3-sphere - Wikipedia

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<span>Coordinate systems on the 3-sphere</span> </div> </a> <button aria-controls="toc-Coordinate_systems_on_the_3-sphere-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Coordinate systems on the 3-sphere subsection</span> </button> <ul id="toc-Coordinate_systems_on_the_3-sphere-sublist" class="vector-toc-list"> <li id="toc-Hyperspherical_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hyperspherical_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Hyperspherical coordinates</span> </div> </a> <ul id="toc-Hyperspherical_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hopf_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hopf_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> 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<div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://el.wikipedia.org/wiki/3-%CF%83%CF%86%CE%B1%CE%AF%CF%81%CE%B1" title="3-σφαίρα – Greek" lang="el" hreflang="el" data-title="3-σφαίρα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/3-esfera" title="3-esfera – Spanish" lang="es" hreflang="es" data-title="3-esfera" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/3-sfero" title="3-sfero – Esperanto" lang="eo" hreflang="eo" data-title="3-sfero" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/3-sph%C3%A8re" title="3-sphère – French" lang="fr" hreflang="fr" data-title="3-sphère" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/3%EC%B0%A8%EC%9B%90_%EC%B4%88%EA%B5%AC" title="3차원 초구 – Korean" lang="ko" hreflang="ko" data-title="3차원 초구" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/3-sfera" title="3-sfera – Italian" lang="it" hreflang="it" data-title="3-sfera" 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://nl.wikipedia.org/wiki/3-sfeer" title="3-sfeer – Dutch" lang="nl" hreflang="nl" data-title="3-sfeer" 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://ja.wikipedia.org/wiki/%E4%B8%89%E6%AC%A1%E5%85%83%E7%90%83%E9%9D%A2" title="三次元球面 – Japanese" lang="ja" hreflang="ja" data-title="三次元球面" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Esfera_tridimensional" title="Esfera tridimensional – Portuguese" lang="pt" hreflang="pt" data-title="Esfera tridimensional" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D1%91%D1%85%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D1%8F_%D1%81%D1%84%D0%B5%D1%80%D0%B0" title="Трёхмерная сфера – Russian" lang="ru" hreflang="ru" data-title="Трёхмерная сфера" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/3-%D1%81%D1%84%D0%B5%D1%80%D0%B0" title="3-сфера – Ukrainian" lang="uk" hreflang="uk" data-title="3-сфера" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%89%E7%B6%AD%E7%90%83%E9%9D%A2" title="三維球面 – Chinese" lang="zh" hreflang="zh" data-title="三維球面" data-language-autonym="中文" data-language-local-name="Chinese" 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0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">June 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hypersphere_coord.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Hypersphere_coord.PNG/220px-Hypersphere_coord.PNG" decoding="async" width="220" height="292" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/32/Hypersphere_coord.PNG 1.5x" data-file-width="320" data-file-height="425" /></a><figcaption><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a> of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is <a href="/wiki/Conformal_map" title="Conformal map">conformal</a>, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect <span class="nowrap">⟨0,0,0,1⟩</span> have infinite radius (= straight line). In this picture, the whole 3D space maps the <i>surface</i> of the hypersphere, whereas in the next picture the 3D space contained the <i>shadow</i> of the bulk hypersphere.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hypersphere.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Hypersphere.png/220px-Hypersphere.png" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Hypersphere.png/330px-Hypersphere.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Hypersphere.png/440px-Hypersphere.png 2x" data-file-width="477" data-file-height="479" /></a><figcaption>Direct projection of <i>3-sphere</i> into 3D space and covered with surface grid, showing structure as stack of 3D spheres (<i>2-spheres</i>)</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>hypersphere</b> or <b>3-sphere</b> is a 4-dimensional analogue of a <a href="/wiki/Sphere" title="Sphere">sphere</a>, and is the 3-dimensional <a href="/wiki/N-sphere" title="N-sphere"><i>n</i>-sphere</a>. In 4-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, it is the set of points equidistant from a fixed central point. The interior of a 3-sphere is a <b>4-ball</b>. </p><p>It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even though it is curved into the 4th dimension. For example, when traveling on a 3-sphere, you can go north and south, east and west, or along a 3rd set of cardinal directions. This means that a 3-sphere is an example of a <a href="/wiki/3-manifold" title="3-manifold">3-manifold</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Coordinates" class="mw-redirect" title="Coordinates">coordinates</a>, a 3-sphere with center <span class="texhtml">(<i>C</i><sub>0</sub>, <i>C</i><sub>1</sub>, <i>C</i><sub>2</sub>, <i>C</i><sub>3</sub>)</span> and radius <span class="texhtml mvar" style="font-style:italic;">r</span> is the set of all points <span class="texhtml">(<i>x</i><sub>0</sub>, <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>)</span> in real, <a href="/wiki/Four-dimensional_space" title="Four-dimensional space">4-dimensional space</a> (<span class="texhtml"><b>R</b><sup>4</sup></span>) such that </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 \sum _{i=0}^{3}(x_{i}-C_{i})^{2}=(x_{0}-C_{0})^{2}+(x_{1}-C_{1})^{2}+(x_{2}-C_{2})^{2}+(x_{3}-C_{3})^{2}=r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{3}(x_{i}-C_{i})^{2}=(x_{0}-C_{0})^{2}+(x_{1}-C_{1})^{2}+(x_{2}-C_{2})^{2}+(x_{3}-C_{3})^{2}=r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63ea96cfc9f55cd91f1f5aafa6c625be53cabf85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:74.333ex; height:7.176ex;" alt="{\displaystyle \sum _{i=0}^{3}(x_{i}-C_{i})^{2}=(x_{0}-C_{0})^{2}+(x_{1}-C_{1})^{2}+(x_{2}-C_{2})^{2}+(x_{3}-C_{3})^{2}=r^{2}.}" /></span></dd></dl> <p>The 3-sphere centered at the origin with radius 1 is called the <b>unit 3-sphere</b> and is usually denoted <span class="texhtml"><i>S</i><sup>3</sup></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 S^{3}=\left\{(x_{0},x_{1},x_{2},x_{3})\in \mathbb {R} ^{4}:x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </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"> <mn>4</mn> </mrow> </msup> <mo>:</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <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> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{3}=\left\{(x_{0},x_{1},x_{2},x_{3})\in \mathbb {R} ^{4}:x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efdefd94a7565635aecce116f39bc17bd24ebcca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.693ex; height:3.343ex;" alt="{\displaystyle S^{3}=\left\{(x_{0},x_{1},x_{2},x_{3})\in \mathbb {R} ^{4}:x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\right\}.}" /></span></dd></dl> <p>It is often convenient to regard <span class="texhtml"><b>R</b><sup>4</sup></span> as the space with 2 <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex dimensions</a> (<span class="texhtml"><b>C</b><sup>2</sup></span>) or the <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> (<span class="texhtml"><b>H</b></span>). The unit 3-sphere is then 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 S^{3}=\left\{(z_{1},z_{2})\in \mathbb {C} ^{2}:|z_{1}|^{2}+|z_{2}|^{2}=1\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{3}=\left\{(z_{1},z_{2})\in \mathbb {C} ^{2}:|z_{1}|^{2}+|z_{2}|^{2}=1\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837943175482619f273657c4cf4ae035306c4612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.467ex; height:4.843ex;" alt="{\displaystyle S^{3}=\left\{(z_{1},z_{2})\in \mathbb {C} ^{2}:|z_{1}|^{2}+|z_{2}|^{2}=1\right\}}" /></span></dd></dl> <p>or </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 S^{3}=\left\{q\in \mathbb {H} :\|q\|=1\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>q</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>q</mi> <mo fence="false" stretchy="false">‖</mo> <mo>=</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{3}=\left\{q\in \mathbb {H} :\|q\|=1\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee9073c7d56ac6cfb926c3f8233f36dbb53babc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.344ex; height:3.176ex;" alt="{\displaystyle S^{3}=\left\{q\in \mathbb {H} :\|q\|=1\right\}.}" /></span></dd></dl> <p>This description as the <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> of <a href="/wiki/Quaternion#Conjugation,_the_norm,_and_reciprocal" title="Quaternion">norm</a> one identifies the 3-sphere with the <a href="/wiki/Versor" title="Versor">versors</a> in the quaternion <a href="/wiki/Division_ring" title="Division ring">division ring</a>. Just as the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> is important for planar <a href="/wiki/Polar_coordinates#Complex_numbers" class="mw-redirect" title="Polar coordinates">polar coordinates</a>, so the 3-sphere is important in the polar view of 4-space involved in quaternion multiplication. See <a href="/wiki/Polar_decomposition#Quaternion_polar_decomposition" title="Polar decomposition">polar decomposition of a quaternion</a> for details of this development of the three-sphere. This view of the 3-sphere is the basis for the study of <a href="/wiki/Elliptic_geometry#Elliptic_space" title="Elliptic geometry">elliptic space</a> as developed by <a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Georges Lemaître</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Elementary_properties">Elementary properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=3" title="Edit section: Elementary properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 3-dimensional surface volume of a 3-sphere of radius <span class="texhtml mvar" style="font-style:italic;">r</span> 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 SV=2\pi ^{2}r^{3}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>V</mi> <mo>=</mo> <mn>2</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SV=2\pi ^{2}r^{3}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61e869f1893c927d74a8d49b2e4abc818184b071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.426ex; height:2.676ex;" alt="{\displaystyle SV=2\pi ^{2}r^{3}\,}" /></span></dd></dl> <p>while the 4-dimensional hypervolume (the content of the 4-dimensional region, or ball, bounded by the 3-sphere) 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 H={\frac {1}{2}}\pi ^{2}r^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H={\frac {1}{2}}\pi ^{2}r^{4}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc5bb30093e09641083826254599b7864c751c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.299ex; height:5.176ex;" alt="{\displaystyle H={\frac {1}{2}}\pi ^{2}r^{4}.}" /></span></dd></dl> <p>Every non-empty intersection of a 3-sphere with a three-dimensional <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point). As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere that reaches its maximal size when the hyperplane cuts right through the "equator" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane. </p><p>In a given three-dimensional hyperplane, a 3-sphere can rotate about an "equatorial plane" (analogous to a 2-sphere rotating about a central axis), in which case it appears to be a 2-sphere whose size is constant. </p> <div class="mw-heading mw-heading3"><h3 id="Topological_properties">Topological properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=4" title="Edit section: Topological properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A 3-sphere is a <a href="/wiki/Compact_space" title="Compact space">compact</a>, <a href="/wiki/Connected_space" title="Connected space">connected</a>, 3-dimensional <a href="/wiki/Manifold" title="Manifold">manifold</a> without boundary. It is also <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>. What this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a>, proved in 2003 by <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a>, provides that the 3-sphere is the only three-dimensional manifold (up to <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>) with these properties. </p><p>The 3-sphere is homeomorphic to the <a href="/wiki/One-point_compactification" class="mw-redirect" title="One-point compactification">one-point compactification</a> of <span class="texhtml"><b>R</b><sup>3</sup></span>. In general, any <a href="/wiki/Topological_space" title="Topological space">topological space</a> that is homeomorphic to the 3-sphere is called a <b>topological 3-sphere</b>. </p><p>The <a href="/wiki/Homology_group" class="mw-redirect" title="Homology group">homology groups</a> of the 3-sphere are as follows: <span class="texhtml">H<sub>0</sub>(<i>S</i><sup>3</sup>, <b>Z</b>)</span> and <span class="texhtml">H<sub>3</sub>(<i>S</i><sup>3</sup>, <b>Z</b>)</span> are both <a href="/wiki/Infinite_cyclic" class="mw-redirect" title="Infinite cyclic">infinite cyclic</a>, while <span class="texhtml">H<sub><i>i</i></sub>(<i>S</i><sup>3</sup>, <b>Z</b>) = {}</span> for all other indices <span class="texhtml mvar" style="font-style:italic;">i</span>. Any topological space with these homology groups is known as a <a href="/wiki/Homology_sphere" title="Homology sphere">homology 3-sphere</a>. Initially <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a> conjectured that all homology 3-spheres are homeomorphic to <span class="texhtml"><i>S</i><sup>3</sup></span>, but then he himself constructed a non-homeomorphic one, now known as the <a href="/wiki/Poincar%C3%A9_homology_sphere" class="mw-redirect" title="Poincaré homology sphere">Poincaré homology sphere</a>. Infinitely many homology spheres are now known to exist. For example, a <a href="/wiki/Dehn_filling" class="mw-redirect" title="Dehn filling">Dehn filling</a> with slope <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>n</i></span></span>⁠</span></span> on any <a href="/wiki/Knot_theory" title="Knot theory">knot</a> in the 3-sphere gives a homology sphere; typically these are not homeomorphic to the 3-sphere. </p><p>As to the <a href="/wiki/Homotopy_groups" class="mw-redirect" title="Homotopy groups">homotopy groups</a>, we have <span class="texhtml">π<sub>1</sub>(<i>S</i><sup>3</sup>) = π<sub>2</sub>(<i>S</i><sup>3</sup>) = {}</span> and <span class="texhtml">π<sub>3</sub>(<i>S</i><sup>3</sup>)</span> is infinite cyclic. The higher-homotopy groups (<span class="texhtml"><i>k</i> ≥ 4</span>) are all <a href="/wiki/Finite_abelian_group" class="mw-redirect" title="Finite abelian group">finite abelian</a> but otherwise follow no discernible pattern. For more discussion see <a href="/wiki/Homotopy_groups_of_spheres" title="Homotopy groups of spheres">homotopy groups of spheres</a>. </p> <table class="wikitable" style="text-align: center; margin: auto;"> <caption>Homotopy groups of <span class="texhtml"><i>S</i><sup>3</sup></span> </caption> <tbody><tr> <td><span class="texhtml mvar" style="font-style:italic;">k</span> </td> <td>0</td> <td>1</td> <td>2</td> <td>3</td> <td>4</td> <td>5</td> <td>6</td> <td>7</td> <td>8</td> <td>9</td> <td>10</td> <td>11</td> <td>12</td> <td>13</td> <td>14</td> <td>15</td> <td>16 </td></tr> <tr> <td><span class="texhtml">π<sub><i>k</i></sub>(<i>S</i><sup>3</sup>)</span> </td> <td>0</td> <td>0</td> <td>0</td> <td><span class="texhtml"><b>Z</b></span></td> <td><span class="texhtml"><b>Z</b><sub>2</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>2</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>12</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>2</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>2</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>3</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>15</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>2</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>2</sub>⊕<b>Z</b><sub>2</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>12</sub>⊕<b>Z</b><sub>2</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>84</sub>⊕<b>Z</b><sub>2</sub>⊕<b>Z</b><sub>2</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>2</sub>⊕<b>Z</b><sub>2</sub></span></td> <td><span class="texhtml"><b>Z</b><sub>6</sub></span> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Geometric_properties">Geometric properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=5" title="Edit section: Geometric properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The 3-sphere is naturally a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>, in fact, a closed <a href="/wiki/Embedded_submanifold" class="mw-redirect" title="Embedded submanifold">embedded submanifold</a> of <span class="texhtml"><b>R</b><sup>4</sup></span>. The <a href="/wiki/Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a> on <span class="texhtml"><b>R</b><sup>4</sup></span> induces a <a href="/wiki/Metric_tensor" title="Metric tensor">metric</a> on the 3-sphere giving it the structure of a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>. As with all spheres, the 3-sphere has constant positive <a href="/wiki/Sectional_curvature" title="Sectional curvature">sectional curvature</a> equal to <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"><i>r</i><sup>2</sup></span></span>⁠</span></span> where <span class="texhtml mvar" style="font-style:italic;">r</span> is the radius. </p><p>Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural <a href="/wiki/Lie_group" title="Lie group">Lie group</a> structure given by quaternion multiplication (see the section below on <a href="#Group_structure">group structure</a>). The only other spheres with such a structure are the 0-sphere and the 1-sphere (see <a href="/wiki/Circle_group" title="Circle group">circle group</a>). </p><p>Unlike the 2-sphere, the 3-sphere admits nonvanishing <a href="/wiki/Vector_field" title="Vector field">vector fields</a> (<a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">sections</a> of its <a href="/wiki/Tangent_bundle" title="Tangent bundle">tangent bundle</a>). One can even find three linearly independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of the 3-sphere. This implies that the 3-sphere is <a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">parallelizable</a>. It follows that the tangent bundle of the 3-sphere is <a href="/wiki/Trivial_bundle" class="mw-redirect" title="Trivial bundle">trivial</a>. For a general discussion of the number of linear independent vector fields on a <span class="texhtml mvar" style="font-style:italic;">n</span>-sphere, see the article <a href="/wiki/Vector_fields_on_spheres" title="Vector fields on spheres">vector fields on spheres</a>. </p><p>There is an interesting <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of the <a href="/wiki/Circle_group" title="Circle group">circle group</a> <span class="texhtml"><b>T</b></span> on <span class="texhtml"><i>S</i><sup>3</sup></span> giving the 3-sphere the structure of a <a href="/wiki/Principal_circle_bundle" class="mw-redirect" title="Principal circle bundle">principal circle bundle</a> known as the <a href="/wiki/Hopf_bundle" class="mw-redirect" title="Hopf bundle">Hopf bundle</a>. If one thinks of <span class="texhtml"><i>S</i><sup>3</sup></span> as a subset of <span class="texhtml"><b>C</b><sup>2</sup></span>, the action 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 (z_{1},z_{2})\cdot \lambda =(z_{1}\lambda ,z_{2}\lambda )\quad \forall \lambda \in \mathbb {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>λ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>λ</mi> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>λ</mi> <mo stretchy="false">)</mo> <mspace width="1em"></mspace> <mi mathvariant="normal">∀</mi> <mi>λ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (z_{1},z_{2})\cdot \lambda =(z_{1}\lambda ,z_{2}\lambda )\quad \forall \lambda \in \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d753d78e5a96ba535dfb43cb2138865e09271646" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.433ex; height:2.843ex;" alt="{\displaystyle (z_{1},z_{2})\cdot \lambda =(z_{1}\lambda ,z_{2}\lambda )\quad \forall \lambda \in \mathbb {T} }" /></span>.</dd></dl> <p>The <a href="/wiki/Orbit_space" class="mw-redirect" title="Orbit space">orbit space</a> of this action is homeomorphic to the two-sphere <span class="texhtml"><i>S</i><sup>2</sup></span>. Since <span class="texhtml"><i>S</i><sup>3</sup></span> is not homeomorphic to <span class="texhtml"><i>S</i><sup>2</sup> × <i>S</i><sup>1</sup></span>, the Hopf bundle is nontrivial. </p> <div class="mw-heading mw-heading2"><h2 id="Topological_construction">Topological construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=6" title="Edit section: Topological construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are several well-known constructions of the three-sphere. Here we describe gluing a pair of three-balls and then the one-point compactification. </p> <div class="mw-heading mw-heading3"><h3 id="Gluing">Gluing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=7" title="Edit section: Gluing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A 3-sphere can be constructed <a href="/wiki/Topology" title="Topology">topologically</a> by <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">"gluing" together</a> the boundaries of a pair of 3-<a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">balls</a>. The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair of 2-spheres be identically equivalent to each other. In analogy with the case of the 2-sphere (see below), the gluing surface is called an equatorial sphere. </p><p>Note that the interiors of the 3-balls are not glued to each other. One way to think of the fourth dimension is as a continuous <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued function</a> of the 3-dimensional coordinates of the 3-ball, perhaps considered to be "temperature". We take the "temperature" to be zero along the gluing 2-sphere and let one of the 3-balls be "hot" and let the other 3-ball be "cold". The "hot" 3-ball could be thought of as the "upper hemisphere" and the "cold" 3-ball could be thought of as the "lower hemisphere". The temperature is highest/lowest at the centers of the two 3-balls. </p><p>This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. </p> <div class="mw-heading mw-heading3"><h3 id="One-point_compactification">One-point compactification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=8" title="Edit section: One-point compactification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean plane. In the same way, removing a single point from the 3-sphere yields three-dimensional space. An extremely useful way to see this is via <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a>. We first describe the lower-dimensional version. </p><p>Rest the south pole of a unit 2-sphere on the <span class="texhtml mvar" style="font-style:italic;">xy</span>-plane in three-space. We map a point <span class="texhtml">P</span> of the sphere (minus the north pole <span class="texhtml">N</span>) to the plane by sending <span class="texhtml">P</span> to the intersection of the line <span class="texhtml">NP</span> with the plane. Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner. (Notice that, since stereographic projection is <a href="/wiki/Conformal_map_projection" title="Conformal map projection">conformal</a>, round spheres are sent to round spheres or to planes.) </p><p>A somewhat different way to think of the one-point compactification is via the <a href="/wiki/Exponential_map_(Riemmanian_geometry)" class="mw-redirect" title="Exponential map (Riemmanian geometry)">exponential map</a>. Returning to our picture of the unit two-sphere sitting on the Euclidean plane: Consider a geodesic in the plane, based at the origin, and map this to a geodesic in the two-sphere of the same length, based at the south pole. Under this map all points of the circle of radius <span class="texhtml mvar" style="font-style:italic;">π</span> are sent to the north pole. Since the open <a href="/wiki/Unit_disk" title="Unit disk">unit disk</a> is homeomorphic to the Euclidean plane, this is again a one-point compactification. </p><p>The exponential map for 3-sphere is similarly constructed; it may also be discussed using the fact that the 3-sphere is the <a href="/wiki/Lie_group" title="Lie group">Lie group</a> of unit quaternions. </p> <div class="mw-heading mw-heading2"><h2 id="Coordinate_systems_on_the_3-sphere">Coordinate systems on the 3-sphere</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=9" title="Edit section: Coordinate systems on the 3-sphere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The four Euclidean coordinates for <span class="texhtml"><i>S</i><sup>3</sup></span> are redundant since they are subject to the condition that <span class="texhtml"><i>x</i><sub>0</sub><sup>2</sup> + <i>x</i><sub>1</sub><sup>2</sup> + <i>x</i><sub>2</sub><sup>2</sup> + <i>x</i><sub>3</sub><sup>2</sup> = 1</span>. As a 3-dimensional manifold one should be able to parameterize <span class="texhtml"><i>S</i><sup>3</sup></span> by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as <a href="/wiki/Latitude" title="Latitude">latitude</a> and <a href="/wiki/Longitude" title="Longitude">longitude</a>). Due to the nontrivial topology of <span class="texhtml"><i>S</i><sup>3</sup></span> it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use <i>at least</i> two <a href="/wiki/Coordinate_chart" class="mw-redirect" title="Coordinate chart">coordinate charts</a>. Some different choices of coordinates are given below. </p> <div class="mw-heading mw-heading3"><h3 id="Hyperspherical_coordinates">Hyperspherical coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=10" title="Edit section: Hyperspherical coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is convenient to have some sort of <a href="/wiki/N-sphere#Spherical_coordinates" title="N-sphere">hyperspherical coordinates</a> on <span class="texhtml"><i>S</i><sup>3</sup></span> in analogy to the usual <a href="/wiki/Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a> on <span class="texhtml"><i>S</i><sup>2</sup></span>. One such choice — by no means unique — is to use <span class="texhtml">(<i>ψ</i>, <i>θ</i>, <i>φ</i>)</span>, where </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 {\begin{aligned}x_{0}&=r\cos \psi \\x_{1}&=r\sin \psi \cos \theta \\x_{2}&=r\sin \psi \sin \theta \cos \varphi \\x_{3}&=r\sin \psi \sin \theta \sin \varphi \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{0}&=r\cos \psi \\x_{1}&=r\sin \psi \cos \theta \\x_{2}&=r\sin \psi \sin \theta \cos \varphi \\x_{3}&=r\sin \psi \sin \theta \sin \varphi \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abfb8f5b7936e79c6e259326a3869aab391c3c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:22.551ex; height:12.009ex;" alt="{\displaystyle {\begin{aligned}x_{0}&=r\cos \psi \\x_{1}&=r\sin \psi \cos \theta \\x_{2}&=r\sin \psi \sin \theta \cos \varphi \\x_{3}&=r\sin \psi \sin \theta \sin \varphi \end{aligned}}}" /></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">ψ</span> and <span class="texhtml mvar" style="font-style:italic;">θ</span> run over the range 0 to <span class="texhtml mvar" style="font-style:italic;">π</span>, and <span class="texhtml mvar" style="font-style:italic;">φ</span> runs over 0 to 2<span class="texhtml mvar" style="font-style:italic;">π</span>. Note that, for any fixed value of <span class="texhtml mvar" style="font-style:italic;">ψ</span>, <span class="texhtml mvar" style="font-style:italic;">θ</span> and <span class="texhtml mvar" style="font-style:italic;">φ</span> parameterize a 2-sphere 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\sin \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\sin \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b8cc02592c4eee1be1384a62a9a1fe7895aff4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.192ex; height:2.509ex;" alt="{\displaystyle r\sin \psi }" /></span>, except for the degenerate cases, when <span class="texhtml mvar" style="font-style:italic;">ψ</span> equals 0 or <span class="texhtml mvar" style="font-style:italic;">π</span>, in which case they describe a point. </p><p>The <a href="/wiki/Metric_tensor" title="Metric tensor">round metric</a> on the 3-sphere in these coordinates is given by<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </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 ds^{2}=r^{2}\left[d\psi ^{2}+\sin ^{2}\psi \left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <mi>d</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ψ</mi> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msup> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=r^{2}\left[d\psi ^{2}+\sin ^{2}\psi \left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right)\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4991f71548a77845bc7d2c59807a9ad2ef53c8b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.605ex; height:3.343ex;" alt="{\displaystyle ds^{2}=r^{2}\left[d\psi ^{2}+\sin ^{2}\psi \left(d\theta ^{2}+\sin ^{2}\theta \,d\varphi ^{2}\right)\right]}" /></span></dd></dl> <p>and the <a href="/wiki/Volume_form" title="Volume form">volume form</a> 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 dV=r^{3}\left(\sin ^{2}\psi \,\sin \theta \right)\,d\psi \wedge d\theta \wedge d\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>V</mi> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ψ</mi> <mspace width="thinmathspace"></mspace> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>ψ</mi> <mo>∧</mo> <mi>d</mi> <mi>θ</mi> <mo>∧</mo> <mi>d</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dV=r^{3}\left(\sin ^{2}\psi \,\sin \theta \right)\,d\psi \wedge d\theta \wedge d\varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765508903796b5489e033cda6c1f686a86e3d680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.996ex; height:3.343ex;" alt="{\displaystyle dV=r^{3}\left(\sin ^{2}\psi \,\sin \theta \right)\,d\psi \wedge d\theta \wedge d\varphi .}" /></span></dd></dl> <p>These coordinates have an elegant description in terms of <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>. Any unit quaternion <span class="texhtml mvar" style="font-style:italic;">q</span> can be written as a <a href="/wiki/Versor" title="Versor">versor</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 q=e^{\tau \psi }=\cos \psi +\tau \sin \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mi>ψ</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ψ</mi> <mo>+</mo> <mi>τ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=e^{\tau \psi }=\cos \psi +\tau \sin \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/895e4e742ca66e17354567a98ca47cb4000a44ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.699ex; height:3.009ex;" alt="{\displaystyle q=e^{\tau \psi }=\cos \psi +\tau \sin \psi }" /></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">τ</span> is a <a href="/wiki/Quaternion#Square_roots_of_−1" title="Quaternion">unit imaginary quaternion</a>; that is, a quaternion that satisfies <span class="texhtml"><i>τ</i><sup>2</sup> = −1</span>. This is the quaternionic analogue of <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>. Now the unit imaginary quaternions all lie on the unit 2-sphere in <span class="texhtml">Im <b>H</b></span> so any such <span class="texhtml mvar" style="font-style:italic;">τ</span> can be written: </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 \tau =(\cos \theta )i+(\sin \theta \cos \varphi )j+(\sin \theta \sin \varphi )k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>i</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>j</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =(\cos \theta )i+(\sin \theta \cos \varphi )j+(\sin \theta \sin \varphi )k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac62207530d941cc8dc7f76ca0c858ac18bb128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.191ex; height:2.843ex;" alt="{\displaystyle \tau =(\cos \theta )i+(\sin \theta \cos \varphi )j+(\sin \theta \sin \varphi )k}" /></span></dd></dl> <p>With <span class="texhtml mvar" style="font-style:italic;">τ</span> in this form, the unit quaternion <span class="texhtml mvar" style="font-style:italic;">q</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 q=e^{\tau \psi }=x_{0}+x_{1}i+x_{2}j+x_{3}k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mi>ψ</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>i</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>j</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=e^{\tau \psi }=x_{0}+x_{1}i+x_{2}j+x_{3}k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee1b604205a4434deaa667b50a045c9ce7747aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.531ex; height:3.009ex;" alt="{\displaystyle q=e^{\tau \psi }=x_{0}+x_{1}i+x_{2}j+x_{3}k}" /></span></dd></dl> <p>where <span class="texhtml"><i>x</i><sub>0,1,2,3</sub></span> are as above. </p><p>When <span class="texhtml mvar" style="font-style:italic;">q</span> is used to describe spatial rotations (cf. <a href="/wiki/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">quaternions and spatial rotations</a>), it describes a rotation about <span class="texhtml mvar" style="font-style:italic;">τ</span> through an angle of <span class="texhtml">2<i>ψ</i></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Hopf_coordinates">Hopf coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=11" title="Edit section: Hopf coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Hopf_Fibration.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Hopf_Fibration.png/250px-Hopf_Fibration.png" decoding="async" width="250" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Hopf_Fibration.png/375px-Hopf_Fibration.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Hopf_Fibration.png/500px-Hopf_Fibration.png 2x" data-file-width="9920" data-file-height="8816" /></a><figcaption>The Hopf fibration can be visualized using a <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a> of <span class="texhtml"><i>S</i><sup>3</sup></span> to <span class="texhtml"><b>R</b><sup>3</sup></span> and then compressing <span class="texhtml"><i>R</i><sup>3</sup></span> to a ball. This image shows points on <span class="texhtml"><i>S</i><sup>2</sup></span> and their corresponding fibers with the same color.</figcaption></figure> <p>For unit radius another choice of hyperspherical coordinates, <span class="texhtml">(<i>η</i>, <i>ξ</i><sub>1</sub>, <i>ξ</i><sub>2</sub>)</span>, makes use of the embedding of <span class="texhtml"><i>S</i><sup>3</sup></span> in <span class="texhtml"><b>C</b><sup>2</sup></span>. In complex coordinates <span class="texhtml">(<i>z</i><sub>1</sub>, <i>z</i><sub>2</sub>) ∈ <b>C</b><sup>2</sup></span> we write </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 {\begin{aligned}z_{1}&=e^{i\,\xi _{1}}\sin \eta \\z_{2}&=e^{i\,\xi _{2}}\cos \eta .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mspace width="thinmathspace"></mspace> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>η</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mspace width="thinmathspace"></mspace> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mi>η</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}z_{1}&=e^{i\,\xi _{1}}\sin \eta \\z_{2}&=e^{i\,\xi _{2}}\cos \eta .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42f80e9da639fed7ecf013291844ac78a7afc583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.509ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}z_{1}&=e^{i\,\xi _{1}}\sin \eta \\z_{2}&=e^{i\,\xi _{2}}\cos \eta .\end{aligned}}}" /></span></dd></dl> <p>This could also be expressed in <span class="texhtml"><b>R</b><sup>4</sup></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 {\begin{aligned}x_{0}&=\cos \xi _{1}\sin \eta \\x_{1}&=\sin \xi _{1}\sin \eta \\x_{2}&=\cos \xi _{2}\cos \eta \\x_{3}&=\sin \xi _{2}\cos \eta .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>η</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>η</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>η</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>η</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{0}&=\cos \xi _{1}\sin \eta \\x_{1}&=\sin \xi _{1}\sin \eta \\x_{2}&=\cos \xi _{2}\cos \eta \\x_{3}&=\sin \xi _{2}\cos \eta .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3de187ae6a9de08f53e53b78bf7964f877849c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:17.251ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}x_{0}&=\cos \xi _{1}\sin \eta \\x_{1}&=\sin \xi _{1}\sin \eta \\x_{2}&=\cos \xi _{2}\cos \eta \\x_{3}&=\sin \xi _{2}\cos \eta .\end{aligned}}}" /></span></dd></dl> <p>Here <span class="texhtml mvar" style="font-style:italic;">η</span> runs over the range 0 to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, and <span class="texhtml"><i>ξ</i><sub>1</sub></span> and <span class="texhtml"><i>ξ</i><sub>2</sub></span> can take any values between 0 and 2<span class="texhtml mvar" style="font-style:italic;">π</span>. These coordinates are useful in the description of the 3-sphere as the <a href="/wiki/Hopf_bundle" class="mw-redirect" title="Hopf bundle">Hopf bundle</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 S^{1}\to S^{3}\to S^{2}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{1}\to S^{3}\to S^{2}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b2f859f2a0545c9f0351fb675c3c17a1cacf43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.99ex; height:2.676ex;" alt="{\displaystyle S^{1}\to S^{3}\to S^{2}.\,}" /></span></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Toroidal_coord.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Toroidal_coord.png/220px-Toroidal_coord.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Toroidal_coord.png/330px-Toroidal_coord.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Toroidal_coord.png/440px-Toroidal_coord.png 2x" data-file-width="1024" data-file-height="768" /></a><figcaption>A diagram depicting the poloidal (<span class="texhtml"><i>ξ</i><sub>1</sub></span>) direction, represented by the red arrow, and the toroidal (<span class="texhtml"><i>ξ</i><sub>2</sub></span>) direction, represented by the blue arrow, although the terms <i>poloidal</i> and <i>toroidal</i> are arbitrary in this <i><a href="/wiki/Flat_torus#Flat_torus" class="mw-redirect" title="Flat torus">flat torus</a></i> case.</figcaption></figure> <p>For any fixed value of <span class="texhtml mvar" style="font-style:italic;">η</span> between 0 and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, the coordinates <span class="texhtml">(<i>ξ</i><sub>1</sub>, <i>ξ</i><sub>2</sub>)</span> parameterize a 2-dimensional <a href="/wiki/Torus" title="Torus">torus</a>. Rings of constant <span class="texhtml"><i>ξ</i><sub>1</sub></span> and <span class="texhtml"><i>ξ</i><sub>2</sub></span> above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when <span class="texhtml mvar" style="font-style:italic;">η</span> equals 0 or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, these coordinates describe a <a href="/wiki/Circle" title="Circle">circle</a>. </p><p>The round metric on the 3-sphere in these coordinates 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 ds^{2}=d\eta ^{2}+\sin ^{2}\eta \,d\xi _{1}^{2}+\cos ^{2}\eta \,d\xi _{2}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>d</mi> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>η</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>η</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=d\eta ^{2}+\sin ^{2}\eta \,d\xi _{1}^{2}+\cos ^{2}\eta \,d\xi _{2}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c128f7c0dadd17bc50da00d5c5751d81bcbb833a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.154ex; height:3.343ex;" alt="{\displaystyle ds^{2}=d\eta ^{2}+\sin ^{2}\eta \,d\xi _{1}^{2}+\cos ^{2}\eta \,d\xi _{2}^{2}}" /></span></dd></dl> <p>and the volume form 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 dV=\sin \eta \cos \eta \,d\eta \wedge d\xi _{1}\wedge d\xi _{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>V</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>η</mi> <mi>cos</mi> <mo>⁡</mo> <mi>η</mi> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>η</mi> <mo>∧</mo> <mi>d</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mi>d</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dV=\sin \eta \cos \eta \,d\eta \wedge d\xi _{1}\wedge d\xi _{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8977127fe751c5411a12db0c76af060861b91f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.73ex; height:2.676ex;" alt="{\displaystyle dV=\sin \eta \cos \eta \,d\eta \wedge d\xi _{1}\wedge d\xi _{2}.}" /></span></dd></dl> <p>To get the interlocking circles of the <a href="/wiki/Hopf_fibration" title="Hopf fibration">Hopf fibration</a>, make a simple substitution in the equations above<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </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 {\begin{aligned}z_{1}&=e^{i\,(\xi _{1}+\xi _{2})}\sin \eta \\z_{2}&=e^{i\,(\xi _{2}-\xi _{1})}\cos \eta .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>η</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mspace width="thinmathspace"></mspace> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mi>η</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}z_{1}&=e^{i\,(\xi _{1}+\xi _{2})}\sin \eta \\z_{2}&=e^{i\,(\xi _{2}-\xi _{1})}\cos \eta .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180981c38da074c69327f642d9e140ba51c1b579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.619ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}z_{1}&=e^{i\,(\xi _{1}+\xi _{2})}\sin \eta \\z_{2}&=e^{i\,(\xi _{2}-\xi _{1})}\cos \eta .\end{aligned}}}" /></span></dd></dl> <p>In this case <span class="texhtml mvar" style="font-style:italic;">η</span>, and <span class="texhtml"><i>ξ</i><sub>1</sub></span> specify which circle, and <span class="texhtml"><i>ξ</i><sub>2</sub></span> specifies the position along each circle. One round trip (0 to 2<span class="texhtml mvar" style="font-style:italic;">π</span>) of <span class="texhtml"><i>ξ</i><sub>1</sub></span> or <span class="texhtml"><i>ξ</i><sub>2</sub></span> equates to a round trip of the torus in the 2 respective directions. </p> <div class="mw-heading mw-heading3"><h3 id="Stereographic_coordinates">Stereographic coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=12" title="Edit section: Stereographic coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another convenient set of coordinates can be obtained via <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a> of <span class="texhtml"><i>S</i><sup>3</sup></span> from a pole onto the corresponding equatorial <span class="texhtml"><b>R</b><sup>3</sup></span> <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a>. For example, if we project from the point <span class="texhtml">(−1, 0, 0, 0)</span> we can write a point <span class="texhtml mvar" style="font-style:italic;">p</span> in <span class="texhtml"><i>S</i><sup>3</sup></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 p=\left({\frac {1-\|u\|^{2}}{1+\|u\|^{2}}},{\frac {2\mathbf {u} }{1+\|u\|^{2}}}\right)={\frac {1+\mathbf {u} }{1-\mathbf {u} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\left({\frac {1-\|u\|^{2}}{1+\|u\|^{2}}},{\frac {2\mathbf {u} }{1+\|u\|^{2}}}\right)={\frac {1+\mathbf {u} }{1-\mathbf {u} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef6ab472f7be41f463433beb82c75cfbf5053911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:37.591ex; height:7.509ex;" alt="{\displaystyle p=\left({\frac {1-\|u\|^{2}}{1+\|u\|^{2}}},{\frac {2\mathbf {u} }{1+\|u\|^{2}}}\right)={\frac {1+\mathbf {u} }{1-\mathbf {u} }}}" /></span></dd></dl> <p>where <span class="texhtml"><b>u</b> = (<i>u</i><sub>1</sub>, <i>u</i><sub>2</sub>, <i>u</i><sub>3</sub>)</span> is a vector in <span class="texhtml"><b>R</b><sup>3</sup></span> and <span class="texhtml">‖<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>u</i></span>‖<sup>2</sup> = <i>u</i><sub>1</sub><sup>2</sup> + <i>u</i><sub>2</sub><sup>2</sup> + <i>u</i><sub>3</sub><sup>2</sup></span>. In the second equality above, we have identified <span class="texhtml mvar" style="font-style:italic;">p</span> with a unit quaternion and <span class="texhtml"><b>u</b> = <i>u</i><sub>1</sub><i>i</i> + <i>u</i><sub>2</sub><i>j</i> + <i>u</i><sub>3</sub><i>k</i></span> with a pure quaternion. (Note that the numerator and denominator commute here even though quaternionic multiplication is generally noncommutative). The inverse of this map takes <span class="texhtml"><i>p</i> = (<i>x</i><sub>0</sub>, <i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, <i>x</i><sub>3</sub>)</span> in <span class="texhtml"><i>S</i><sup>3</sup></span> to </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 \mathbf {u} ={\frac {1}{1+x_{0}}}\left(x_{1},x_{2},x_{3}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ={\frac {1}{1+x_{0}}}\left(x_{1},x_{2},x_{3}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b60ff5cdd51bf46d8239ff8984471330eb46affe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.257ex; height:5.509ex;" alt="{\displaystyle \mathbf {u} ={\frac {1}{1+x_{0}}}\left(x_{1},x_{2},x_{3}\right).}" /></span></dd></dl> <p>We could just as well have projected from the point <span class="texhtml">(1, 0, 0, 0)</span>, in which case the point <span class="texhtml mvar" style="font-style:italic;">p</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 p=\left({\frac {-1+\|v\|^{2}}{1+\|v\|^{2}}},{\frac {2\mathbf {v} }{1+\|v\|^{2}}}\right)={\frac {-1+\mathbf {v} }{1+\mathbf {v} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\left({\frac {-1+\|v\|^{2}}{1+\|v\|^{2}}},{\frac {2\mathbf {v} }{1+\|v\|^{2}}}\right)={\frac {-1+\mathbf {v} }{1+\mathbf {v} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d4c73a33030e86a495389954eeaa9506fb8333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; margin-left: -0.089ex; width:40.729ex; height:7.509ex;" alt="{\displaystyle p=\left({\frac {-1+\|v\|^{2}}{1+\|v\|^{2}}},{\frac {2\mathbf {v} }{1+\|v\|^{2}}}\right)={\frac {-1+\mathbf {v} }{1+\mathbf {v} }}}" /></span></dd></dl> <p>where <span class="texhtml"><b>v</b> = (<i>v</i><sub>1</sub>, <i>v</i><sub>2</sub>, <i>v</i><sub>3</sub>)</span> is another vector in <span class="texhtml"><b>R</b><sup>3</sup></span>. The inverse of this map takes <span class="texhtml mvar" style="font-style:italic;">p</span> to </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 \mathbf {v} ={\frac {1}{1-x_{0}}}\left(x_{1},x_{2},x_{3}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\frac {1}{1-x_{0}}}\left(x_{1},x_{2},x_{3}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8516cf677b3354e3ab352f4f70cbcf951219c4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.182ex; height:5.509ex;" alt="{\displaystyle \mathbf {v} ={\frac {1}{1-x_{0}}}\left(x_{1},x_{2},x_{3}\right).}" /></span></dd></dl> <p>Note that the <span class="texhtml"><b>u</b></span> coordinates are defined everywhere but <span class="texhtml">(−1, 0, 0, 0)</span> and the <span class="texhtml"><b>v</b></span> coordinates everywhere but <span class="texhtml">(1, 0, 0, 0)</span>. This defines an <a href="/wiki/Atlas_(topology)" title="Atlas (topology)">atlas</a> on <span class="texhtml"><i>S</i><sup>3</sup></span> consisting of two <a href="/wiki/Chart_(topology)" class="mw-redirect" title="Chart (topology)">coordinate charts</a> or "patches", which together cover all of <span class="texhtml"><i>S</i><sup>3</sup></span>. Note that the transition function between these two charts on their overlap 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 \mathbf {v} ={\frac {1}{\|u\|^{2}}}\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\frac {1}{\|u\|^{2}}}\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e41a097bc4f2af4dbd5cbc03fd86e2cb87cb483f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.54ex; height:6.009ex;" alt="{\displaystyle \mathbf {v} ={\frac {1}{\|u\|^{2}}}\mathbf {u} }" /></span></dd></dl> <p>and vice versa. </p> <div class="mw-heading mw-heading2"><h2 id="Group_structure">Group structure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=13" title="Edit section: Group structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When considered as the set of <a href="/wiki/Unit_quaternion" class="mw-redirect" title="Unit quaternion">unit quaternions</a>, <span class="texhtml"><i>S</i><sup>3</sup></span> inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, <span class="texhtml"><i>S</i><sup>3</sup></span> takes on the structure of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>. Moreover, since quaternionic multiplication is <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a>, <span class="texhtml"><i>S</i><sup>3</sup></span> can be regarded as a real <a href="/wiki/Lie_group" title="Lie group">Lie group</a>. It is a <a href="/wiki/Nonabelian_group" class="mw-redirect" title="Nonabelian group">nonabelian</a>, <a href="/wiki/Compact_space" title="Compact space">compact</a> Lie group of dimension 3. When thought of as a Lie group, <span class="texhtml"><i>S</i><sup>3</sup></span> is often denoted <span class="texhtml"><a href="/wiki/Symplectic_group" title="Symplectic group">Sp(1)</a></span> or <span class="texhtml">U(1, <b>H</b>)</span>. </p><p>It turns out that the only <a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">spheres</a> that admit a Lie group structure are <span class="texhtml"><a href="/wiki/Unit_circle" title="Unit circle"><i>S</i><sup>1</sup></a></span>, thought of as the set of unit <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, and <span class="texhtml"><i>S</i><sup>3</sup></span>, the set of unit quaternions (The degenerate case <span class="texhtml"><i>S</i><sup>0</sup></span> which consists of the real numbers 1 and −1 is also a Lie group, albeit a 0-dimensional one). One might think that <span class="texhtml"><i>S</i><sup>7</sup></span>, the set of unit <a href="/wiki/Octonion" title="Octonion">octonions</a>, would form a Lie group, but this fails since octonion multiplication is <a href="/wiki/Associative" class="mw-redirect" title="Associative">nonassociative</a>. The octonionic structure does give <span class="texhtml"><i>S</i><sup>7</sup></span> one important property: <i><a href="/wiki/Parallelizability" class="mw-redirect" title="Parallelizability">parallelizability</a></i>. It turns out that the only spheres that are parallelizable are <span class="texhtml"><i>S</i><sup>1</sup></span>, <span class="texhtml"><i>S</i><sup>3</sup></span>, and <span class="texhtml"><i>S</i><sup>7</sup></span>. </p><p>By using a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> representation of the quaternions, <span class="texhtml"><b>H</b></span>, one obtains a matrix representation of <span class="texhtml"><i>S</i><sup>3</sup></span>. One convenient choice is given by the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</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}+x_{2}i+x_{3}j+x_{4}k\mapsto {\begin{pmatrix}\;\;\,x_{1}+ix_{2}&x_{3}+ix_{4}\\-x_{3}+ix_{4}&x_{1}-ix_{2}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>i</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mi>j</mi> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mi>k</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thinmathspace"></mspace> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}+x_{2}i+x_{3}j+x_{4}k\mapsto {\begin{pmatrix}\;\;\,x_{1}+ix_{2}&x_{3}+ix_{4}\\-x_{3}+ix_{4}&x_{1}-ix_{2}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c80fab95b9a3f58847e9e6799e74f268c5cfa4f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.414ex; height:6.176ex;" alt="{\displaystyle x_{1}+x_{2}i+x_{3}j+x_{4}k\mapsto {\begin{pmatrix}\;\;\,x_{1}+ix_{2}&x_{3}+ix_{4}\\-x_{3}+ix_{4}&x_{1}-ix_{2}\end{pmatrix}}.}" /></span></dd></dl> <p>This map gives an <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a> <a href="/wiki/Algebra_homomorphism" class="mw-redirect" title="Algebra homomorphism">algebra homomorphism</a> from <span class="texhtml"><b>H</b></span> to the set of 2 × 2 complex matrices. It has the property that the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of a quaternion <span class="texhtml mvar" style="font-style:italic;">q</span> is equal to the <a href="/wiki/Square_root" title="Square root">square root</a> of the <a href="/wiki/Determinant" title="Determinant">determinant</a> of the matrix image of <span class="texhtml mvar" style="font-style:italic;">q</span>. </p><p>The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a> <span class="texhtml">SU(2)</span>. Thus, <span class="texhtml"><i>S</i><sup>3</sup></span> as a Lie group is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to <span class="texhtml">SU(2)</span>. </p><p>Using our Hopf coordinates <span class="texhtml">(<i>η</i>, <i>ξ</i><sub>1</sub>, <i>ξ</i><sub>2</sub>)</span> we can then write any element of <span class="texhtml">SU(2)</span> in the form </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 {\begin{pmatrix}e^{i\,\xi _{1}}\sin \eta &e^{i\,\xi _{2}}\cos \eta \\-e^{-i\,\xi _{2}}\cos \eta &e^{-i\,\xi _{1}}\sin \eta \end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mspace width="thinmathspace"></mspace> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>η</mi> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mspace width="thinmathspace"></mspace> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mi>η</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace"></mspace> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mi>η</mi> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mspace width="thinmathspace"></mspace> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>η</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}e^{i\,\xi _{1}}\sin \eta &e^{i\,\xi _{2}}\cos \eta \\-e^{-i\,\xi _{2}}\cos \eta &e^{-i\,\xi _{1}}\sin \eta \end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163bcbe00c81e623233ec44ea8ee8b4f6eac6b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.005ex; height:6.509ex;" alt="{\displaystyle {\begin{pmatrix}e^{i\,\xi _{1}}\sin \eta &e^{i\,\xi _{2}}\cos \eta \\-e^{-i\,\xi _{2}}\cos \eta &e^{-i\,\xi _{1}}\sin \eta \end{pmatrix}}.}" /></span></dd></dl> <p>Another way to state this result is if we express the matrix representation of an element of <span class="texhtml">SU(2)</span> as an exponential of a linear combination of the Pauli matrices. It is seen that an arbitrary element <span class="texhtml"><i>U</i> ∈ SU(2)</span> can be written 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 U=\exp \left(\sum _{i=1}^{3}\alpha _{i}J_{i}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\exp \left(\sum _{i=1}^{3}\alpha _{i}J_{i}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8359b50c8fb66e415c91cb25f617d03bfd843294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.881ex; height:7.509ex;" alt="{\displaystyle U=\exp \left(\sum _{i=1}^{3}\alpha _{i}J_{i}\right).}" /></span><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></dd></dl> <p>The condition that the determinant of <span class="texhtml mvar" style="font-style:italic;">U</span> is +1 implies that the coefficients <span class="texhtml"><i>α</i><sub>1</sub></span> are constrained to lie on a 3-sphere. </p> <div class="mw-heading mw-heading2"><h2 id="In_literature">In literature</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=14" title="Edit section: In literature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Edwin_Abbott_Abbott" title="Edwin Abbott Abbott">Edwin Abbott Abbott</a>'s <i><a href="/wiki/Flatland" title="Flatland">Flatland</a></i>, published in 1884, and in <i><a href="/wiki/Sphereland" title="Sphereland">Sphereland</a></i>, a 1965 sequel to <i>Flatland</i> by <a href="/wiki/Dionys_Burger" title="Dionys Burger">Dionys Burger</a>, the 3-sphere is referred to as an <b>oversphere</b>, and a 4-sphere is referred to as a <b>hypersphere</b>. </p><p>Writing in the <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a></i>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Mark A. Peterson describes three different ways of visualizing 3-spheres and points out language in <i><a href="/wiki/The_Divine_Comedy" class="mw-redirect" title="The Divine Comedy">The Divine Comedy</a></i> that suggests <a href="/wiki/Dante_Alighieri" title="Dante Alighieri">Dante</a> viewed the Universe in the same way; <a href="/wiki/Carlo_Rovelli" title="Carlo Rovelli">Carlo Rovelli</a> supports the same idea.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In <i>Art Meets Mathematics in the Fourth Dimension</i>,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Stephen L. Lipscomb develops the concept of the hypersphere dimensions as it relates to art, architecture, and mathematics. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Unit_circle" title="Unit circle">1-sphere</a>, <a href="/wiki/Sphere" title="Sphere">2-sphere</a>, <a href="/wiki/N-sphere" title="N-sphere"><i>n</i>-sphere</a></li> <li><a href="/wiki/Tesseract" title="Tesseract">tesseract</a>, <a href="/wiki/Polychoron" class="mw-redirect" title="Polychoron">polychoron</a>, <a href="/wiki/Simplex" title="Simplex">simplex</a></li> <li><a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a></li> <li><a href="/wiki/Hopf_bundle" class="mw-redirect" title="Hopf bundle">Hopf bundle</a>, <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a></li> <li><a href="/wiki/Poincar%C3%A9_homology_sphere" class="mw-redirect" title="Poincaré homology sphere">Poincaré sphere</a></li> <li><a href="/wiki/Reeb_foliation" title="Reeb foliation">Reeb foliation</a></li> <li><a href="/wiki/Clifford_torus" title="Clifford torus">Clifford torus</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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="#cite_ref-1">^</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="CITEREFLemaître1948" class="citation journal cs1"><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître, Georges</a> (1948). "Quaternions et espace elliptique". <i>Acta</i>. <b>12</b>. <a href="/wiki/Pontifical_Academy_of_Sciences" title="Pontifical Academy of Sciences">Pontifical Academy of Sciences</a>: <span class="nowrap">57–</span>78.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta&rft.atitle=Quaternions+et+espace+elliptique&rft.volume=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E57-%3C%2Fspan%3E78&rft.date=1948&rft.aulast=Lema%C3%AEtre&rft.aufirst=Georges&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLandauLifshitz1988" class="citation book cs1"><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, Lev D.</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, Evgeny M.</a> (1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=X18PF4oKyrUC"><i>Classical Theory of Fields</i></a>. <a href="/wiki/Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Course of Theoretical Physics</a>. Vol. 2 (7th ed.). Moscow: <a href="/wiki/Nauka_(publisher)" title="Nauka (publisher)">Nauka</a>. p. 385. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-5-02-014420-0" title="Special:BookSources/978-5-02-014420-0"><bdi>978-5-02-014420-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Theory+of+Fields&rft.place=Moscow&rft.series=Course+of+Theoretical+Physics&rft.pages=385&rft.edition=7th&rft.pub=Nauka&rft.date=1988&rft.isbn=978-5-02-014420-0&rft.aulast=Landau&rft.aufirst=Lev+D.&rft.au=Lifshitz%2C+Evgeny+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DX18PF4oKyrUC&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span> </span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBanchoff" class="citation web cs1">Banchoff, Thomas. <a rel="nofollow" class="external text" href="http://www.geom.uiuc.edu/~banchoff/script/b3d/hypertorus.html">"The Flat Torus in the Three-Sphere"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Flat+Torus+in+the+Three-Sphere&rft.aulast=Banchoff&rft.aufirst=Thomas&rft_id=http%3A%2F%2Fwww.geom.uiuc.edu%2F~banchoff%2Fscript%2Fb3d%2Fhypertorus.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchwichtenberg2015" class="citation book cs1">Schwichtenberg, Jakob (2015). <i>Physics from symmetry</i>. Cham: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-19201-7" title="Special:BookSources/978-3-319-19201-7"><bdi>978-3-319-19201-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/910917227">910917227</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+from+symmetry&rft.place=Cham&rft.pub=Springer&rft.date=2015&rft_id=info%3Aoclcnum%2F910917227&rft.isbn=978-3-319-19201-7&rft.aulast=Schwichtenberg&rft.aufirst=Jakob&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPeterson1979" class="citation journal cs1">Peterson, Mark A. (1979). <a rel="nofollow" class="external text" href="https://archive.today/20130223083042/http://link.aip.org/link/ajpias/v47/i12/p1031/s1">"Dante and the 3-sphere"</a>. <i>American Journal of Physics</i>. <b>47</b> (12): <span class="nowrap">1031–</span>1035. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979AmJPh..47.1031P">1979AmJPh..47.1031P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.11968">10.1119/1.11968</a>. Archived from <a rel="nofollow" class="external text" href="http://link.aip.org/link/ajpias/v47/i12/p1031/s1">the original</a> on 23 February 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Dante+and+the+3-sphere&rft.volume=47&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1031-%3C%2Fspan%3E1035&rft.date=1979&rft_id=info%3Adoi%2F10.1119%2F1.11968&rft_id=info%3Abibcode%2F1979AmJPh..47.1031P&rft.aulast=Peterson&rft.aufirst=Mark+A.&rft_id=http%3A%2F%2Flink.aip.org%2Flink%2Fajpias%2Fv47%2Fi12%2Fp1031%2Fs1&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRovelli2021" class="citation book cs1">Rovelli, Carlo (9 September 2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=000_EAAAQBAJ&pg=PA40"><i>General Relativity: The Essentials</i></a>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-00-901369-7" title="Special:BookSources/978-1-00-901369-7"><bdi>978-1-00-901369-7</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">13 September</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Relativity%3A+The+Essentials&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2021-09-09&rft.isbn=978-1-00-901369-7&rft.aulast=Rovelli&rft.aufirst=Carlo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D000_EAAAQBAJ%26pg%3DPA40&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLipscomb2014" class="citation book cs1">Lipscomb, Stephen (2014). <a rel="nofollow" class="external text" href="https://www.springer.com/gp/book/9783319062532"><i>Art meets mathematics in the fourth dimension</i></a> (2 ed.). Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-06254-9" title="Special:BookSources/978-3-319-06254-9"><bdi>978-3-319-06254-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/893872366">893872366</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Art+meets+mathematics+in+the+fourth+dimension&rft.place=Berlin&rft.edition=2&rft.pub=Springer&rft.date=2014&rft_id=info%3Aoclcnum%2F893872366&rft.isbn=978-3-319-06254-9&rft.aulast=Lipscomb&rft.aufirst=Stephen&rft_id=https%3A%2F%2Fwww.springer.com%2Fgp%2Fbook%2F9783319062532&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Further_reading">Further reading</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=17" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHenderson2001" class="citation book cs1"><a href="/wiki/David_W._Henderson" title="David W. Henderson">Henderson, David W.</a> (2001). "Chapter 20: 3-spheres and hyperbolic 3-spaces". <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180619070158/http://pi.math.cornell.edu/~henderson/books/eg00/"><i>Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces</i></a> (second ed.). Prentice-Hall. Archived from <a rel="nofollow" class="external text" href="http://www.math.cornell.edu/~henderson/books/eg00">the original</a> on 2018-06-19.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+20%3A+3-spheres+and+hyperbolic+3-spaces&rft.btitle=Experiencing+Geometry%3A+In+Euclidean%2C+Spherical%2C+and+Hyperbolic+Spaces&rft.edition=second&rft.pub=Prentice-Hall&rft.date=2001&rft.aulast=Henderson&rft.aufirst=David+W.&rft_id=http%3A%2F%2Fwww.math.cornell.edu%2F~henderson%2Fbooks%2Feg00&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeeks1985" class="citation book cs1"><a href="/wiki/Jeffrey_Weeks_(mathematician)" title="Jeffrey Weeks (mathematician)">Weeks, Jeffrey R.</a> (1985). "Chapter 14: The Hypersphere". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Lurp6nB4LtQC)"><i>The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds</i></a>. <q>A Warning on terminology: Our two-sphere is defined in three-dimensional space, where it is the boundary of a three-dimensional ball. This terminology is standard among mathematicians, but not among physicists. So don't be surprised if you find people calling the two-sphere a three-sphere.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+14%3A+The+Hypersphere&rft.btitle=The+Shape+of+Space%3A+How+to+Visualize+Surfaces+and+Three-dimensional+Manifolds&rft.date=1985&rft.aulast=Weeks&rft.aufirst=Jeffrey+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLurp6nB4LtQC%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZamboj2021" class="citation journal cs1">Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". <i>Journal of Computational Design and Engineering</i>. <b>8</b> (3): <span class="nowrap">836–</span>854. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2003.09236v2">2003.09236v2</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fjcde%2Fqwab018">10.1093/jcde/qwab018</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Design+and+Engineering&rft.atitle=Synthetic+construction+of+the+Hopf+fibration+in+a+double+orthogonal+projection+of+4-space&rft.volume=8&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E836-%3C%2Fspan%3E854&rft.date=2021-01-08&rft_id=info%3Aarxiv%2F2003.09236v2&rft_id=info%3Adoi%2F10.1093%2Fjcde%2Fqwab018&rft.aulast=Zamboj&rft.aufirst=Michal&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=3-sphere&action=edit&section=18" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Hypersphere"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Hypersphere.html">"Hypersphere"</a>. <i><a href="/wiki/MathWorld" 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=Hypersphere&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FHypersphere.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A3-sphere" class="Z3988"></span></span> <i>Note</i>: This article uses the alternate naming scheme for spheres in which a sphere in <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional space is termed an <span class="texhtml mvar" style="font-style:italic;">n</span>-sphere.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=3-sphere&oldid=1268549529">https://en.wikipedia.org/w/index.php?title=3-sphere&oldid=1268549529</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Four-dimensional_geometry" title="Category:Four-dimensional geometry">Four-dimensional geometry</a></li><li><a href="/wiki/Category:Algebraic_topology" title="Category:Algebraic topology">Algebraic topology</a></li><li><a href="/wiki/Category:Geometric_topology" title="Category:Geometric topology">Geometric topology</a></li><li><a href="/wiki/Category:Analytic_geometry" title="Category:Analytic geometry">Analytic geometry</a></li><li><a href="/wiki/Category:Quaternions" title="Category:Quaternions">Quaternions</a></li><li><a href="/wiki/Category:Spheres" title="Category:Spheres">Spheres</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_June_2016" title="Category:Articles lacking in-text citations from June 2016">Articles lacking in-text citations from June 2016</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 10 January 2025, at 09:15<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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