Unit square

<h2 id="cartesian-coordinates">Cartesian coordinates</h2>
<p>In a <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">Cartesian coordinate system</a> with coordinates (<i>x</i>, <i>y</i>), a unit square is defined as a <a href="/facts/Square_(geometry)/JHJeMvmL">square</a> consisting of the points where both x and y lie in a closed <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> from 0 to 1.
</p><p>That is, a unit square is the <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> <i>I</i> × <i>I</i>, where I denotes the closed unit interval.
</p>
<h2 id="complex-coordinates">Complex coordinates</h2>
<p>The unit square can also be thought of as a subset of the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>, the topological space formed by the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>.
In this view, the four corners of the unit square are at the four complex numbers 0, 1, i, and 1 + <i>i</i>.
</p>
<h2 id="rational-distance-problem">Rational distance problem</h2>

Unsolved problem in mathematics
Is there a point in the plane at a rational distance from all four corners of a unit square?
<a href="/facts/List_of_unsolved_problems_in_mathematics/RroxGPpP">More unsolved problems in mathematics</a>

<p>It is not known whether any point in the plane is a <a href="/facts/Rational_number/VJQmyY05">rational</a> distance from all four vertices of the unit square.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Unit_circle/VsTffjPK">Unit circle</a></li>
<li><a href="/facts/Unit_cube/S52tnk6m">Unit cube</a></li>
<li><a href="/facts/Unit_sphere/5UeoTcug">Unit sphere</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/UnitSquare.html">"Unit square"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Horn, Alastair N. (1991), "IFSs and the Interactive Design of Tiling Structures", in Crilly, A. J.; Earnshow, R. A.; Jones, H. (eds.), Fractals and Chaos, Springer-Verlag, p. 136, doi:10.1007/978-1-4612-3034-2, ISBN 978-1-4612-7770-5 <a href="978-1-4612-7770-5" target="_blank">978-1-4612-7770-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Guy, Richard K. (1991), Unsolved Problems in Number Theory, Problem Books in Mathematics, vol. 1 (2nd ed.), Springer-Verlag, pp. 181–185, doi:10.1007/978-1-4899-3585-4, ISBN 978-1-4899-3587-8 <a href="978-1-4899-3587-8" target="_blank">978-1-4899-3587-8</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Unit square open-in-new

Unit square