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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arc_length"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Arc length</span> </div> </a> <ul id="toc-Arc_length-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triangle_congruence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Triangle_congruence"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Triangle congruence</span> </div> </a> <ul id="toc-Triangle_congruence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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 Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Compressed_sensing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compressed_sensing"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Compressed sensing</span> </div> </a> <ul id="toc-Compressed_sensing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differences_of_frequency_distributions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differences_of_frequency_distributions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Differences of frequency distributions</span> </div> </a> <ul id="toc-Differences_of_frequency_distributions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links_2" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" 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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. Available in 24 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-24" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">24 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Manhattan-metriek" title="Manhattan-metriek – Afrikaans" lang="af" hreflang="af" data-title="Manhattan-metriek" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%87%D9%86%D8%AF%D8%B3%D8%A9_%D8%B3%D9%8A%D8%A7%D8%B1%D8%A9_%D8%A7%D9%84%D8%A3%D8%AC%D8%B1%D8%A9" title="هندسة سيارة الأجرة – Arabic" lang="ar" hreflang="ar" data-title="هندسة سيارة الأجرة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Geometria_del_taxista" title="Geometria del taxista – Catalan" lang="ca" hreflang="ca" data-title="Geometria del taxista" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Manhattansk%C3%A1_metrika" title="Manhattanská metrika – Czech" lang="cs" hreflang="cs" data-title="Manhattanská metrika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Manhattan-Metrik" title="Manhattan-Metrik – German" lang="de" hreflang="de" data-title="Manhattan-Metrik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Geometr%C3%ADa_del_taxista" title="Geometría del taxista – Spanish" lang="es" hreflang="es" data-title="Geometría del taxista" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87_%D9%85%D9%86%D9%87%D8%AA%D9%86" title="فاصله منهتن – Persian" lang="fa" hreflang="fa" data-title="فاصله منهتن" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Distance_de_Manhattan" title="Distance de Manhattan – French" lang="fr" hreflang="fr" data-title="Distance de Manhattan" 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/%EB%A7%A8%ED%95%B4%ED%8A%BC_%EA%B1%B0%EB%A6%AC" title="맨해튼 거리 – Korean" lang="ko" hreflang="ko" data-title="맨해튼 거리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Jarak_Manhattan" title="Jarak Manhattan – Indonesian" lang="id" hreflang="id" data-title="Jarak Manhattan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Geometria_del_taxi" title="Geometria del taxi – Italian" lang="it" hreflang="it" data-title="Geometria del taxi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%99%D7%AA_%D7%A0%D7%94%D7%92%D7%99_%D7%94%D7%9E%D7%95%D7%A0%D7%99%D7%95%D7%AA" title="גאומטריית נהגי המוניות – Hebrew" lang="he" hreflang="he" data-title="גאומטריית נהגי המוניות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%9F%E0%B4%BE%E0%B4%95%E0%B5%8D%E0%B4%B8%E0%B4%BF%E0%B4%95%E0%B4%BE%E0%B4%AC%E0%B5%8D_%E0%B4%9C%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%AE%E0%B4%BF%E0%B4%A4%E0%B4%BF" title="ടാക്സികാബ് ജ്യാമിതി – Malayalam" lang="ml" hreflang="ml" data-title="ടാക്സികാബ് ജ്യാമിതി" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Manhattan-metriek" title="Manhattan-metriek – Dutch" lang="nl" hreflang="nl" data-title="Manhattan-metriek" 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/%E3%83%9E%E3%83%B3%E3%83%8F%E3%83%83%E3%82%BF%E3%83%B3%E8%B7%9D%E9%9B%A2" title="マンハッタン距離 – Japanese" lang="ja" hreflang="ja" data-title="マンハッタン距離" 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/Geometria_do_t%C3%A1xi" title="Geometria do táxi – Portuguese" lang="pt" hreflang="pt" data-title="Geometria do táxi" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Distan%C8%9B%C4%83_Manhattan" title="Distanță Manhattan – Romanian" lang="ro" hreflang="ro" data-title="Distanță Manhattan" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D0%B5_%D0%B3%D0%BE%D1%80%D0%BE%D0%B4%D1%81%D0%BA%D0%B8%D1%85_%D0%BA%D0%B2%D0%B0%D1%80%D1%82%D0%B0%D0%BB%D0%BE%D0%B2" title="Расстояние городских кварталов – Russian" lang="ru" hreflang="ru" data-title="Расстояние городских кварталов" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Manhattan_distance" title="Manhattan distance – Simple English" lang="en-simple" hreflang="en-simple" data-title="Manhattan distance" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Manhattangeometri" title="Manhattangeometri – Swedish" lang="sv" hreflang="sv" data-title="Manhattangeometri" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BD%D0%B3%D0%B5%D1%82%D1%82%D0%B5%D0%BD%D1%81%D1%8C%D0%BA%D0%B0_%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BA%D0%B0" title="Мангеттенська метрика – Ukrainian" lang="uk" hreflang="uk" data-title="Мангеттенська метрика" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kho%E1%BA%A3ng_c%C3%A1ch_Manhattan" title="Khoảng cách Manhattan – Vietnamese" lang="vi" hreflang="vi" data-title="Khoảng cách Manhattan" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%9B%BC%E5%85%8B%E9%A0%93%E8%B7%9D%E9%9B%A2" title="曼克頓距離 – Cantonese" lang="yue" hreflang="yue" data-title="曼克頓距離" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9B%BC%E5%93%88%E9%A0%93%E8%B7%9D%E9%9B%A2" title="曼哈頓距離 – Chinese" lang="zh" hreflang="zh" data-title="曼哈頓距離" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q573509#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> 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</div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to 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typeof="mw:File/Thumb"><a href="/wiki/File:Manhattan_distance.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/260px-Manhattan_distance.svg.png" decoding="async" width="260" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/390px-Manhattan_distance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/520px-Manhattan_distance.svg.png 2x" data-file-width="283" data-file-height="283" /></a><figcaption>In taxicab geometry, the lengths of the red, blue, green, and yellow paths all equal <span class="texhtml">12</span>, the taxicab distance between the opposite corners, and all four paths are shortest paths. Instead, in Euclidean geometry, the red, blue, and yellow paths still have length <span class="texhtml">12</span> but the green path is the unique shortest path, with length equal to the Euclidean distance between the opposite corners, <span class="texhtml">6√2 ≈ 8.49</span>.</figcaption></figure> <p><b>Taxicab geometry</b> or <b>Manhattan geometry</b> is <a href="/wiki/Geometry" title="Geometry">geometry</a> where the familiar <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> is ignored, and the <a href="/wiki/Distance" title="Distance">distance</a> between two <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> is instead defined to be the sum of the <a href="/wiki/Absolute_difference" title="Absolute difference">absolute differences</a> of their respective <a href="/wiki/Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinates</a>, a distance function (or <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>) called the <i>taxicab distance</i>, <i>Manhattan distance</i>, or <i>city block distance</i>. The name refers to the island of <a href="/wiki/Manhattan" title="Manhattan">Manhattan</a>, or generically any planned city with a <a href="/wiki/Rectangular_grid" class="mw-redirect" title="Rectangular grid">rectangular grid</a> of streets, in which a taxicab can only travel along grid directions. In taxicab geometry, the distance between any two points equals the length of their shortest grid path. This different definition of distance also leads to a different definition of the length of a curve, for which a <a href="/wiki/Line_segment" title="Line segment">line segment</a> between any two points has the same length as a grid path between those points rather than its Euclidean length. </p><p>The taxicab distance is also sometimes known as <i>rectilinear distance</i> or <span class="texhtml"><i>L</i><sup>1</sup></span> distance (see <a href="/wiki/Lp_space" title="Lp space"><i>L<sup>p</sup></i> space</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> This geometry has been used in <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a> since the 18th century, and is often referred to as <a href="/wiki/Lasso_(statistics)" title="Lasso (statistics)">LASSO</a>. Its geometric interpretation dates to <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a> of the 19th century and is due to <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a>. </p><p>In the two-<a href="/wiki/Dimension" title="Dimension">dimensional</a> <a href="/wiki/Real_coordinate_space" title="Real coordinate space">real coordinate space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d349b099a2e00103b347c5f640a30e0af2a6ee18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{2},}"></span> the taxicab distance between two points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1})}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{2},y_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{2},y_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52d44e16a796acee486af49af05f678566d181a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{2},y_{2})}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3960547778409b4a265ae0a4699fd427af8ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.263ex; height:2.843ex;" alt="{\displaystyle \left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|}"></span>. That is, it is the sum of the <a href="/wiki/Absolute_value" title="Absolute value">absolute values</a> of the differences in both coordinates. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Taxicab_geometry&action=edit&section=1" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The taxicab distance, <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 d_{\text{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\text{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4d5eb2e94e044af816d58dfda284471b49c4037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.628ex; height:2.509ex;" alt="{\displaystyle d_{\text{T}}}"></span>, between two points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =(p_{1},p_{2},\dots ,p_{n}){\text{ and }}\mathbf {q} =(q_{1},q_{2},\dots ,q_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =(p_{1},p_{2},\dots ,p_{n}){\text{ and }}\mathbf {q} =(q_{1},q_{2},\dots ,q_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/769212eb9991464f8efe460e64a6115deeb28747" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.323ex; height:2.843ex;" alt="{\displaystyle \mathbf {p} =(p_{1},p_{2},\dots ,p_{n}){\text{ and }}\mathbf {q} =(q_{1},q_{2},\dots ,q_{n})}"></span> in an <i>n</i>-dimensional <a href="/wiki/Real_coordinate_space" title="Real coordinate space">real coordinate space</a> with fixed <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a>, is the sum of the lengths of the projections of the <a href="/wiki/Line_segment" title="Line segment">line segment</a> between the points onto the <a href="/wiki/Coordinate_axes" class="mw-redirect" title="Coordinate axes">coordinate axes</a>. More formally,<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\text{T}}(\mathbf {p} ,\mathbf {q} )=\left\|\mathbf {p} -\mathbf {q} \right\|_{\text{T}}=\sum _{i=1}^{n}\left|p_{i}-q_{i}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow> <mo symmetric="true">‖</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>|</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\text{T}}(\mathbf {p} ,\mathbf {q} )=\left\|\mathbf {p} -\mathbf {q} \right\|_{\text{T}}=\sum _{i=1}^{n}\left|p_{i}-q_{i}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e84f3874e1eb36a8881a8248cf83037b455360cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.736ex; height:6.843ex;" alt="{\displaystyle d_{\text{T}}(\mathbf {p} ,\mathbf {q} )=\left\|\mathbf {p} -\mathbf {q} \right\|_{\text{T}}=\sum _{i=1}^{n}\left|p_{i}-q_{i}\right|}"></span>For example, in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>, the taxicab distance between <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 {p} =(p_{1},p_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =(p_{1},p_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8264bf7cc18cde2dbf382a0befc761c5629308a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.874ex; height:2.843ex;" alt="{\displaystyle \mathbf {p} =(p_{1},p_{2})}"></span> and <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 {q} =(q_{1},q_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} =(q_{1},q_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadede69f21d954ef6befd75d03e75769b91a837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.54ex; height:2.843ex;" alt="{\displaystyle \mathbf {q} =(q_{1},q_{2})}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|p_{1}-q_{1}\right|+\left|p_{2}-q_{2}\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|p_{1}-q_{1}\right|+\left|p_{2}-q_{2}\right|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48c6b7a51283f697fec276f0923e36d0c34c84f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.772ex; height:2.843ex;" alt="{\displaystyle \left|p_{1}-q_{1}\right|+\left|p_{2}-q_{2}\right|.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Taxicab_geometry&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>L</i><sup>1</sup> metric was used in <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a>, as a measure of <a href="/wiki/Goodness_of_fit" title="Goodness of fit">goodness of fit</a>, in 1757 by <a href="/wiki/Roger_Joseph_Boscovich" title="Roger Joseph Boscovich">Roger Joseph Boscovich</a>.<sup id="cite_ref-Stigler19862_2-0" class="reference"><a href="#cite_note-Stigler19862-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The interpretation of it as a distance between points in a geometric space dates to the late 19th century and the development of <a href="/wiki/Non-Euclidean_geometries" class="mw-redirect" title="Non-Euclidean geometries">non-Euclidean geometries</a>. Notably it appeared in 1910 in the works of both <a href="/wiki/Frigyes_Riesz" title="Frigyes Riesz">Frigyes Riesz</a> and <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a>. The formalization of <a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup>p</sup> spaces</a>, which include taxicab geometry as a special case, is credited to Riesz.<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> In developing the <a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">geometry of numbers</a>, <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a> established his <a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski inequality</a>, stating that these spaces define <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector spaces</a>.<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> </p><p>The name <i>taxicab geometry</i> was introduced by <a href="/wiki/Karl_Menger" title="Karl Menger">Karl Menger</a> in a 1952 booklet <i>You Will Like Geometry</i>, accompanying a geometry exhibit intended for the general public at the <a href="/wiki/Museum_of_Science_and_Industry_(Chicago)" title="Museum of Science and Industry (Chicago)">Museum of Science and Industry</a> in Chicago.<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> </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=Taxicab_geometry&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Thought of as an additional structure layered on <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, taxicab distance depends on the <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">orientation</a> of the coordinate system and is changed by Euclidean <a href="/wiki/Rotation" title="Rotation">rotation</a> of the space, but is unaffected by <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> or axis-aligned <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a>. Taxicab geometry satisfies all of <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's axioms</a> (a formalization of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>) except that the congruence of angles cannot be defined to precisely match the Euclidean concept, and under plausible definitions of congruent taxicab angles, the <a href="/wiki/Congruence_(geometry)#Determining_congruence" title="Congruence (geometry)">side-angle-side axiom</a> is not satisfied as in general triangles with two taxicab-congruent sides and a taxicab-congruent angle between them are not <a href="/wiki/Congruence_(geometry)#Congruence_of_triangles" title="Congruence (geometry)">congruent triangles</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Spheres">Spheres</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Taxicab_geometry&action=edit&section=4" title="Edit section: Spheres"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:TaxicabGeometryCircle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/TaxicabGeometryCircle.svg/180px-TaxicabGeometryCircle.svg.png" decoding="async" width="180" height="496" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/TaxicabGeometryCircle.svg/270px-TaxicabGeometryCircle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/TaxicabGeometryCircle.svg/360px-TaxicabGeometryCircle.svg.png 2x" data-file-width="214" data-file-height="590" /></a><figcaption>Grid points on a circle in taxicab geometry as the grid is made finer</figcaption></figure> <p>In any <a href="/wiki/Metric_space" title="Metric space">metric space</a>, a <a href="/wiki/Sphere#Metric_spaces" title="Sphere">sphere</a> is a set of points at a fixed distance, the <i><a href="/wiki/Radius" title="Radius">radius</a></i>, from a specific <i><a href="/wiki/Centre_(geometry)" title="Centre (geometry)">center</a></i> point. Whereas a Euclidean sphere is round and rotationally symmetric, under the taxicab distance, the shape of a sphere is a <a href="/wiki/Cross-polytope" title="Cross-polytope">cross-polytope</a>, the <i>n</i>-dimensional generalization of a <a href="/wiki/Regular_octahedron" class="mw-redirect" title="Regular octahedron">regular octahedron</a>, whose points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"></span> satisfy the equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\text{T}}(\mathbf {p} ,\mathbf {c} )=\sum _{i=1}^{n}|p_{i}-c_{i}|=r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>p</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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\text{T}}(\mathbf {p} ,\mathbf {c} )=\sum _{i=1}^{n}|p_{i}-c_{i}|=r,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d551a33ea4362f5e49edbb9292dafba19e4519ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.688ex; height:6.843ex;" alt="{\displaystyle d_{\text{T}}(\mathbf {p} ,\mathbf {c} )=\sum _{i=1}^{n}|p_{i}-c_{i}|=r,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8798d172f59e21f2ce193a3118d4063d19353ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {c} }"></span> is the center and <i>r</i> is the radius. Points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"></span> on the <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a>, a sphere of radius 1 centered at the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>, satisfy the equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d_{\text{T}}(\mathbf {p} ,\mathbf {0} )=\sum _{i=1}^{n}|p_{i}|=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d_{\text{T}}(\mathbf {p} ,\mathbf {0} )=\sum _{i=1}^{n}|p_{i}|=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5217ec9407f2598f25b5f9f747dffd898a4b8f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.303ex; height:3.176ex;" alt="{\textstyle d_{\text{T}}(\mathbf {p} ,\mathbf {0} )=\sum _{i=1}^{n}|p_{i}|=1.}"></span> </p><p>In two dimensional taxicab geometry, the sphere (called a <i><a href="/wiki/Circle" title="Circle">circle</a></i>) is a <a href="/wiki/Square" title="Square">square</a> oriented diagonally to the coordinate axes. The image to the right shows in red the set of all points on a square grid with a fixed distance from the blue center. As the grid is made finer, the red points become more numerous, and in the limit tend to a continuous tilted square. Each side has taxicab length 2<i>r</i>, so the <a href="/wiki/Circumference" title="Circumference">circumference</a> is 8<i>r</i>. Thus, in taxicab geometry, the value of the analog of the circle constant <a href="/wiki/Pi" title="Pi">π</a>, the ratio of circumference to <a href="/wiki/Diameter" title="Diameter">diameter</a>, is equal to 4. </p><p>A closed <i><a href="/wiki/Ball_(mathematics)#In_general_metric_spaces" title="Ball (mathematics)">ball</a></i> (or closed <i><a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a></i> in the 2-dimensional case) is a filled-in sphere, the set of points at distance less than or equal to the radius from a specific center. For <a href="/wiki/Cellular_automata" class="mw-redirect" title="Cellular automata">cellular automata</a> on a square grid, a taxicab <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disk</a> is the <a href="/wiki/Von_Neumann_neighborhood" title="Von Neumann neighborhood">von Neumann neighborhood</a> of range <i>r</i> of its center. </p><p>A circle of radius <i>r</i> for the <a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a> (<a href="/wiki/Lp_space" title="Lp space">L<sub>∞</sub> metric</a>) on a plane is also a square with side length 2<i>r</i> parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between L<sub>1</sub> and L<sub>∞</sub> metrics does not generalize to higher dimensions. </p><p>Whenever each pair in a collection of these circles has a nonempty intersection, there exists an intersection point for the whole collection; therefore, the Manhattan distance forms an <a href="/wiki/Injective_metric_space" title="Injective metric space">injective metric space</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Arc_length">Arc length</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Taxicab_geometry&action=edit&section=5" title="Edit section: Arc length"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span> be a <a href="/wiki/Differentiable_function" title="Differentiable function">continuously differentiable</a> function. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> be the taxicab <a href="/wiki/Arc_length" title="Arc length">arc length</a> of the <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> on some interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span>. Take a <a href="/wiki/Partition_of_an_interval" title="Partition of an interval">partition</a> of the interval into equal infinitesimal subintervals, and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a369483db37c9ede89254f876794b6373c400435" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.826ex; height:2.509ex;" alt="{\displaystyle \Delta s_{i}}"></span> be the taxicab length of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{\text{th}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>th</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{\text{th}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cc827ec109594f9da6862138d76775cd733866" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.588ex; height:2.676ex;" alt="{\displaystyle i^{\text{th}}}"></span> subarc. Then<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><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s_{i}=\Delta x_{i}+\Delta y_{i}=\Delta x_{i}+|f(x_{i})-f(x_{i-1})|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s_{i}=\Delta x_{i}+\Delta y_{i}=\Delta x_{i}+|f(x_{i})-f(x_{i-1})|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d662985c31c2debd77323cc243db076d5a8b514" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.025ex; height:2.843ex;" alt="{\displaystyle \Delta s_{i}=\Delta x_{i}+\Delta y_{i}=\Delta x_{i}+|f(x_{i})-f(x_{i-1})|.}"></span> </p><p>By the <a href="/wiki/Mean_value_theorem" title="Mean value theorem">mean value theorem</a>, there exists some point <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_{i}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4aad224eaab545bb11af7a02363e59addaad6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:2.843ex;" alt="{\displaystyle x_{i}^{*}}"></span> between <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> and <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_{i-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db345bb67bd140474742faf5d2fff314daa04e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.23ex; height:2.009ex;" alt="{\displaystyle x_{i-1}}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{i})-f(x_{i-1})=f'(x_{i}^{*})dx_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{i})-f(x_{i-1})=f'(x_{i}^{*})dx_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83db3e73ace182ebfbdbb19bff20ea9ab9e07838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.018ex; height:3.176ex;" alt="{\displaystyle f(x_{i})-f(x_{i-1})=f'(x_{i}^{*})dx_{i}}"></span>.<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> Then the previous equation can be written </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s_{i}=\Delta x_{i}+|f'(x_{i}^{*})|\Delta x_{i}=\Delta x_{i}(1+|f'(x_{i}^{*})|).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s_{i}=\Delta x_{i}+|f'(x_{i}^{*})|\Delta x_{i}=\Delta x_{i}(1+|f'(x_{i}^{*})|).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8d21354ab1336eb53fec55d5ff0fc8087f59a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.502ex; height:3.176ex;" alt="{\displaystyle \Delta s_{i}=\Delta x_{i}+|f'(x_{i}^{*})|\Delta x_{i}=\Delta x_{i}(1+|f'(x_{i}^{*})|).}"></span> </p><p> Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> is given as the sum of every partition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> as they get <a href="/wiki/Arbitrarily_large" title="Arbitrarily large">arbitrarily small</a>.</p><figure typeof="mw:File/Thumb"><a href="/wiki/File:Three_monotone_increasing_or_decreasing_curves_with_same_endpoints.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Three_monotone_increasing_or_decreasing_curves_with_same_endpoints.png/221px-Three_monotone_increasing_or_decreasing_curves_with_same_endpoints.png" decoding="async" width="221" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Three_monotone_increasing_or_decreasing_curves_with_same_endpoints.png/332px-Three_monotone_increasing_or_decreasing_curves_with_same_endpoints.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Three_monotone_increasing_or_decreasing_curves_with_same_endpoints.png/442px-Three_monotone_increasing_or_decreasing_curves_with_same_endpoints.png 2x" data-file-width="668" data-file-height="658" /></a><figcaption>Curves defined by monotone increasing or decreasing functions have the same taxicab arc length as long as they share the same endpoints.</figcaption></figure><p><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}s&=\lim _{n\rightarrow \infty }\sum _{i=1}^{n}\Delta x_{i}(1+|f'(x_{i}^{*})|)\\&=\int _{a}^{b}1+|f'(x)|\,dx\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> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi mathvariant="normal">Δ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s&=\lim _{n\rightarrow \infty }\sum _{i=1}^{n}\Delta x_{i}(1+|f'(x_{i}^{*})|)\\&=\int _{a}^{b}1+|f'(x)|\,dx\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/747c0c949be9722bd24e0c0b6e7163f4236793a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:30.712ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}s&=\lim _{n\rightarrow \infty }\sum _{i=1}^{n}\Delta x_{i}(1+|f'(x_{i}^{*})|)\\&=\int _{a}^{b}1+|f'(x)|\,dx\end{aligned}}}"></span> </p><p>To test this, take the taxicab circle of <a href="/wiki/Radius" title="Radius">radius</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> centered at the origin. Its curve in the first <a href="/wiki/Quadrant_(plane_geometry)" title="Quadrant (plane geometry)">quadrant</a> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=-x+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=-x+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da138bd28c1cc5bb876452c9e30d875d30a70b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.543ex; height:2.843ex;" alt="{\displaystyle f(x)=-x+r}"></span> whose length is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\int _{0}^{r}1+|-1|dx=2r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\int _{0}^{r}1+|-1|dx=2r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/087ce6d12b169e2b94e99736c212b180d019eb99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.168ex; height:5.843ex;" alt="{\displaystyle s=\int _{0}^{r}1+|-1|dx=2r}"></span> </p><p>Multiplying this value by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"></span> to account for the remaining quadrants gives <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 8r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8dcfa16125a9636e338b236833257b608bcba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.211ex; height:2.176ex;" alt="{\displaystyle 8r}"></span>, which agrees with the <a href="/wiki/Circumference" title="Circumference">circumference</a> of a taxicab circle.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Now take the <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a> circle of radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> centered at the origin, which is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\sqrt {r^{2}-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\sqrt {r^{2}-x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3218f29a2d00d33d4716ace8f06eb9399235871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.167ex; height:3.509ex;" alt="{\displaystyle f(x)={\sqrt {r^{2}-x^{2}}}}"></span>. Its arc length in the first quadrant is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s&=\int _{0}^{r}1+|{\frac {-x}{\sqrt {r^{2}-x^{2}}}}|dx\\&=x+{\sqrt {r^{2}-x^{2}}}{\bigg |}_{0}^{r}\\&=r-(-r)\\&=2r\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> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>x</mi> </mrow> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>r</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s&=\int _{0}^{r}1+|{\frac {-x}{\sqrt {r^{2}-x^{2}}}}|dx\\&=x+{\sqrt {r^{2}-x^{2}}}{\bigg |}_{0}^{r}\\&=r-(-r)\\&=2r\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8fdaf6450fab9be1c6b76a9fee532d31fbab4b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:27.095ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}s&=\int _{0}^{r}1+|{\frac {-x}{\sqrt {r^{2}-x^{2}}}}|dx\\&=x+{\sqrt {r^{2}-x^{2}}}{\bigg |}_{0}^{r}\\&=r-(-r)\\&=2r\end{aligned}}}"></span> </p><p>Accounting for the remaining quadrants gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\times 2r=8r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <mn>2</mn> <mi>r</mi> <mo>=</mo> <mn>8</mn> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times 2r=8r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc0ece94fb8ce888a944477f9da28f51add34ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.524ex; height:2.176ex;" alt="{\displaystyle 4\times 2r=8r}"></span> again. Therefore, the <a href="/wiki/Circumference" title="Circumference">circumference</a> of the taxicab circle and the <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a> circle in the taxicab <a href="/wiki/Metric_space" title="Metric space">metric</a> are equal.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In fact, for any function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> that is monotonic and <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a> with a continuous <a href="/wiki/Derivative" title="Derivative">derivative</a> over an interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span>, the arc length of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b-a)+\mid f(b)-f(a)\mid }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>∣</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">∣</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b-a)+\mid f(b)-f(a)\mid }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b87d354d673be52e91028bc676f4802651a214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.513ex; height:2.843ex;" alt="{\displaystyle (b-a)+\mid f(b)-f(a)\mid }"></span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Triangle_congruence">Triangle congruence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Taxicab_geometry&action=edit&section=6" title="Edit section: Triangle congruence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Congruencetriangletaxicab.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Congruencetriangletaxicab.png/391px-Congruencetriangletaxicab.png" decoding="async" width="391" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Congruencetriangletaxicab.png/587px-Congruencetriangletaxicab.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Congruencetriangletaxicab.png/782px-Congruencetriangletaxicab.png 2x" data-file-width="1696" data-file-height="856" /></a><figcaption>Two taxicab right isoceles triangles. Three angles and two legs are congruent, but the triangles are not congruent. Therefore, ASASA is not a congruence theorem in taxicab geometry.</figcaption></figure> <p>Two triangles are congruent if and only if three corresponding sides are equal in distance and three corresponding angles are equal in measure. There are several theorems that guarantee <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">triangle congruence</a> in Euclidean geometry, namely Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), Side-Angle-Side (SAS), and Side-Side-Side (SSS). In taxicab geometry, however, only SASAS guarantees triangle congruence.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Take, for example, two right isosceles taxicab triangles whose angles measure 45-90-45. The two legs of both triangles have a taxicab length 2, but the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuses</a> are not congruent. This counterexample eliminates AAS, ASA, and SAS. It also eliminates AASS, AAAS, and even ASASA. Having three congruent angles and two sides does not guarantee triangle congruence in taxicab geometry. Therefore, the only triangle congruence theorem in taxicab geometry is SASAS, where all three corresponding sides must be congruent and at least two corresponding angles must be congruent.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This result is mainly due to the fact that the length of a line segment depends on its orientation in taxicab geometry. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Taxicab_geometry&action=edit&section=7" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Compressed_sensing">Compressed sensing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Taxicab_geometry&action=edit&section=8" title="Edit section: Compressed sensing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In solving an <a href="/wiki/Underdetermined_system" title="Underdetermined system">underdetermined system</a> of linear equations, the <a href="/wiki/Regularization_(mathematics)" title="Regularization (mathematics)">regularization</a> term for the parameter vector is expressed in terms of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/361ddd720474aa41cb05453e03424fb7999d3b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.024ex; height:2.509ex;" alt="{\displaystyle \ell _{1}}"></span> norm (taxicab geometry) of the vector.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> This approach appears in the signal recovery framework called <a href="/wiki/Compressed_sensing" title="Compressed sensing">compressed sensing</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Differences_of_frequency_distributions">Differences of frequency distributions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Taxicab_geometry&action=edit&section=9" title="Edit section: Differences of frequency distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Taxicab geometry can be used to assess the differences in discrete frequency distributions. For example, in <a href="/wiki/RNA_splicing" title="RNA splicing">RNA splicing</a> positional distributions of <a href="/wiki/Hexamers" class="mw-redirect" title="Hexamers">hexamers</a>, which plot the probability of each hexamer appearing at each given <a href="/wiki/Nucleotide" title="Nucleotide">nucleotide</a> near a splice site, can be compared with L1-distance. Each position distribution can be represented as a vector where each entry represents the likelihood of the hexamer starting at a certain nucleotide. A large L1-distance between the two vectors indicates a significant difference in the nature of the distributions while a small distance denotes similarly shaped distributions. This is equivalent to measuring the area between the two distribution curves because the area of each segment is the absolute difference between the two curves' likelihoods at that point. When summed together for all segments, it provides the same measure as L1-distance.<sup id="cite_ref-lim2_14-0" class="reference"><a href="#cite_note-lim2-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </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=Taxicab_geometry&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski_distance_examples.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Minkowski_distance_examples.svg/220px-Minkowski_distance_examples.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Minkowski_distance_examples.svg/330px-Minkowski_distance_examples.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Minkowski_distance_examples.svg/440px-Minkowski_distance_examples.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard</figcaption></figure> <ul><li><a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li> <li><a href="/wiki/Hamming_distance" title="Hamming distance">Hamming distance</a> – The number of bits differing between two strings of binary digits</li> <li><a href="/wiki/Lee_distance" title="Lee distance">Lee distance</a></li> <li><a href="/wiki/Orthogonal_convex_hull" title="Orthogonal convex hull">Orthogonal convex hull</a> – Minimal superset that intersects each axis-parallel line in an interval</li> <li><a href="/wiki/Staircase_paradox" title="Staircase paradox">Staircase paradox</a> – The paradox that the limit of the lengths of finer and finer "staircase curves" does not tend to the length of the diagonal line segment the curves tend towards</li></ul> <div style="clear:both;" class=""></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=Taxicab_geometry&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/4365387/29780">Taxicab metric with stoplights</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=Taxicab_geometry&action=edit&section=12" 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 mw-references-columns"><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="CITEREFBlack" class="citation web cs1">Black, Paul E. <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/manhattanDistance.html">"Manhattan distance"</a>. <i>Dictionary of Algorithms and Data Structures</i><span class="reference-accessdate">. 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(July 5, 2011). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3131313">"Using positional distribution to identify splicing elements and predict pre-mRNA processing defects in human genes"</a>. <i>Proceedings of the National Academy of Sciences of the United States of America</i>. <b>108</b> (27): 11093–11098. <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/2011PNAS..10811093H">2011PNAS..10811093H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.1101135108">10.1073/pnas.1101135108</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3131313">3131313</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21685335">21685335</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences+of+the+United+States+of+America&rft.atitle=Using+positional+distribution+to+identify+splicing+elements+and+predict+pre-mRNA+processing+defects+in+human+genes&rft.volume=108&rft.issue=27&rft.pages=11093-11098&rft.date=2011-07-05&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3131313%23id-name%3DPMC&rft_id=info%3Apmid%2F21685335&rft_id=info%3Adoi%2F10.1073%2Fpnas.1101135108&rft_id=info%3Abibcode%2F2011PNAS..10811093H&rft.aulast=Lim&rft.aufirst=Kian+Huat&rft.au=Ferraris%2C+Luciana&rft.au=Filloux%2C+Madeleine+E.&rft.au=Raphael%2C+Benjamin+J.&rft.au=Fairbrother%2C+William+G.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3131313&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATaxicab+geometry" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Taxicab_geometry&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardner1997" class="citation book cs1"><a href="/wiki/Martin_Gardner" title="Martin Gardner">Gardner, Martin</a> (1997). <a rel="nofollow" class="external text" href="https://archive.org/details/springer_10.1007-978-0-387-30389-5/page/n164/">"10. Taxicab Geometry"</a>. <i>The Last Recreations</i>. Copernicus. pp. 159–176. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94929-1" title="Special:BookSources/0-387-94929-1"><bdi>0-387-94929-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=10.+Taxicab+Geometry&rft.btitle=The+Last+Recreations&rft.pages=159-176&rft.pub=Copernicus&rft.date=1997&rft.isbn=0-387-94929-1&rft.aulast=Gardner&rft.aufirst=Martin&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspringer_10.1007-978-0-387-30389-5%2Fpage%2Fn164%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATaxicab+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrause1975" class="citation book cs1">Krause, Eugene F. (1975). <i>Taxicab Geometry</i>. Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0201039346" title="Special:BookSources/0201039346"><bdi>0201039346</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Taxicab+Geometry&rft.pub=Addison-Wesley&rft.date=1975&rft.isbn=0201039346&rft.aulast=Krause&rft.aufirst=Eugene+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATaxicab+geometry" class="Z3988"></span> Reprinted by Dover (1986), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-25202-7" title="Special:BookSources/0-486-25202-7">0-486-25202-7</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links_2">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Taxicab_geometry&action=edit&section=14" 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-Taxicab_Metric"><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/TaxicabMetric.html">"Taxicab Metric"</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=Taxicab+Metric&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTaxicabMetric.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATaxicab+geometry" class="Z3988"></span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMalkevitch2007" class="citation web cs1">Malkevitch, Joe (October 1, 2007). <a rel="nofollow" class="external text" href="http://www.ams.org/publicoutreach/feature-column/fcarc-taxi">"Taxi!"</a>. <i>American Mathematical Society</i><span class="reference-accessdate">. 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href="/wiki/Measurable_space" title="Measurable space">Measurable space</a>/<a href="/wiki/Measurable_function" title="Measurable function">function</a></li> <li><a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a></li> <li><a href="/wiki/Sequence_space" title="Sequence space">Sequence spaces</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L1_space" class="mw-redirect" title="L1 space"><i>L</i><sup>1</sup> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integrable_function" class="mw-redirect" title="Integrable function">Integrable function</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a class="mw-selflink selflink">Taxicab geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L2_space" class="mw-redirect" title="L2 space"><i>L</i><sup>2</sup> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bessel%27s_inequality" title="Bessel's inequality">Bessel's</a></li> <li><a href="/wiki/Cauchy%E2%80%93Schwarz_inequality" title="Cauchy–Schwarz inequality">Cauchy–Schwarz</a></li> <li><a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></li> <li><a href="/wiki/Parseval%27s_identity" title="Parseval's identity">Parseval's identity</a></li> <li><a href="/wiki/Polarization_identity" title="Polarization identity">Polarization identity</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li> <li><a href="/wiki/Square-integrable_function" title="Square-integrable function">Square-integrable function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/L-infinity" title="L-infinity"><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 L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}"></span> spaces</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bounded_function" title="Bounded function">Bounded function</a></li> <li><a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a></li> <li><a href="/wiki/Infimum_and_supremum" title="Infimum and supremum">Infimum and supremum</a> <ul><li><a href="/wiki/Essential_infimum_and_essential_supremum" title="Essential infimum and essential supremum">Essential</a></li></ul></li> <li><a href="/wiki/Uniform_norm" title="Uniform norm">Uniform norm</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Maps</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Almost_everywhere" title="Almost everywhere">Almost everywhere</a></li> <li><a href="/wiki/Convergence_almost_everywhere" class="mw-redirect" title="Convergence almost everywhere">Convergence almost everywhere</a></li> <li><a href="/wiki/Convergence_in_measure" title="Convergence in measure">Convergence in measure</a></li> <li><a href="/wiki/Function_space" title="Function space">Function space</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Locally_integrable_function" title="Locally integrable function">Locally integrable function</a></li> <li><a href="/wiki/Measurable_function" title="Measurable function">Measurable function</a></li> <li><a href="/wiki/Symmetric_decreasing_rearrangement" title="Symmetric decreasing rearrangement">Symmetric decreasing rearrangement</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Inequalities</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Babenko%E2%80%93Beckner_inequality" title="Babenko–Beckner inequality">Babenko–Beckner</a></li> <li><a href="/wiki/Chebyshev%27s_inequality" title="Chebyshev's inequality">Chebyshev's</a></li> <li><a href="/wiki/Clarkson%27s_inequalities" title="Clarkson's inequalities">Clarkson's</a></li> <li><a href="/wiki/Hanner%27s_inequalities" title="Hanner's inequalities">Hanner's</a></li> <li><a href="/wiki/Hausdorff%E2%80%93Young_inequality" title="Hausdorff–Young inequality">Hausdorff–Young</a></li> <li><a href="/wiki/H%C3%B6lder%27s_inequality" title="Hölder's inequality">Hölder's</a></li> <li><a href="/wiki/Markov%27s_inequality" title="Markov's inequality">Markov's</a></li> <li><a href="/wiki/Minkowski_inequality" title="Minkowski inequality">Minkowski</a></li> <li><a href="/wiki/Young%27s_convolution_inequality" title="Young's convolution inequality">Young's convolution</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_analysis" title="Category:Theorems in analysis">Results</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Marcinkiewicz_interpolation_theorem" title="Marcinkiewicz interpolation theorem">Marcinkiewicz interpolation theorem</a></li> <li><a href="/wiki/Plancherel_theorem" title="Plancherel theorem">Plancherel theorem</a></li> <li><a href="/wiki/Riemann%E2%80%93Lebesgue_lemma" title="Riemann–Lebesgue lemma">Riemann–Lebesgue</a></li> <li><a href="/wiki/Riesz%E2%80%93Fischer_theorem" title="Riesz–Fischer theorem">Riesz–Fischer theorem</a></li> <li><a href="/wiki/Riesz%E2%80%93Thorin_theorem" title="Riesz–Thorin theorem">Riesz–Thorin theorem</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><span style="font-size:85%;">For <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a></span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isoperimetric_inequality" title="Isoperimetric inequality">Isoperimetric inequality</a></li> <li><a href="/wiki/Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a> <ul><li><a href="/wiki/Milman%27s_reverse_Brunn%E2%80%93Minkowski_inequality" title="Milman's reverse Brunn–Minkowski inequality">Milman's reverse</a></li></ul></li> <li><a href="/wiki/Minkowski%E2%80%93Steiner_formula" title="Minkowski–Steiner formula">Minkowski–Steiner formula</a></li> <li><a href="/wiki/Pr%C3%A9kopa%E2%80%93Leindler_inequality" title="Prékopa–Leindler inequality">Prékopa–Leindler inequality</a></li> <li><a href="/wiki/Vitale%27s_random_Brunn%E2%80%93Minkowski_inequality" title="Vitale's random Brunn–Minkowski inequality">Vitale's random Brunn–Minkowski inequality</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications & related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Lorentz_space" title="Lorentz space">Lorentz space</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Quasinorm" title="Quasinorm">Quasinorm</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev space</a></li> <li><a href="/wiki/*-algebra" title="*-algebra">*-algebra</a> <ul><li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann</a></li></ul></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" 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=Taxicab_geometry&oldid=1259671220">https://en.wikipedia.org/w/index.php?title=Taxicab_geometry&oldid=1259671220</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:Digital_geometry" title="Category:Digital geometry">Digital geometry</a></li><li><a href="/wiki/Category:Distance" title="Category:Distance">Distance</a></li><li><a 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