# Dissection problem

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{{short description|Geometric problems involving the partition of a figure}}
{{CS1 config|mode=cs2}}
In [geometry](/source/geometry), a '''dissection problem ''' is the problem of partitioning a geometric [figure](/source/shape) (such as a [polytope](/source/polytope) or [ball](/source/Ball_(mathematics))) into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a '''dissection''' (of one polytope into another). It is usually required that the dissection use only a finite number of pieces. Additionally, to avoid [set-theoretic](/source/set-theoretic) issues related to the [Banach–Tarski paradox](/source/Banach%E2%80%93Tarski_paradox) and [Tarski's circle-squaring problem](/source/Tarski's_circle-squaring_problem), the pieces are typically required to be [well-behaved](/source/Pathological_(mathematics)). For instance, they may be restricted to being the [closures](/source/Closure_(topology)) of disjoint [open set](/source/open_set)s.

==Polygon dissection problem==
The [Bolyai–Gerwien theorem](/source/Bolyai%E2%80%93Gerwien_theorem) states that any [polygon](/source/polygon) may be dissected into any other polygon of the same area, using interior-disjoint polygonal pieces.  It is not true, however, that any [polyhedron](/source/polyhedron) has a dissection into any other polyhedron of the same volume using polyhedral pieces (see [Dehn invariant](/source/Dehn_invariant)). This process ''is'' possible, however, for any two [honeycombs](/source/Honeycomb_(geometry)) (such as [cube](/source/cube)) in three dimension and any two [zonohedra](/source/zonohedra) of equal volume (in any dimension).

A partition into [triangle](/source/triangle)s of equal area is called an [equidissection](/source/equidissection). Most polygons cannot be equidissected, and those that can often have restrictions on the possible numbers of triangles. For example, [Monsky's theorem](/source/Monsky's_theorem) states that there is no odd equidissection of a [square](/source/square).<ref>{{Citation |last=Stein |first=Sherman K. |date=March 2004 |title=Cutting a Polygon into Triangles of Equal Areas |journal=The Mathematical Intelligencer |volume=26 |issue=1 |pages=17–21 |doi=10.1007/BF02985395 |zbl=1186.52015|s2cid=117930135 }}</ref>

==Equilateral-triangle squaring problem==
thumb|Four-piece hinged dissection of an equilateral triangle into a square
Among [dissection puzzle](/source/dissection_puzzle)s, an example is the Haberdasher's Puzzle, posed by puzzle writer [Henry Dudeney](/source/Henry_Dudeney) in 1902.<ref>{{citation |first=Henry E. |last=Dudeney |title=Puzzles and Prizes |year=1902 |journal=[Weekly Dispatch](/source/Sunday_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.</ref>  It seeks a dissection from [equilateral triangle](/source/equilateral_triangle) into a [square](/source/square). Dudeney provided a [hinged dissection](/source/hinged_dissection) with four pieces. In 2024, [Erik Demaine](/source/Erik_Demaine), Tonan Kamata, and Ryuhei Uehara published a preprint claiming to prove that no dissection with fewer pieces exists.<ref>{{cite arXiv|title=Dudeney's Dissection is Optimal|first1=Erik D.|last1=Demaine|first2=Tonan|last2=Kamata|first3=Ryuhei|last3=Uehara|eprint=2412.03865|class=cs.CG|date=December 5, 2024|mode=cs2}}</ref><ref>{{cite journal
|journal=Scientific American
|title=Mathematicians Find Proof to 122-Year-Old Triangle-to-Square Puzzle
|author=Lyndie Chiou
|date=2025-03-27
|url=https://www.scientificamerican.com/article/mathematicians-find-proof-to-122-year-old-triangle-to-square-puzzle/
|editor=Clara Moskowitz
}}</ref>
This work was selected as one of [Scientific American](/source/Scientific_American)'s "The 10 Biggest Math Breakthroughs of 2025."<ref>{{Cite web |url=https://www.scientificamerican.com/article/the-top-10-math-discoveries-of-2025/?fbclid=IwY2xjawPN65FleHRuA2FlbQIxMABicmlkETFySUZ5ajB4YVp1eWVtMnBWc3J0YwZhcHBfaWQQMjIyMDM5MTc4ODIwMDg5MgABHplSIphSz9w3EiLatIwVUbI2rLI9rLXKSWM5enZQMggbcZfrQOo99dMayn1Q_aem_j256cI7dZx_pXv0O9oeoQQ |title=The 10 Biggest Math Breakthroughs of 2025 |website=SCIAM |publisher=Scientific American |date=2025-12-19 |accessdate=2026-01-09 |author=Clara Moskowitz |editor=Andrea Thompson}}</ref>。

==See also==
*[Hilbert's third problem](/source/Hilbert's_third_problem)

==References==
{{Reflist}}

==External links==
*[David Eppstein](/source/David_Eppstein), [http://www.ics.uci.edu/~eppstein/junkyard/distile/ Dissection Tiling].
*Oisín Flynn-Connolly, [https://irma.math.unistra.fr/~dotsenko/teaching/files/MA341C-1819/341CR6-2.pdf One square and an odd number of triangles]
Category:Discrete geometry
Category:Euclidean geometry
Category:Geometric dissection
Category:Polygons
Category:Polyhedra
Category:Polytopes
Category:Mathematical problems

{{geometry-stub}}

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Adapted from the Wikipedia article [Dissection problem](https://en.wikipedia.org/wiki/Dissection_problem) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Dissection_problem?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.