imported>David Eppstein: /* Equilateral-triangle squaring problem */ Image violates both WP:WATERMARK and MOS:ANIMATION; swap out with another image that has neither problem

2025-04-02T05:05:22Z

Equilateral-triangle squaring problem: Image violates both WP:WATERMARK and MOS:ANIMATION; swap out with another image that has neither problem

New page

{{short description|Geometric problems involving the partition of a figure}}
{{CS1 config|mode=cs2}}
In [[geometry]], a '''dissection problem ''' is the problem of partitioning a geometric [[shape|figure]] (such as a [[polytope]] or [[Ball (mathematics)|ball]]) 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]] issues related to the [[Banach–Tarski paradox]] and [[Tarski's circle-squaring problem]], the pieces are typically required to be [[Pathological (mathematics)|well-behaved]]. For instance, they may be restricted to being the [[Closure (topology)|closures]] of disjoint [[open set]]s.

==Polygon dissection problem==
The [[Bolyai–Gerwien theorem]] states that any [[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]] has a dissection into any other polyhedron of the same volume using polyhedral pieces (see [[Dehn invariant]]). This process ''is'' possible, however, for any two [[Honeycomb (geometry)|honeycombs]] (such as [[cube]]) in three dimension and any two [[zonohedra]] of equal volume (in any dimension).

A partition into [[triangle]]s of equal area is called an [[equidissection]]. Most polygons cannot be equidissected, and those that can often have restrictions on the possible numbers of triangles. For example, [[Monsky's theorem]] states that there is no odd equidissection of a [[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==
[[File:Hinged haberdasher square.svg|thumb|Four-piece hinged dissection of an equilateral triangle into a square]]
Among [[dissection puzzle]]s, an example is the Haberdasher's Puzzle, posed by puzzle writer [[Henry Dudeney]] in 1902.<ref>{{citation |first=Henry E. |last=Dudeney |title=Puzzles and Prizes |year=1902 |journal=[[Sunday Dispatch|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.</ref> It seeks a dissection from [[equilateral triangle]] into a [[square]]. Dudeney provided a [[hinged dissection]] with four pieces. In 2024, [[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>

==See also==
*[[Hilbert's third problem]]

==References==
{{Reflist}}

==External links==
*[[David Eppstein]], [http://www.ics.uci.edu/~eppstein/junkyard/distile/ Dissection Tiling].

[[Category:Discrete geometry]]
[[Category:Euclidean geometry]]
[[Category:Geometric dissection]]
[[Category:Polygons]]
[[Category:Polyhedra]]
[[Category:Polytopes]]
[[Category:Mathematical problems]]

{{geometry-stub}}

Dissection problem - Revision history

imported>David Eppstein: /* Equilateral-triangle squaring problem */ Image violates both WP:WATERMARK and MOS:ANIMATION; swap out with another image that has neither problem