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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Unit cube</span></span>
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<p>A <b>unit cube</b>, more formally a <b>cube of side 1</b>, is a <a href="Cube" title="Cube">cube</a> whose sides are 1 unit long.<sup id="cite_ref-pcm_1-0" class="reference"><a href="#cite_note-pcm-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cbm_2-0" class="reference"><a href="#cite_note-cbm-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.<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>
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<div class="mw-heading mw-heading2"><h2 id="Unit_hypercube">Unit hypercube</h2></div>
<p>The term <i>unit cube</i> or <b>unit hypercube</b> is also used for <a href="Hypercube" title="Hypercube">hypercubes</a>, or "cubes" in <a href="N-dimensional_space" class="mw-redirect" title="N-dimensional space"><i>n</i>-dimensional spaces</a>, for values of <i>n</i> other than 3 and edge length 1.<sup id="cite_ref-pcm_1-1" class="reference"><a href="#cite_note-pcm-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cbm_2-1" class="reference"><a href="#cite_note-cbm-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>Sometimes the term "unit cube" refers in specific to the set [0, 1]<sup><i>n</i></sup> of all <i>n</i>-tuples of numbers in the interval [0, 1].<sup id="cite_ref-pcm_1-2" class="reference"><a href="#cite_note-pcm-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>The length of the longest diagonal of a unit hypercube of <i>n</i> dimensions is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}}">
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<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}}</annotation>
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</math></span><img src="./2a2994734eae382ce30100fb17b9447fd8e99f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}}" loading="lazy"></span>, the square root of <i>n</i> and the (Euclidean) length of the vector (1,1,1,....1,1) in <i>n</i>-dimensional space.<sup id="cite_ref-cbm_2-2" class="reference"><a href="#cite_note-cbm-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Doubling_the_cube" title="Doubling the cube">Doubling the cube</a></li>
<li><a href="K-cell_(mathematics)" class="mw-redirect" title="K-cell (mathematics)"><i>k</i>-cell</a></li>
<li><a href="Robbins_constant" class="mw-redirect" title="Robbins constant">Robbins constant</a>, the average distance between two random points in a unit cube</li>
<li><a href="Tychonoff_cube" title="Tychonoff cube">Tychonoff cube</a>, an infinite-dimensional analogue of the unit cube</li>
<li><a href="Unit_square" title="Unit square">Unit square</a></li>
<li><a href="Unit_sphere" title="Unit sphere">Unit sphere</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFBall2010" class="citation cs2">Ball, Keith (2010), "High-dimensional geometry and its probabilistic analogues", in <a href="Timothy_Gowers" title="Timothy Gowers">Gowers, Timothy</a> (ed.), <i><a href="The_Princeton_Companion_to_Mathematics" title="The Princeton Companion to Mathematics">The Princeton Companion to Mathematics</a></i>, Princeton University Press, pp. <span class="nowrap">670–</span>680, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781400830398</bdi></cite>. See in particular <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZOfUsvemJDMC&pg=PA671">p. 671</a>.</span>
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<li id="cite_note-cbm-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-cbm_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-cbm_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-cbm_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFGardner2001" class="citation cs2">Gardner, Martin (2001), "Chapter 13: Hypercubes", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=orz0SDEakpYC&pg=PA162"><i>The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems : Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics</i></a>, W. W. Norton & Company, pp. <span class="nowrap">162–</span>174, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780393020236</bdi></cite>.</span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation cs2"><i>Geometry: Reteaching Masters</i>, Holt Rinehart & Winston, 2001, p. 74, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780030543289</bdi></cite>.</span>
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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><span class="citation mathworld" id="Reference-Mathworld-Unit_cube"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/UnitCube.html">"Unit cube"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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