R} ^{n}.}"> </noscript><span class="lazy-image-placeholder" style="width: 21.689ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86661b34262250238b6fdc87225ad70f9253195b" data-alt="{\displaystyle x=(x_{1},...x_{n})\in \mathbb {R} ^{n}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> Define the usual <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{p}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> ℓ<!-- ℓ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ell _{p}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8265d5df21cb4dab10dd3cd69a19895d649f5b45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.029ex; height:2.843ex;" alt="{\displaystyle \ell _{p}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.029ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8265d5df21cb4dab10dd3cd69a19895d649f5b45" data-alt="{\displaystyle \ell _{p}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-norm for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\geq 1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo> ≥<!-- ≥ --> </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p\geq 1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93c1353080a21b276e79058d82c19c40310653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\geq 1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.52ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93c1353080a21b276e79058d82c19c40310653" data-alt="{\displaystyle p\geq 1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> as:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{p}={\biggl (}\sum _{k=1}^{n}|x_{k}|^{p}{\biggr )}^{1/p}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mi> x </mi> <msub> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> </mrow> <munderover> <mo> ∑<!-- ∑ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mi> p </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|x\|_{p}={\biggl (}\sum _{k=1}^{n}|x_{k}|^{p}{\biggr )}^{1/p}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92bc596f9581bd37035e70db6c290d5d9544a127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.837ex; height:7.176ex;" alt="{\displaystyle \|x\|_{p}={\biggl (}\sum _{k=1}^{n}|x_{k}|^{p}{\biggr )}^{1/p}}"> </noscript><span class="lazy-image-placeholder" style="width: 22.837ex;height: 7.176ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92bc596f9581bd37035e70db6c290d5d9544a127" data-alt="{\displaystyle \|x\|_{p}={\biggl (}\sum _{k=1}^{n}|x_{k}|^{p}{\biggr )}^{1/p}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mi> x </mi> <msub> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|x\|_{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64198280094b8bdaac094dd8f5681d62b2b4d767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.709ex; height:2.843ex;" alt="{\displaystyle \|x\|_{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.709ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64198280094b8bdaac094dd8f5681d62b2b4d767" data-alt="{\displaystyle \|x\|_{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the usual <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hilbert_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hilbert space">Hilbert space</a> norm. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mi> x </mi> <msub> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|x\|_{1}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ca585aadc8cee1acad7d68fdab962d3a18ae7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.709ex; height:2.843ex;" alt="{\displaystyle \|x\|_{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.709ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50ca585aadc8cee1acad7d68fdab962d3a18ae7e" data-alt="{\displaystyle \|x\|_{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is called the Hamming norm, or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> ℓ<!-- ℓ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ell _{1}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/361ddd720474aa41cb05453e03424fb7999d3b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.024ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/361ddd720474aa41cb05453e03424fb7999d3b02" data-alt="{\displaystyle \ell _{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-norm. The condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\geq 1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> p </mi> <mo> ≥<!-- ≥ --> </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle p\geq 1} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93c1353080a21b276e79058d82c19c40310653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\geq 1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.52ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf93c1353080a21b276e79058d82c19c40310653" data-alt="{\displaystyle p\geq 1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is necessary in the definition of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{p}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> ℓ<!-- ℓ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ell _{p}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8265d5df21cb4dab10dd3cd69a19895d649f5b45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.029ex; height:2.843ex;" alt="{\displaystyle \ell _{p}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.029ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8265d5df21cb4dab10dd3cd69a19895d649f5b45" data-alt="{\displaystyle \ell _{p}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> norm, as the unit ball in any normed space must be <a href="https://en-m-wikipedia-org.translate.goog/wiki/Convex_set?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Convex set">convex</a> as a consequence of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Triangle inequality">triangle inequality</a>. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|x\|_{\infty }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mi> x </mi> <msub> <mo fence="false" stretchy="false"> ‖<!-- ‖ --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞<!-- ∞ --> </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|x\|_{\infty }} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52addef01b4eb8d8e0c79f11dc907e86837e23e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle \|x\|_{\infty }}"> </noscript><span class="lazy-image-placeholder" style="width: 5.53ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52addef01b4eb8d8e0c79f11dc907e86837e23e5" data-alt="{\displaystyle \|x\|_{\infty }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> denote the max-norm or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{\infty }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> ℓ<!-- ℓ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞<!-- ∞ --> </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ell _{\infty }} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1989b3bca9a0361579a2b9aec109b16c9daa7d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.845ex; height:2.509ex;" alt="{\displaystyle \ell _{\infty }}"> </noscript><span class="lazy-image-placeholder" style="width: 2.845ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1989b3bca9a0361579a2b9aec109b16c9daa7d18" data-alt="{\displaystyle \ell _{\infty }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-norm of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" data-alt="{\displaystyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <p>Note that for the one-dimensional circumferences <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{p}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle C_{p}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc37470431fbf62081b69ba870ad3f855178361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.721ex; height:2.843ex;" alt="{\displaystyle C_{p}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.721ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc37470431fbf62081b69ba870ad3f855178361" data-alt="{\displaystyle C_{p}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> of the two-dimensional unit balls, we have:</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}=4{\sqrt {2}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 4 </mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle C_{1}=4{\sqrt {2}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f833aa82efb99684516c211120502b83414b8757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.075ex; height:3.009ex;" alt="{\displaystyle C_{1}=4{\sqrt {2}}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.075ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f833aa82efb99684516c211120502b83414b8757" data-alt="{\displaystyle C_{1}=4{\sqrt {2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the minimum value. </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}=2\pi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mn> 2 </mn> <mi> π<!-- π --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle C_{2}=2\pi } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627321c9a32dc994b64bc0fbcf28802b182b2adb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.309ex; height:2.509ex;" alt="{\displaystyle C_{2}=2\pi }"> </noscript><span class="lazy-image-placeholder" style="width: 8.309ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627321c9a32dc994b64bc0fbcf28802b182b2adb" data-alt="{\displaystyle C_{2}=2\pi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\infty }=8}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> C </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞<!-- ∞ --> </mi> </mrow> </msub> <mo> = </mo> <mn> 8 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle C_{\infty }=8} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/283fd81ef6eeb7eb728041bba1df5178cec8a9d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.798ex; height:2.509ex;" alt="{\displaystyle C_{\infty }=8}"> </noscript><span class="lazy-image-placeholder" style="width: 7.798ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/283fd81ef6eeb7eb728041bba1df5178cec8a9d4" data-alt="{\displaystyle C_{\infty }=8}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the maximum value. </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Generalizations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <div class="mw-heading mw-heading3"> <h3 id="Metric_spaces">Metric spaces</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Metric spaces" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>All three of the above definitions can be straightforwardly generalized to a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Metric_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Metric space">metric space</a>, with respect to a chosen origin. However, topological considerations (interior, closure, border) need not apply in the same way (e.g., in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ultrametric?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Ultrametric">ultrametric</a> spaces, all of the three are simultaneously open and closed sets), and the unit sphere may even be empty in some metric spaces.</p> <div class="mw-heading mw-heading3"> <h3 id="Quadratic_forms">Quadratic forms</h3><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Quadratic forms" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> </noscript><span class="lazy-image-placeholder" style="width: 1.787ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" data-alt="{\displaystyle V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is a linear space with a real <a href="https://en-m-wikipedia-org.translate.goog/wiki/Quadratic_form?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quadratic form">quadratic form</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:V\to \mathbb {R} ,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo> : </mo> <mi> V </mi> <mo stretchy="false"> →<!-- → --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F:V\to \mathbb {R} ,} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70ac6ed5747aadeb434bee0cd669b5cda4650ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.509ex;" alt="{\displaystyle F:V\to \mathbb {R} ,}"> </noscript><span class="lazy-image-placeholder" style="width: 11.404ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c70ac6ed5747aadeb434bee0cd669b5cda4650ea" data-alt="{\displaystyle F:V\to \mathbb {R} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p\in V:F(p)=1\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <mi> p </mi> <mo> ∈<!-- ∈ --> </mo> <mi> V </mi> <mo> : </mo> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> p </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 1 </mn> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{p\in V:F(p)=1\}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75de7c967a5a9bb4181eab0f69ee011419cee3ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.04ex; height:2.843ex;" alt="{\displaystyle \{p\in V:F(p)=1\}}"> </noscript><span class="lazy-image-placeholder" style="width: 19.04ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75de7c967a5a9bb4181eab0f69ee011419cee3ab" data-alt="{\displaystyle \{p\in V:F(p)=1\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> may be called the unit sphere<sup id="cite_ref-3" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> or <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hyperboloid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Relation_to_the_sphere" title="Hyperboloid">unit quasi-sphere</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> V </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V.} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.434ex; height:2.176ex;" alt="{\displaystyle V.}"> </noscript><span class="lazy-image-placeholder" style="width: 2.434ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2661a49b86bd1a5548e527bbfb068aa9f59978" data-alt="{\displaystyle V.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> For example, the quadratic form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-y^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> −<!-- − --> </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x^{2}-y^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f07ef330c7bbc2bd1c835de4093c922c1ce1d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.439ex; height:3.009ex;" alt="{\displaystyle x^{2}-y^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 7.439ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7f07ef330c7bbc2bd1c835de4093c922c1ce1d6" data-alt="{\displaystyle x^{2}-y^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, when set equal to one, produces the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Unit_hyperbola?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unit hyperbola">unit hyperbola</a>, which plays the role of the "unit circle" in the plane of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Split-complex_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Split-complex number">split-complex numbers</a>. Similarly, the quadratic form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x^{2}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.384ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" data-alt="{\displaystyle x^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> yields a pair of lines for the unit sphere in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dual_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dual number">dual number</a> plane.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ball_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ball (mathematics)">Ball</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/N-sphere?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="N-sphere"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-sphere</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Sphere?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sphere">Sphere</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Superellipse?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Superellipse">Superellipse</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Unit_circle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unit circle">Unit circle</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Unit_disk?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unit disk">Unit disk</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Unit_tangent_bundle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unit tangent bundle">Unit tangent bundle</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Unit_square?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Unit square">Unit square</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Notes_and_references">Notes and references</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Notes and references" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style> <div class="reflist"> <div class="mw-references-wrap"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=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</a></span></li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFManin2006" class="citation journal cs1">Manin, Yuri I. (2006). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.ams.org/bull/2006-43-02/S0273-0979-06-01081-0/S0273-0979-06-01081-0.pdf">"The notion of dimension in geometry and algebra"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of the American Mathematical Society</i>. <b>43</b> (2): 139–161. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1090%252FS0273-0979-06-01081-0">10.1090/S0273-0979-06-01081-0</a><span class="reference-accessdate">. Retrieved <span class="nowrap">17 December</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=The+notion+of+dimension+in+geometry+and+algebra&rft.volume=43&rft.issue=2&rft.pages=139-161&rft.date=2006&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-06-01081-0&rft.aulast=Manin&rft.aufirst=Yuri+I.&rft_id=https%3A%2F%2Fwww.ams.org%2Fbull%2F2006-43-02%2FS0273-0979-06-01081-0%2FS0273-0979-06-01081-0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnit+sphere" class="Z3988"></span></span></li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">^</a></b></span> <span class="reference-text">Takashi Ono (1994) <i>Variations on a Theme of Euler: quadratic forms, elliptic curves, and Hopf maps</i>, chapter 5: Quadratic spherical maps, page 165, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Plenum_Press?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Plenum Press">Plenum Press</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-306-44789-4?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-306-44789-4">0-306-44789-4</a></span></li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="https://en-m-wikipedia-org.translate.goog/wiki/Closed_unit_ball?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-4">^</a></b></span> <span class="reference-text">F. Reese Harvey (1990) <i>Spinors and calibrations</i>, "Generalized Spheres", page 42, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Academic_Press?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Academic Press">Academic Press</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-12-329650-1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-12-329650-1">0-12-329650-1</a></span></li> </ol> </div> </div> <ul> <li>Mahlon M. Day (1958) <i>Normed Linear Spaces</i>, page 24, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Springer-Verlag?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>.</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDezaDeza2006" class="citation cs2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Elena_Deza?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Elena Deza">Deza, E.</a>; Deza, M. (2006), <i>Dictionary of Distances</i>, Elsevier, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-444-52087-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-444-52087-2"><bdi>0-444-52087-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dictionary+of+Distances&rft.pub=Elsevier&rft.date=2006&rft.isbn=0-444-52087-2&rft.aulast=Deza&rft.aufirst=E.&rft.au=Deza%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnit+sphere" class="Z3988"></span>. Reviewed in <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.scribd.com/doc/2668595/Newsletter-of-the-European-Mathematical-Society-20070664-featuring-Let-Platonism-Die"><i>Newsletter of the European Mathematical Society</i> <b>64</b> (June 2007)</a>, p. 57. This book is organized as a list of distances of many types, each with a brief description.</li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="External_links">External links</h2><span class="mw-editsection"> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Unit_sphere&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style> <style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style> <div class="side-box side-box-right plainlinks sistersitebox"> <style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"> <span class="noviewer" typeof="mw:File"><span> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" data-file-width="512" data-file-height="512"> </noscript><span class="lazy-image-placeholder" style="width: 40px;height: 40px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" data-alt="" data-width="40" data-height="40" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-class="mw-file-element"> </span></span></span> </div> <div class="side-box-text plainlist"> Look up <i><b><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wiktionary.org/wiki/unit_sphere" class="extiw" title="wiktionary:unit sphere">unit sphere</a></b></i> in Wiktionary, the free dictionary. </div> </div> </div> <ul> <li><span class="citation mathworld" id="Reference-Mathworld-Unit_sphere"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Eric_W._Weisstein?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathworld.wolfram.com/UnitSphere.html">"Unit sphere"</a>. <i><a href="https://en-m-wikipedia-org.translate.goog/wiki/MathWorld?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Unit+sphere&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FUnitSphere.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnit+sphere" class="Z3988"></span></span></li> </ul><!-- NewPP limit report Parsed by mw‐web.codfw.main‐f69cdc8f6‐djcsx Cached time: 20241122144514 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.413 seconds Real time usage: 0.618 seconds Preprocessor visited node count: 1691/1000000 Post‐expand include size: 11357/2097152 bytes Template argument size: 1267/2097152 bytes Highest expansion depth: 16/100 Expensive parser function count: 4/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 22207/5000000 bytes Lua time usage: 0.164/10.000 seconds Lua memory usage: 4103463/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 345.086 1 -total 40.44% 139.544 1 Template:Reflist 28.52% 98.429 1 Template:Short_description 25.79% 89.002 1 Template:Cite_journal 16.81% 58.003 2 Template:Pagetype 9.00% 31.062 2 Template:Isbn 8.07% 27.852 1 Template:Wiktionary 7.53% 25.993 6 Template:Main_other 7.45% 25.716 1 Template:Sister_project 6.93% 23.910 1 Template:Side_box --> <!-- Saved in parser cache with key enwiki:pcache:27033090:|#|:idhash:canonical and timestamp 20241122144514 and revision id 1226082845. 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data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Eenheidsbol" title="Eenheidsbol – Dutch" lang="nl" hreflang="nl" data-title="Eenheidsbol" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E5%258D%2598%25E4%25BD%258D%25E7%2590%2583%25E9%259D%25A2" title="単位球面 – Japanese" lang="ja" hreflang="ja" data-title="単位球面" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li> <li class="interlanguage-link interwiki-no mw-list-item"><a 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href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/Birim_k%25C3%25BCre" title="Birim küre – Turkish" lang="tr" hreflang="tr" data-title="Birim küre" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259E%25D0%25B4%25D0%25B8%25D0%25BD%25D0%25B8%25D1%2587%25D0%25BD%25D0%25B0_%25D1%2581%25D1%2584%25D0%25B5%25D1%2580%25D0%25B0" title="Одинична сфера – Ukrainian" lang="uk" hreflang="uk" data-title="Одинична сфера" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li> <li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/H%25C3%25ACnh_c%25E1%25BA%25A7u_%25C4%2591%25C6%25A1n_v%25E1%25BB%258B" title="Hình cầu đơn vị – Vietnamese" lang="vi" hreflang="vi" data-title="Hình cầu đơn vị" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E5%258D%2595%25E4%25BD%258D%25E7%2590%2583%25E9%259D%25A2" title="单位球面 – Chinese" lang="zh" hreflang="zh" data-title="单位球面" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; 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