Dissection problem

<h2 id="polygon-dissection-problem">Polygon dissection problem</h2>
<p>The <a href="/facts/Bolyai%25E2%2580%2593Gerwien_theorem/QbGcuvmS">Bolyai–Gerwien theorem</a> states that any <a href="/facts/Polygon/yAq7RjPz">polygon</a> may be dissected into any other polygon of the same area, using interior-disjoint polygonal pieces.  It is not true, however, that any <a href="/facts/Polyhedron/UC0p9FTF">polyhedron</a> has a dissection into any other polyhedron of the same volume using polyhedral pieces (see <a href="/facts/Dehn_invariant/eahMFAAb">Dehn invariant</a>). This process <i>is</i> possible, however, for any two <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycombs</a> (such as <a href="/facts/Cube/65lUaAqN">cube</a>) in three dimension and any two <a href="/facts/Zonohedra/w16kH9H3">zonohedra</a> of equal volume (in any dimension).
</p><p>A partition into <a href="/facts/Triangle/cRXUwsMn">triangles</a> of equal area is called an <a href="/facts/Equidissection/ad58TGBX">equidissection</a>. Most polygons cannot be equidissected, and those that can often have restrictions on the possible numbers of triangles. For example, <a href="/facts/Monsky%2527s_theorem/tgYxLBJD">Monsky's theorem</a> states that there is no odd equidissection of a <a href="/facts/Square/JHJeMvmL">square</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
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<h2 id="equilateral-triangle-squaring-problem">Equilateral-triangle squaring problem</h2>

<p>Among <a href="/facts/Dissection_puzzle/tuKnFpL5">dissection puzzles</a>, an example is the Haberdasher's Puzzle, posed by puzzle writer <a href="/facts/Henry_Dudeney/YSSBQgoK">Henry Dudeney</a> in 1902.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>  It seeks a dissection from <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a> into a <a href="/facts/Square/JHJeMvmL">square</a>. Dudeney provided a <a href="/facts/Hinged_dissection/2Eq6mkIx">hinged dissection</a> with four pieces. In 2024, <a href="/facts/Erik_Demaine/qby54dWx">Erik Demaine</a>, Tonan Kamata, and Ryuhei Uehara published a preprint claiming to prove that no dissection with fewer pieces exists.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
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<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Hilbert%2527s_third_problem/UcS6SWVR">Hilbert's third problem</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/David_Eppstein/hIjTU6nC">David Eppstein</a>, <a href="http://www.ics.uci.edu/~eppstein/junkyard/distile/">Dissection Tiling</a>.</li></ul>

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

<ol>
<li id="fn:1"><p>Stein, Sherman K. (March 2004), "Cutting a Polygon into Triangles of Equal Areas", The Mathematical Intelligencer, 26 (1): 17–21, doi:10.1007/BF02985395, S2CID 117930135, Zbl 1186.52015 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Dudeney, Henry E. (1902), "Puzzles and Prizes", Weekly Dispatch - The puzzle appeared in the April 6 issue of this column. A discussion followed on April 20, and the solution appeared on May 4. <a href="/wiki/Sunday_Dispatch" target="_blank">/wiki/Sunday_Dispatch</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Demaine, Erik D.; Kamata, Tonan; Uehara, Ryuhei (December 5, 2024), "Dudeney's Dissection is Optimal", arXiv:2412.03865 [cs.CG]{{cite arXiv}}:  CS1 maint: overridden setting (link) <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Lyndie Chiou (2025-03-27), Clara Moskowitz (ed.), "Mathematicians Find Proof to 122-Year-Old Triangle-to-Square Puzzle", Scientific American <a href="https://www.scientificamerican.com/article/mathematicians-find-proof-to-122-year-old-triangle-to-square-puzzle/" target="_blank">https://www.scientificamerican.com/article/mathematicians-find-proof-to-122-year-old-triangle-to-square-puzzle/</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Dissection problem open-in-new

Dissection problem