CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> Euclidean space in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Euclidean space </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/14090/#Item_2" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Euclidean spaces</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="analysis">Analysis</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/analysis">analysis</a></strong> (<a class="existingWikiWord" href="/nlab/show/differential+calculus">differential</a>/<a class="existingWikiWord" href="/nlab/show/integral+calculus">integral</a> <a class="existingWikiWord" href="/nlab/show/calculus">calculus</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a>, <a class="existingWikiWord" href="/nlab/show/topology">topology</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/epsilontic+analysis">epsilontic analysis</a></p> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+analysis">infinitesimal analysis</a></p> <p><a class="existingWikiWord" href="/nlab/show/computable+analysis">computable analysis</a></p> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">Introduction</a></em></p> <h2 id="basic_concepts">Basic concepts</h2> <p><a class="existingWikiWord" href="/nlab/show/triangle+inequality">triangle inequality</a></p> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/normed+vector+space">normed vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>, <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> <p><a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a></p> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a>, <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit of a sequence</a></p> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compactness</a>, <a class="existingWikiWord" href="/nlab/show/sequentially+compact+space">sequential compactness</a></p> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/integration">integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> <h2 id="basic_facts">Basic facts</h2> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> <p>…</p> <h2 id="theorems">Theorems</h2> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> <p>…</p> </div></div> <h4 id="geometry">Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong> / <strong><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </div></div> <h4 id="trigonometry">Trigonometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/trigonometry">trigonometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangle">triangle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/angle">angle</a>, <a class="existingWikiWord" href="/nlab/show/arc+length">arc length</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trigonometric+function">trigonometric function</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cosine">cosine</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sine">sine</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+function">tangent</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cotangent">cotangent</a></p> </li> <li> <p><span class="newWikiWord">secant<a href="/nlab/new/secant">?</a></span></p> </li> <li> <p><span class="newWikiWord">cosecant<a href="/nlab/new/cosecant">?</a></span></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverse+trigonometric+function">inverse trigonometric function</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arccos">arccos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arcsin">arcsin</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arctan">arctan</a></p> </li> <li> <p>…</p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trigonometric+identity">trigonometric identity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Euler%27s+formula">Euler's formula</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperbolic+function">hyperbolic function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+function">elliptic function</a></p> </li> </ul> </div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="euclidean_spaces">Euclidean spaces</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#remarks_on_terminology'>Remarks on terminology</a></li> <li><a href='#euclidean_spaces_with_infinitesimals'>Euclidean spaces with infinitesimals</a></li> <li><a href='#in_constructive_mathematics'>In constructive mathematics</a></li> <ul> <li><a href='#in_predicative_constructive_mathematics'>In predicative constructive mathematics</a></li> </ul> <li><a href='#lengths_and_angles'>Lengths and angles</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The concept of <em>Euclidean space</em> in <a class="existingWikiWord" href="/nlab/show/analysis">analysis</a>, <a class="existingWikiWord" href="/nlab/show/topology">topology</a>, <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> and specifically <em><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a></em>, and <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is a fomalization in modern terms of the <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> studied in <a href="#Euclid300BC">Euclid 300BC</a>, equipped with the <a class="existingWikiWord" href="/nlab/show/extra+structure">structures</a> that Euclid recognised his spaces as having.</p> <p>In the strict sense of the word, Euclidean space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">E^n</annotation></semantics></math> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is, up to <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> whose underlying <a class="existingWikiWord" href="/nlab/show/set">set</a> is the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> and whose <a class="existingWikiWord" href="/nlab/show/distance">distance</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> is given by the <a class="existingWikiWord" href="/nlab/show/Euclidean+norm">Euclidean norm</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>Eucl</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">‖</mo></mrow><mo>=</mo><msqrt><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>y</mi> <mi>i</mi></msub><mo>−</mo><msub><mi>x</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow></msqrt><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_{Eucl}(x,y) \coloneqq {\Vert x-y\Vert} = \sqrt{ \sum_{i = 1}^n (y_i - x_i)^2 } \,. </annotation></semantics></math></div> <p>In <a href="#Euclid300BC">Euclid 300BC</a> this is considered for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math>; and it is considered not in terms of <a class="existingWikiWord" href="/nlab/show/coordinate+functions">coordinate functions</a> as above, but via <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> of <em><a class="existingWikiWord" href="/nlab/show/synthetic+geometry">synthetic geometry</a></em>.</p> <p>This means that in a Euclidean space one may construct for instance the <a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a> around any point, or the shortest <a class="existingWikiWord" href="/nlab/show/curve">curve</a> connecting any two points. These are the operations studied in (<a href="#Euclid300BC">Euclid 300BC</a>), see at <em><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a></em>.</p> <p>Of course these operations may be considered in <em>every</em> (other) <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, too, see at <em><a class="existingWikiWord" href="/nlab/show/non-Euclidean+geometry">non-Euclidean geometry</a></em>. <a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a> is distinguished notably from <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a> or <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a> by the fact that it satisfies the <a class="existingWikiWord" href="/nlab/show/parallel+postulate">parallel postulate</a>.</p> <p>In regarding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mi>n</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><msub><mi>d</mi> <mi>Eucl</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^n = (\mathbb{R}^n, d_{Eucl})</annotation></semantics></math> (only) as a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, some <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a> still carried by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is disregarded, such as its <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> structure, hence its <a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a> structure and its canonical <a class="existingWikiWord" href="/nlab/show/inner+product+space">inner product space</a> structure. Sometimes “Euclidean space” is used to refer to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">E^n</annotation></semantics></math> with that further extra structure remembered, which might then be called <em><a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a></em>.</p> <p>Retaining the <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> on top of the <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> structure means that on top of <a class="existingWikiWord" href="/nlab/show/distances">distances</a> one may also speak of <a class="existingWikiWord" href="/nlab/show/angles">angles</a> in a Euclidean space.</p> <p>Then of course <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> carries also non-canonical <a class="existingWikiWord" href="/nlab/show/inner+product+space">inner product space</a> structures, not corresponding to the <a class="existingWikiWord" href="/nlab/show/Euclidean+norm">Euclidean norm</a>. Regarding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">E^n</annotation></semantics></math> as equipped with these one says that it is a <strong>pseudo-Euclidean space</strong>. These are now, again in the sense of <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>, the local model spaces for <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo-Riemannian geometry</a>.</p> <p>Finally one could generalize and allow the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> to be countably infinite, and regard separable <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a> as generalized Euclidean spaces.</p> <h2 id="remarks_on_terminology">Remarks on terminology</h2> <p>Arguably, the spaces studied by Euclid were not really modelled on inner product spaces, as the distances were lengths, not <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> (which, if non-negative, are <em>ratios</em> of lengths). So we should say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> has an inner product valued in some <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/line">line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> (or rather, in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">L^2</annotation></semantics></math>). Of course, Euclid did not use the inner product (which takes negative values) directly, but today we can recover it from what Euclid did discuss: lengths (valued in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>) and angles (dimensionless).</p> <p>Since the days of <a class="existingWikiWord" href="/nlab/show/Ren%C3%A9+Descartes">René Descartes</a>, it is common to identify a Euclidean space with a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>, that is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> the dimension. But Euclid's spaces had no coordinates; and in any case, what we do with them is still coordinate-independent.</p> <h2 id="euclidean_spaces_with_infinitesimals">Euclidean spaces with infinitesimals</h2> <p>Instead of working in the real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> real vector spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, one could instead work in a <a class="existingWikiWord" href="/nlab/show/Archimedean+ordered+Artinian+local+ring">Archimedean ordered Artinian local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>ℝ</mi> </mrow> <annotation encoding="application/x-tex">\mathbb{R}</annotation> </semantics> </math>-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/rank">rank</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/infinitesimals">infinitesimals</a>, and so the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> have <a class="existingWikiWord" href="/nlab/show/infinitesimals">infinitesimals</a> as well. Nevertheless, it is still possible to define the Euclidean distance function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>; the only difference is that the distance function is a <a class="existingWikiWord" href="/nlab/show/pseudometric">pseudometric</a> rather than a <a class="existingWikiWord" href="/nlab/show/metric">metric</a> here.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, the quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> by its <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> of non-invertible elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> itself, and the canonical function used in defining the <a class="existingWikiWord" href="/nlab/show/quotient+ring">quotient ring</a> is the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\Re:A \to \mathbb{R}</annotation></semantics></math> which takes a number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math> to its purely real component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\Re(a) \in \mathbb{R}</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an ordered <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-algebra, there is a <a class="existingWikiWord" href="/nlab/show/strictly+monotone">strictly monotone</a> <a class="existingWikiWord" href="/nlab/show/ring+homomorphism">ring homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">h:\mathbb{R} \to A</annotation></semantics></math>.</p> <p>The real numbers have <a class="existingWikiWord" href="/nlab/show/lattice">lattice</a> structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>min</mi><mo>:</mo><mi>ℝ</mi><mo>×</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\min:\mathbb{R} \times \mathbb{R} \to \mathbb{R}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>max</mi><mo>:</mo><mi>ℝ</mi><mo>×</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\max:\mathbb{R} \times \mathbb{R} \to \mathbb{R}</annotation></semantics></math>. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has a distance function given by the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>A</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\rho:A \times A \to \mathbb{R}</annotation></semantics></math>, defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>max</mi><mo stretchy="false">(</mo><mi>ℜ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℜ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mi>min</mi><mo stretchy="false">(</mo><mi>ℜ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℜ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(a, b) \coloneqq \max(\Re(a), \Re(b)) - \min(\Re(a), \Re(b))</annotation></semantics></math></div> <p>as well as an <a class="existingWikiWord" href="/nlab/show/absolute+value">absolute value</a> given by the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>:</mo><mi>A</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\vert-\vert:A \to \mathbb{R}</annotation></semantics></math>, defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>a</mi><mo stretchy="false">|</mo><mo>≔</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\vert a \vert \coloneqq \rho(a, 0)</annotation></semantics></math></div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>min</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>max</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\min(a, b) \leq \max(a, b)</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/pseudometric">pseudometric</a> and multiplicative <a class="existingWikiWord" href="/nlab/show/seminorm">seminorm</a> are always non-negative. In addition, by definition, the pseudometric takes any two elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">b \in A</annotation></semantics></math> whose difference <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>−</mo><mi>b</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">a - b \in I</annotation></semantics></math> is an infinitesimal to zero <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\rho(a, b) = 0</annotation></semantics></math>.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/Euclidean+field">Euclidean field</a>, it has a <a class="existingWikiWord" href="/nlab/show/metric+square+root+function">metric square root function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mo lspace="verythinmathspace" rspace="0em">−</mo></msqrt><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sqrt{-}:[0, \infty) \to [0, \infty)</annotation></semantics></math>. Every <a class="existingWikiWord" href="/nlab/show/rank">rank</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> with basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>:</mo><mi mathvariant="normal">Fin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v:\mathrm{Fin}(n) \to V</annotation></semantics></math> thus has a <strong>Euclidean pseudometric</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>V</mi></msub><mo>:</mo><mi>V</mi><mo>×</mo><mi>V</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\rho_V:V \times V \to K</annotation></semantics></math> defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>V</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><msqrt><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>∈</mo><mi mathvariant="normal">Fin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></munder><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>b</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\rho_V(a, b) \coloneqq \sqrt{\sum_{i \in \mathrm{Fin}(n)} \rho(a_i, b_i)^2}</annotation></semantics></math></div> <p>for module elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">a \in V</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">b \in V</annotation></semantics></math> and scalars <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a_i \in A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mi>i</mi></msub><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">b_i \in A</annotation></semantics></math> for index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="normal">Fin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i \in \mathrm{Fin}(n)</annotation></semantics></math>, where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>∈</mo><mi mathvariant="normal">Fin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></munder><msub><mi>a</mi> <mi>i</mi></msub><msub><mi>v</mi> <mi>i</mi></msub><mspace width="1em"></mspace><mi>b</mi><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>∈</mo><mi mathvariant="normal">Fin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></munder><msub><mi>b</mi> <mi>i</mi></msub><msub><mi>v</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a = \sum_{i \in \mathrm{Fin}(n)} a_i v_i \quad b = \sum_{i \in \mathrm{Fin}(n)} b_i v_i</annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/ordered+field">ordered field</a>, then this reduces down to the Euclidean metric defined above.</p> <h2 id="in_constructive_mathematics">In constructive mathematics</h2> <p>In <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> used to define Euclidean spaces are the <a class="existingWikiWord" href="/nlab/show/Dedekind+real+numbers">Dedekind real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℝ</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{R}_{D}</annotation></semantics></math>, as those are the only ones that are <a class="existingWikiWord" href="/nlab/show/Dedekind+complete">Dedekind complete</a>, in the sense of not having any gaps in the <a class="existingWikiWord" href="/nlab/show/dense+linear+order">dense linear order</a>. The Dedekind real numbers are also the real numbers that are geometrically contractible: whose <a class="existingWikiWord" href="/nlab/show/shape">shape</a> is <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopically</a> <a class="existingWikiWord" href="/nlab/show/contractible">contractible</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo stretchy="false">(</mo><msub><mi>ℝ</mi> <mi>D</mi></msub><mo stretchy="false">)</mo><mo>≅</mo><mi>𝟙</mi></mrow><annotation encoding="application/x-tex">\esh(\mathbb{R}_D) \cong \mathbb{1}</annotation></semantics></math>.</p> <h3 id="in_predicative_constructive_mathematics">In predicative constructive mathematics</h3> <p>In <a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative</a> constructive mathematics, the <a class="existingWikiWord" href="/nlab/show/Dedekind+real+numbers">Dedekind real numbers</a> are defined relative to a universe <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding="application/x-tex">\mathcal{U}</annotation></semantics></math>, and thus there are many different such Dedekind real numbers that could be used to define Euclidean spaces, one <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℝ</mi> <mi>𝒰</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{R}_\mathcal{U}</annotation></semantics></math> for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding="application/x-tex">\mathcal{U}</annotation></semantics></math>. However, each set of Dedekind real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℝ</mi> <mi>𝒰</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{R}_\mathcal{U}</annotation></semantics></math> would be large relative to the sets in the universe <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding="application/x-tex">\mathcal{U}</annotation></semantics></math>.</p> <p>If the predicative constructive foundations does not have universes, then there doesn’t exist any <a class="existingWikiWord" href="/nlab/show/dense+linear+order">dense linear order</a> that is actually <a class="existingWikiWord" href="/nlab/show/Dedekind+complete">Dedekind complete</a> in the usual sense, and so the usual definition of Euclidean space does not work. Some mathematicians have proposed to use <a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> <a class="existingWikiWord" href="/nlab/show/sigma-frame"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>σ</mi> </mrow> <annotation encoding="application/x-tex">\sigma</annotation> </semantics> </math>-frame</a>, for defining the real numbers, in place of the large set of all <a class="existingWikiWord" href="/nlab/show/propositions">propositions</a> in a universe <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">Prop</mi> <mi>𝒰</mi></msub></mrow><annotation encoding="application/x-tex">\mathrm{Prop}_\mathcal{U}</annotation></semantics></math>, but the real numbers in that case are only <a class="existingWikiWord" href="/nlab/show/sigma+Dedekind+complete"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>Σ</mi> </mrow> <annotation encoding="application/x-tex">\Sigma</annotation> </semantics> </math>-Dedekind complete</a>, which is a weaker condition than being <a class="existingWikiWord" href="/nlab/show/Dedekind+complete">Dedekind complete</a>. Furthermore, Lešnik showed that for any two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-frames <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mo>′</mo></msup></mrow><annotation encoding="application/x-tex">\Sigma^{'}</annotation></semantics></math> that embed into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">Prop</mi> <mi>𝒰</mi></msub></mrow><annotation encoding="application/x-tex">\mathrm{Prop}_\mathcal{U}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>⊆</mo><msup><mi>Σ</mi> <mo>′</mo></msup></mrow><annotation encoding="application/x-tex">\Sigma \subseteq \Sigma^{'}</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>-Dedekind completion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mo>′</mo></msup></mrow><annotation encoding="application/x-tex">\Sigma^{'}</annotation></semantics></math>-Dedekind completion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊆</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \subseteq B</annotation></semantics></math>, so the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>-Dedekind real numbers are not complete.</p> <h2 id="lengths_and_angles">Lengths and angles</h2> <p>Given two points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> of a Euclidean space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, their difference <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x - y</annotation></semantics></math> belongs to the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, where it has a norm</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">‖</mo></mrow><mo>=</mo><msqrt><mrow><mo stretchy="false">⟨</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo stretchy="false">⟩</mo></mrow></msqrt><mo>.</mo></mrow><annotation encoding="application/x-tex"> {\|x - y\|} = \sqrt{\langle{x - y, x - y}\rangle} .</annotation></semantics></math></div> <p>This real number (or properly, element of the line <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>) is the <strong>distance</strong> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, or the <strong>length</strong> of the line segment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>x</mi><mi>y</mi></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{x y}</annotation></semantics></math>. This distance function makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> into an (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>-valued) <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>.</p> <p>Given three points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">x, y, z</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>≠</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">x, y \ne z</annotation></semantics></math> (so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo>−</mo><mi>z</mi><mo stretchy="false">‖</mo></mrow><mo>,</mo><mrow><mo stretchy="false">‖</mo><mi>y</mi><mo>−</mo><mi>z</mi><mo stretchy="false">‖</mo></mrow><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">{\|x - z\|}, {\|y - z\|} \ne 0</annotation></semantics></math>), we can form the ratio</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mo stretchy="false">⟨</mo><mrow><mi>x</mi><mo>−</mo><mi>z</mi><mo>,</mo><mi>y</mi><mo>−</mo><mi>z</mi></mrow><mo stretchy="false">⟩</mo></mrow><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo>−</mo><mi>z</mi><mo stretchy="false">‖</mo></mrow><mrow><mo stretchy="false">‖</mo><mi>y</mi><mo>−</mo><mi>z</mi><mo stretchy="false">‖</mo></mrow></mrow></mfrac><mo>,</mo></mrow><annotation encoding="application/x-tex"> \frac{\langle{x - z, y - z}\rangle}{{\|x - z\|} {\|y - z\|}} ,</annotation></semantics></math></div> <p>which is a (dimensionless) real number. By the Cauchy–Schwartz inequality, this number lies between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>, so it's the <a class="existingWikiWord" href="/nlab/show/cosine">cosine</a> of a unique angle measure between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> radians. This is the measure of the <strong>angle</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∠</mo><mi>x</mi><mi>z</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">\angle x z y</annotation></semantics></math>. In a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-dimensional Euclidean space, we can interpret <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∠</mo><mi>x</mi><mi>z</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">\angle x z y</annotation></semantics></math> as a signed angle (so taking values anywhere on the <a class="existingWikiWord" href="/nlab/show/unit+circle">unit circle</a>) if we fix an <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> <p>Conversely, knowing angles and lengths, we may recover the inner product on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>;</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mrow><mi>x</mi><mo>−</mo><mi>z</mi><mo>,</mo><mi>y</mi><mo>−</mo><mi>z</mi></mrow><mo stretchy="false">⟩</mo><mo>=</mo><mrow><mo stretchy="false">‖</mo><mover><mrow><mi>x</mi><mi>z</mi></mrow><mo>¯</mo></mover><mo stretchy="false">‖</mo></mrow><mrow><mo stretchy="false">‖</mo><mover><mrow><mi>y</mi><mi>z</mi></mrow><mo>¯</mo></mover><mo stretchy="false">‖</mo></mrow><mi>cos</mi><mo>∠</mo><mi>x</mi><mi>z</mi><mi>y</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> \langle{x - z, y - z}\rangle = {\|\overline{x z}\|} {\|\overline{y z}\|} \cos \angle x z y ,</annotation></semantics></math></div> <p>and other inner products are recovered by linearity. (We must then use the axioms of <a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a> to prove that this is well defined and actually an inner product.) It’s actually possible to recover the inner product and angles from lengths alone; this is discussed at <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+field">Euclidean field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+G-space">Euclidean G-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super-Euclidean+space">super-Euclidean space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+geometry">synthetic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+algebra">geometric algebra</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="Euclid300BC"><a class="existingWikiWord" href="/nlab/show/Euclid">Euclid</a>, <em><a class="existingWikiWord" href="/nlab/show/Elements">Elements</a></em>, 300BC</li> </ul> <p>Textbook accounts:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Igor+R.+Shafarevich">Igor R. Shafarevich</a>, <a class="existingWikiWord" href="/nlab/show/Alexey+O.+Remizov">Alexey O. Remizov</a>: §7 in: <em>Linear Algebra and Geometry</em> (2012) [<a href="https://doi.org/10.1007/978-3-642-30994-6">doi:10.1007/978-3-642-30994-6</a>, <a href="https://maa.org/press/maa-reviews/linear-algebra-and-geometry">MAA-review</a>]</li> </ul> <p>On the use of the Dedekind real numbers in constructive and predicative constructive mathematics, such as for Euclidean spaces:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, <a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+%E2%80%93+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a> (2013)</li> <li><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, Mathematical Structures in Computer Science Vol 28 (6) (2018): 856-941 (<a href="https://arxiv.org/abs/1509.07584">arXiv:1509.07584</a>, <a href="https://doi.org/10.1017/S0960129517000147">doi:10.1017/S0960129517000147</a>)</li> <li>Davorin Lešnik, Synthetic Topology and Constructive Metric Spaces, (<a href="https://arxiv.org/abs/2104.10399">arxiv:2104.10399</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 5, 2023 at 15:03:54. See the <a href="/nlab/history/Euclidean+space" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Euclidean+space" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/14090/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/Euclidean+space/18" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Euclidean+space" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Euclidean+space" accesskey="S" class="navlink" id="history" rel="nofollow">History (18 revisions)</a> <a href="/nlab/show/Euclidean+space/cite" style="color: black">Cite</a> <a href="/nlab/print/Euclidean+space" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Euclidean+space" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>