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<!doctype html> < html lang = en-AU dir = ltr class = scroll-smooth data-default-appearance = light data-auto-appearance = true > < head > < meta charset = utf-8 > < meta name = viewport content = "width=device-width,initial-scale=1" > < meta name = theme-color content = "rgb(255,255,255)" > < title > Mathematical notation · Congo< / title >
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< meta name = title content = "Mathematical notation · Congo" > < script type = text/javascript src = /congo/js/appearance.min.022d0ebc3b46a335eb1c7ef79b7f2de143d7cd5156d433638592ef1ce5f8554e.js integrity = "sha256-Ai0OvDtGozXrHH73m38t4UPXzVFW1DNjhZLvHOX4VU4=" > < / script > < link type = text/css rel = stylesheet href = /congo/css/main.bundle.min.2dd44849efa9d0ef68e8cdede2901f86dec79026811f5cf6b25aa2b8cd8ee63e.css integrity = "sha256-LdRISe+p0O9o6M3t4pAfht7HkCaBH1z2slqiuM2O5j4=" > < script defer type = text/javascript id = script-bundle src = /congo/js/main.bundle.min.f29ffdffd9ab4cc95250c3c7196b2d5dae8ee6ef0a4139451073f90183ae7e31.js integrity = "sha256-8p/9/9mrTMlSUMPHGWstXa6O5u8KQTlFEHP5AYOufjE=" data-copy = Copy data-copied = Copied > < / script > < meta name = description content = "
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A brief sample of mathematical notation in Congo.
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">< link rel = canonical href = https://jpanther.github.io/congo/samples/mathematical-notation/ > < link rel = apple-touch-icon sizes = 180x180 href = /congo/apple-touch-icon.png > < link rel = icon type = image/png sizes = 32x32 href = /congo/favicon-32x32.png > < link rel = icon type = image/png sizes = 16x16 href = /congo/favicon-16x16.png > < link rel = manifest href = /congo/site.webmanifest > < meta property = "og:title" content = "Mathematical notation" > < meta property = "og:description" content = "A brief sample of mathematical notation in Congo." > < meta property = "og:type" content = "article" > < meta property = "og:url" content = "https://jpanther.github.io/congo/samples/mathematical-notation/" > < meta property = "og:image" content = "https://jpanther.github.io/congo/samples/mathematical-notation/feature-artturi-jalli-gYrYa37fAKI-unsplash.jpg" > < meta property = "article:section" content = "samples" > < meta property = "article:published_time" content = "2019-03-08T00:00:00+00:00" > < meta property = "article:modified_time" content = "2019-03-08T00:00:00+00:00" > < meta name = twitter:card content = "summary_large_image" > < meta name = twitter:image content = "https://jpanther.github.io/congo/samples/mathematical-notation/feature-artturi-jalli-gYrYa37fAKI-unsplash.jpg" > < meta name = twitter:title content = "Mathematical notation" > < meta name = twitter:description content = "A brief sample of mathematical notation in Congo." > < script type = application/ld+json > { "@context" : "https://schema.org" , "@type" : "Article" , "articleSection" : "Content Samples" , "name" : "Mathematical notation" , "headline" : "Mathematical notation" , "description" : "A brief sample of mathematical notation in Congo." , "abstract" : "\u003cp\u003eKaTeX can be used to render mathematical notation within articles.\u003c\/p\u003e" , "inLanguage" : "en-AU" , "url" : "https:\/\/jpanther.github.io\/congo\/samples\/mathematical-notation\/" , "author" : { "@type" : "Person" , "name" : "Congo" } , "copyrightYear" : "2019" , "dateCreated" : "2019-03-08T00:00:00\u002b00:00" , "datePublished" : "2019-03-08T00:00:00\u002b00:00" , "dateModified" : "2019-03-08T00:00:00\u002b00:00" , "keywords" : [ "sample" , "katex" , "maths" , "shortcodes" ] , "mainEntityOfPage" : "true" , "wordCount" : "143" } < / script > < meta name = author content = "Congo" > < link href = https://twitter.com/ rel = me > < link href = https://facebook.com/ rel = me > < link href = https://linkedin.com/ rel = me > < link href = https://youtube.com/ rel = me > < link type = text/css rel = stylesheet href = /congo/lib/katex/katex.min.7c7c1a59e6eec00ecd485e5083f69fe14783c7d5ea52962938682e6e8df25aef9bd88411e14790c1b6f8d938bb81502336d56823f23d4c4a187aef689b0702ea.css integrity = "sha512-fHwaWebuwA7NSF5Qg/af4UeDx9XqUpYpOGgubo3yWu+b2IQR4UeQwbb42Ti7gVAjNtVoI/I9TEoYeu9omwcC6g==" > < script defer src = /congo/lib/katex/katex.min.2d037120c479ad7bfba3e6f597cf8dd4464c7e11bb88567d1e19db2644e9e338cdaf95afb2def902a51e143c5e4546bb979bc40f2522022a7ffebf8414b2c2c8.js integrity = "sha512-LQNxIMR5rXv7o+b1l8+N1EZMfhG7iFZ9HhnbJkTp4zjNr5Wvst75AqUeFDxeRUa7l5vEDyUiAip//r+EFLLCyA==" > < / script > < script defer src = /congo/lib/katex/auto-render.min.8968ae052e67b7aafad1f0b3dba35dd19a9ed276e4d594c841b9772afee462c5fec8a314147ce3687dbe02733abe9d97b3e80d99a0405562634a6b8fc3be847e.js integrity = "sha512-iWiuBS5nt6r60fCz26Nd0Zqe0nbk1ZTIQbl3Kv7kYsX+yKMUFHzjaH2+AnM6vp2Xs+gNmaBAVWJjSmuPw76Efg==" onload = renderMathInElement(document.body) > < / script > < / head > < body class = "flex flex-col h-screen px-6 m-auto text-lg leading-7 max-w-7xl bg-neutral text-neutral-900 dark:bg-neutral-800 dark:text-neutral sm:px-14 md:px-24 lg:px-32" > < div id = the-top class = "absolute flex self-center" > < a class = "px-3 py-1 text-sm -translate-y-8 rounded-b-lg bg-primary-200 focus:translate-y-0 dark:bg-neutral-600" href = #main-content > < span class = "font-bold pe-2 text-primary-600 dark:text-primary-400" > ↓ < / span > Skip to main content< / a > < / div > < header class = "py-6 font-semibold text-neutral-900 dark:text-neutral print:hidden sm:py-10" > < nav class = "flex items-start justify-between sm:items-center" > < div class = "flex flex-row items-center" > < a class = "decoration-primary-500 hover:underline hover:decoration-2 hover:underline-offset-2" rel = me href = /congo/ > Congo< / a > < / div > < ul class = "flex flex-col list-none text-end sm:f
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< span class = "transition-colors group-dark:hover:text-primary-400 group-hover:text-primary-600" > < span class = "relative inline-block align-text-bottom px-1 icon" > < svg aria-hidden = "true" focusable = "false" data-prefix = "fas" data-icon = "search" class = "svg-inline--fa fa-search fa-w-16" role = "img" xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M505 442.7 405.3 343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3 44-79.7 44-128C416 93.1 322.9.0 208 0S0 93.1.0 208s93.1 208 208 208c48.3.0 92.7-16.4 128-44v16.3c0 6.4 2.5 12.5 7 17l99.7 99.7c9.4 9.4 24.6 9.4 33.9.0l28.3-28.3c9.4-9.4 9.4-24.6.1-34zM208 336c-70.7.0-128-57.2-128-128 0-70.7 57.2-128 128-128 70.7.0 128 57.2 128 128 0 70.7-57.2 128-128 128z" / > < / svg > < / span > < / span > < span class = "decoration-primary-500 group-hover:underline group-hover:decoration-2 group-hover:underline-offset-2" > < / span > < / button > < / li > < li class = "group mb-1 sm:mb-0 sm:me-7 sm:last:me-0.5" > < div class = "relative group" > < button class = "flex items-center justify-end w-full transition-colors group-dark:hover:text-primary-400 group-hover:text-primary-600" >
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< a href = https://jpanther.github.io/congo/docs/shortcodes/#katex > < code > katex< / code > shortcode< / a > within the article. Any KaTeX syntax on that page will then be automatically rendered.< / p > < p > Use the online reference of < a href = https://katex.org/docs/supported.html target = _blank rel = noreferrer > supported TeX functions< / a > for the available syntax.< / p > < h2 id = inline-notation class = "relative group" > Inline notation < span class = "absolute top-0 w-6 transition-opacity opacity-0 -start-6 not-prose group-hover:opacity-100" > < a class = "group-hover:text-primary-300 dark:group-hover:text-neutral-700" style = text-decoration-line:none!important href = #inline-notation aria-label = Anchor > #< / a > < / span > < / h2 > < p > Inline notation can be generated by wrapping the expression in < code > \\(< / code > and < code > \\)< / code > delimiters.< / p > < p > < strong > Example:< / strong > < / p > < div class = highlight > < pre tabindex = 0 class = chroma > < code class = language-tex data-lang = tex > < span class = line > < span class = cl > < span class = c > % KaTeX inline notation
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< / span > < / span > < / span > < span class = line > < span class = cl > < span class = c > < / span > Inline notation: < span class = k > \\< / span > (< span class = k > \varphi< / span > = < span class = k > \dfrac< / span > < span class = nb > {< / span > 1+< span class = k > \sqrt< / span > 5< span class = nb > }{< / span > 2< span class = nb > }< / span > = 1.6180339887…< span class = k > \\< / span > )
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< / span > < / span > < / code > < / pre > < / div > < p > Inline notation: \(\varphi = \dfrac{1+\sqrt5}{2}= 1.6180339887…\)< / p > < h2 id = block-notation class = "relative group" > Block notation < span class = "absolute top-0 w-6 transition-opacity opacity-0 -start-6 not-prose group-hover:opacity-100" > < a class = "group-hover:text-primary-300 dark:group-hover:text-neutral-700" style = text-decoration-line:none!important href = #block-notation aria-label = Anchor > #< / a > < / span > < / h2 > < p > Alternatively, block notation can be generated using < code > $$< / code > delimiters. This will output the expression in its own HTML block.< / p > < p > < strong > Example:< / strong > < / p > < div class = highlight > < pre tabindex = 0 class = chroma > < code class = language-tex data-lang = tex > < span class = line > < span class = cl > < span class = c > % KaTeX block notation
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< / span > < / span > < / span > < span class = line > < span class = cl > < span class = c > < / span > < span class = sb > $$< / span > < span class = nb >
< / span > < / span > < / span > < span class = line > < span class = cl > < span class = nb > < / span > < span class = nv > \varphi< / span > < span class = nb > < / span > < span class = o > =< / span > < span class = nb > < / span > < span class = m > 1< / span > < span class = o > +< / span > < span class = nv > \frac< / span > < span class = nb > {< / span > < span class = m > 1< / span > < span class = nb > } {< / span > < span class = m > 1< / span > < span class = o > +< / span > < span class = nv > \frac< / span > < span class = nb > {< / span > < span class = m > 1< / span > < span class = nb > } {< / span > < span class = m > 1< / span > < span class = o > +< / span > < span class = nv > \frac< / span > < span class = nb > {< / span > < span class = m > 1< / span > < span class = nb > } {< / span > < span class = m > 1< / span > < span class = o > +< / span > < span class = nv > \cdots< / span > < span class = nb > } } }
< / span > < / span > < / span > < span class = line > < span class = cl > < span class = nb > < / span > < span class = s > $$< / span >
< / span > < / span > < / code > < / pre > < / div > < p > $$
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\varphi = 1+\frac{1} {1+\frac{1} {1+\frac{1} {1+\cdots} } }
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$$< / p > < / div > < / section > < footer class = "pt-8 max-w-prose print:hidden" > < div class = flex > < img class = "!mb-0 !mt-0 me-4 h-24 w-24 rounded-full" width = 96 height = 96 alt = Congo src = /congo/img/author_hu052d24d60828c1cca42a6aefe45b8b44_54195_192x192_fill_q75_box_center.jpg loading = lazy > < div class = place-self-center > < div class = "text-[0.6rem] uppercase leading-3 text-neutral-500 dark:text-neutral-400" > Author< / div > < div class = "font-semibold leading-6 text-neutral-800 dark:text-neutral-300" > Congo< / div > < div class = "text-sm text-neutral-700 dark:text-neutral-400" > This is an example author bio, and although there’ s a stock photo of a dog here, this article was actually created by a human. 🐶< / div > < div class = "text-2xl sm:text-lg" > < div class = "flex flex-wrap text-neutral-400 dark:text-neutral-500" > < a class = "px-1 transition-transform hover:scale-125 hover:text-primary-700 dark:hover:text-primary-400" style = will-change:transform href = https://twitter.com/ target = _blank aria-label = X-Twitter rel = "me noopener noreferrer" > < span class = "relative inline-block align-text-bottom px-1 icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M389.2 48h70.6L305.6 224.2 487 464H345L233.7 318.6 106.5 464H35.8L200.7 275.5 26.8 48H172.4L272.9 180.9 389.2 48zM364.4 421.8h39.1L151.1 88h-42L364.4 421.8z" / > < / svg > < / span > < / a > < a class = "px-1 transition-transform hover:scale-125 hover:text-primary-700 dark:hover:text-primary-400" style = will-change:transform href = https://facebook.com/ target = _blank aria-label = Facebook rel = "me noopener noreferrer" > < span class = "relative inline-block align-text-bottom px-1 icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M504 256C504 119 393 8 256 8S8 119 8 256c0 123.78 90.69 226.38 209.25 245V327.69h-63V256h63v-54.64c0-62.15 37-96.48 93.67-96.48 27.14.0 55.52 4.84 55.52 4.84v61h-31.28c-30.8.0-40.41 19.12-40.41 38.73V256h68.78l-11 71.69h-57.78V501C413.31 482.38 504 379.78 504 256z" / > < / svg > < / span > < / a > < a class = "px-1 transition-transform hover:scale-125 hover:text-primary-700 dark:hover:text-primary-400" style = will-change:transform href = https://linkedin.com/ target = _blank aria-label = Linkedin rel = "me noopener noreferrer" > < span class = "relative inline-block align-text-bottom px-1 icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 448 512" > < path fill = "currentcolor" d = "M416 32H31.9C14.3 32 0 46.5.0 64.3v383.4C0 465.5 14.3 480 31.9 480H416c17.6.0 32-14.5 32-32.3V64.3c0-17.8-14.4-32.3-32-32.3zM135.4 416H69V202.2h66.5V416zm-33.2-243c-21.3.0-38.5-17.3-38.5-38.5S80.9 96 102.2 96c21.2.0 38.5 17.3 38.5 38.5.0 21.3-17.2 38.5-38.5 38.5zm282.1 243h-66.4V312c0-24.8-.5-56.7-34.5-56.7-34.6.0-39.9 27-39.9 54.9V416h-66.4V202.2h63.7v29.2h.9c8.9-16.8 30.6-34.5 62.9-34.5 67.2.0 79.7 44.3 79.7 101.9V416z" / > < / svg > < / span > < / a > < a class = "px-1 transition-transform hover:scale-125 hover:text-primary-700 dark:hover:text-primary-400" style = will-change:transform href = https://youtube.com/ target = _blank aria-label = Youtube rel = "me noopener noreferrer" > < span class = "relative inline-block align-text-bottom px-1 icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 576 512" > < path fill = "currentcolor" d = "M549.655 124.083c-6.281-23.65-24.787-42.276-48.284-48.597C458.781 64 288 64 288 64S117.22 64 74.629 75.486c-23.497 6.322-42.003 24.947-48.284 48.597-11.412 42.867-11.412 132.305-11.412 132.305s0 89.438 11.412 132.305c6.281 23.65 24.787 41.5 48.284 47.821C117.22 448 288 448 288 448s170.78.0 213.371-11.486c23.497-6.321 42.003-24.171 48.284-47.821 11.412-42.867 11.412-132.305 11.412-132.305s0-89.438-11.412-132.305zm-317.51 213.508V175.185l142.739 81.205-142.739 81.201z" / > < / svg > < / span > < / a > < / div > < / div > < / div > < / div > < div class = pt-8 > < hr class = "border-dotted border-neutral-300 dark:border-neutral-600" > < div class = "flex justify-between pt-3" > < span > < a class = "group flex" href = /congo/samples/charts/ > < span class = "me-2 text-neutral-700 transition-transform group-hover:-translate-x-[2px] group-hover:text-primary-600 dark:text-neutral dark:group-hover:text-primary-400" > < span class = "ltr:inline rtl:hid
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< span class = "flex flex-col" > < span class = "mt-[0.1rem] leading-6 group-hover:underline group-hover:decoration-primary-500" > Charts< / span >
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< span class = "mt-[0.1rem] text-xs text-neutral-500 dark:text-neutral-400" > < time datetime = "2019-03-06 00:00:00 +0000 UTC" > 6 March 2019< / time >
< / span > < / span > < / a > < / span > < span > < a class = "group flex text-right" href = /congo/samples/placeholder-text/ > < span class = "flex flex-col" > < span class = "mt-[0.1rem] leading-6 group-hover:underline group-hover:decoration-primary-500" > Placeholder Text< / span >
< span class = "mt-[0.1rem] text-xs text-neutral-500 dark:text-neutral-400" > < time datetime = "2019-03-09 00:00:00 +0000 UTC" > 9 March 2019< / time >
< / span > < / span > < span class = "ms-2 text-neutral-700 transition-transform group-hover:-translate-x-[2px] group-hover:text-primary-600 dark:text-neutral dark:group-hover:text-primary-400" > < span class = "ltr:inline rtl:hidden" > → < / span > < span class = "ltr:hidden rtl:inline" > ← < / span > < / span > < / a > < / span > < / div > < / div > < / footer > < / article > < / main > < footer class = "py-10 print:hidden" > < div class = "flex items-center justify-between" > < div > < p class = "text-sm text-neutral-500 dark:text-neutral-400" > © 2023 Congo contributors< / p > < p class = "text-xs text-neutral-500 dark:text-neutral-400" > Powered by < a class = "hover:underline hover:decoration-primary-400 hover:text-primary-500" href = https://gohugo.io/ target = _blank rel = "noopener noreferrer" > Hugo< / a > & < a class = "hover:underline hover:decoration-primary-400 hover:text-primary-500" href = https://github.com/jpanther/congo target = _blank rel = "noopener noreferrer" > Congo< / a > < / p > < / div > < div class = "flex flex-row items-center" > < div class = "me-14 cursor-pointer text-sm text-neutral-700 hover:text-primary-600 dark:text-neutral dark:hover:text-primary-400" > < button id = appearance-switcher-0 type = button aria-label = "appearance switcher" > < div class = "flex items-center justify-center w-12 h-12 dark:hidden" title = "Switch to dark appearance" > < span class = "relative inline-block align-text-bottom px-1 icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M32 256C32 132.2 132.3 32 255.8 32c11.36.0 29.7 1.668 40.9 3.746 9.616 1.777 11.75 14.63 3.279 19.44C245 86.5 211.2 144.6 211.2 207.8c0 109.7 99.71 193 208.3 172.3 9.561-1.805 16.28 9.324 10.11 16.95C387.9 448.6 324.8 480 255.8 480 132.1 480 32 379.6 32 256z" / > < / svg > < / span > < / div > < div class = "items-center justify-center hidden w-12 h-12 dark:flex" title = "Switch to light appearance" > < span class = "relative inline-block align-text-bottom px-1 icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M256 159.1c-53.02.0-95.1 42.98-95.1 95.1s41.2 96.9 95.1 96.9 95.1-42.98 95.1-95.1S309 159.1 256 159.1zM509.3 347l-63.2-91.9 63.15-91.01c6.332-9.125 1.104-21.74-9.826-23.72l-109-19.7-19.7-109c-1.975-10.93-14.59-16.16-23.72-9.824L256 65.89 164.1 2.736c-9.125-6.332-21.74-1.107-23.72 9.824L121.6 121.6 12.56 141.3C1.633 143.2-3.596 155.9 2.736 164.1L65.89 256 2.74 347.01c-6.332 9.125-1.105 21.74 9.824 23.72l109 19.7 19.7 109c1.975 10.93 14.59 16.16 23.72 9.824L256 446.1l91.01 63.15c9.127 6.334 21.75 1.107 23.72-9.822l19.7-109 109-19.7C510.4 368.8 515.6 356.1 509.3 347zM256 383.1c-70.69.0-127.1-57.31-127.1-127.1.0-70.69 57.31-127.1 127.1-127.1S383.1 186.2 383.1 256c0 70.7-56.4 127.1-127.1 127.1z" / > < / svg > < / span > < / div > < / button > < / div > < / div > < / div > < / footer > < div id = search-wrapper class = "invisible fixed inset-0 z-50 flex h-screen w-screen cursor-default flex-col bg-neutral-500/50 p-4 backdrop-blur-sm dark:bg-neutral-900/50 sm:p-6 md:p-[10vh] lg:p-[12vh]" data-url = https://jpanther.github.io/congo/ > < div id = search-modal class = "flex flex-col w-full max-w-3xl min-h-0 mx-auto border rounded-md shadow-lg top-20 border-neutral-200 bg-neutral dark:border-neutral-700 dark:bg-neutral-800" > < header class = "relative z-10 flex items-center justify-between flex-none px-2" > < form class = "flex items-center flex-auto min-w-0" > < div class = "flex items-center justify-center w-8 h-8 text-neutral-400" > < span class = "relative inline-block align-text-bottom px-1 icon" > < svg aria-hidden = "true" focusable = "false" data-prefix = "fas" data-icon = "search" class = "svg-inline--fa fa-search fa-w-16" role = "img" xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 512 512" > < path fill = "currentcolor" d = "M505 442.7 405.3 343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3 44-79.7 44-128C416 93.1 322.9.0 208 0S0 93.1.0 208s93.1 208 208 208c48.3.0 92.7-16.4 128-44v16.3c0 6.4 2.5 12.5 7 17l99.7 99.7c9.4 9.4 24.6 9.4 33.9.0l28.3-28.3c9.4-9.4 9.4-24.6.1-34zM208 336c-70.7.0-128-57.2-128-128 0-70.7 57.2-128 128-128 70.7.0 128 57.2 128 128 0 70.7-57.2 128-128 128z" / > < / svg > < / span > < / div > < input type = search id = search-query class = "flex flex-auto h-12 mx-1 bg-transparent appearance-none focus:outlin
2023-06-04 00:38:53 +00:00
< span class = "relative inline-block align-text-bottom px-1 icon" > < svg xmlns = "http://www.w3.org/2000/svg" viewBox = "0 0 320 512" > < path fill = "currentcolor" d = "M310.6 361.4c12.5 12.5 12.5 32.75.0 45.25C304.4 412.9 296.2 416 288 416s-16.38-3.125-22.62-9.375L160 301.3 54.63 406.6C48.38 412.9 40.19 416 32 416S15.63 412.9 9.375 406.6c-12.5-12.5-12.5-32.75.0-45.25l105.4-105.4L9.375 150.6c-12.5-12.5-12.5-32.75.0-45.25s32.75-12.5 45.25.0L160 210.8l105.4-105.4c12.5-12.5 32.75-12.5 45.25.0s12.5 32.75.0 45.25l-105.4 105.4L310.6 361.4z" / > < / svg > < / span > < / button > < / header > < section class = "flex-auto px-2 overflow-auto" > < ul id = search-results > < / ul > < / section > < / div > < / div > < / div > < / body > < / html >
