Up to

<h2 id="examples">Examples</h2>
<h3>Tetris</h3>

<p>Consider the seven <a href="/facts/Tetris/djLfTK0n">Tetris</a> pieces (I, J, L, O, S, T, Z), known mathematically as the <a href="/facts/Tetromino/QR8rjoaw">tetrominoes</a>. If you consider all the possible rotations of these pieces — for example, if you consider the "I" oriented vertically to be distinct from the "I" oriented horizontally — then you find there are 19 distinct possible shapes to be displayed on the screen. (These 19 are the so-called "fixed" tetrominoes.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>) But if rotations are not considered distinct — so that we treat both "I vertically" and "I horizontally" indifferently as "I" — then there are only seven. We say that "there are seven <a href="/facts/Tetromino/QR8rjoaw">tetrominoes</a>, up to rotation". One could also say that "there are five tetrominoes, up to rotation and reflection", which accounts for the fact that L reflected gives J, and S reflected gives Z.
</p>
<h3>Eight queens</h3>

<p>In the <a href="/facts/Eight_queens_puzzle/XDx2HwlB">eight queens puzzle</a>, if the queens are considered to be distinct (e.g. if they are colored with eight different colors), then there are 3709440 distinct solutions. Normally, however, the queens are considered to be interchangeable, and one usually says "there are 3,709,440 / 8! = 92 unique solutions up to <a href="/facts/Permutation/JSw9ukmv">permutation</a> of the queens", or that "there are 92 solutions modulo the names of the queens", signifying that two different arrangements of the queens are considered equivalent if the queens have been permuted, as long as the set of occupied squares remains the same.
</p><p>If, in addition to treating the queens as identical, <a href="/facts/Rotation/yChkz49x">rotations</a> and <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflections</a> of the board were allowed, we would have only 12 distinct solutions "up to <a href="/facts/Symmetry/hpu7DK8p">symmetry</a> and the naming of the queens". For more, see <a href="/facts/Eight_queens_puzzle/XDx2HwlB">Eight queens puzzle § Solutions</a>.
</p>
<h3>Polygons</h3>
<p>The <a href="/facts/Regular_polygon/ws89Nhpt">regular n-gon</a>, for a fixed n, is unique up to <a href="/facts/Similarity_(geometry)/Sbd6hEAN">similarity</a>. In other words, the "similarity" equivalence relation over the regular n-gons (for a fixed n) has only one equivalence class; it is impossible to produce two regular n-gons which are not similar to each other.
</p>
<h3>Group theory</h3>
<p>In <a href="/facts/Group_theory/NfowcI5H">group theory</a>, one may have a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">acting</a> on a set X, in which case, one might say that two elements of X are equivalent "up to the group action"—if they lie in the same <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">orbit</a>.
</p><p>Another typical example is the statement that "there are two different <a href="/facts/Group_(mathematics)/IQTYqINt">groups</a> of order 4 up to <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>", or "modulo isomorphism, there are two groups of order 4". This means that, if one considers <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphic</a> groups "equivalent", there are only two equivalence classes of groups of order 4.
</p>
<h3>Nonstandard analysis</h3>
<p>A <a href="/facts/Hyperreal_number/Kvf09qpo">hyperreal</a> x and its <a href="/facts/Standard_part_function/fTuL5IXj">standard part</a> st(<i>x</i>) are equal up to an <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a> difference.
</p>
<h2 id="see-also">See also</h2>

Look up <i>up to</i> in Wiktionary, the free dictionary.

<ul><li><a href="/facts/Abuse_of_notation/D4W6n85F">Abuse of notation</a></li>
<li><a href="/facts/Adequality/nq5odBwa">Adequality</a></li>
<li><a href="/facts/Essentially_unique/P69grpIK">Essentially unique</a></li>
<li><a href="/facts/List_of_mathematical_jargon/fTRoAtn2">List of mathematical jargon</a></li>
<li><a href="/facts/Modulo_(jargon)/lg6V4FF9">Modulo (jargon)</a></li>
<li><a href="/facts/Quotient_group/HlwpHtjb">Quotient group</a></li>
<li><a href="/facts/Quotient_set/1d2sh0TB">Quotient set</a></li></ul>

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

<ol>
<li id="fn:1"><p>Nekovář, Jan (2011). "Mathematical English (a brief summary)" (PDF). Institut de mathématiques de Jussieu – Paris Rive Gauche. Retrieved 2024-02-08. <a href="https://webusers.imj-prg.fr/~jan.nekovar/co/en/en.pdf" target="_blank">https://webusers.imj-prg.fr/~jan.nekovar/co/en/en.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Weisstein, Eric W. "Tetromino". MathWorld. Retrieved 2023-09-26. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

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