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词条 Monsky's theorem
释义

  1. Proof

  2. Optimal dissections

  3. Generalizations

  4. References

In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area.[1] In other words, a square does not have an odd equidissection.

The problem was posed by Fred Richman in the American Mathematical Monthly in 1965, and was proved by Paul Monsky in 1970.[2][3][4]

Proof

Monsky's proof combines combinatorial and algebraic techniques, and in outline is as follows:

  1. Take the square to be the unit square with vertices at (0,0), (0,1), (1,0) and (1,1). If there is a dissection into n triangles of equal area then the area of each triangle is 1/n.
  2. Colour each point in the square with one of three colours, depending on the 2-adic valuation of its coordinates.
  3. Show that a straight line can contain points of only two colours.
  4. Use Sperner's lemma to show that every triangulation of the square into triangles meeting edge-to-edge must contain at least one triangle whose vertices have three different colours.
  5. Conclude from the straight-line property that a tricolored triangle must also exist in every dissection of the square into triangles, not necessarily meeting edge-to-edge.
  6. Use Cartesian geometry to show that the 2-adic valuation of the area of a triangle whose vertices have three different colours is greater than 1. So every dissection of the square into triangles must contain at least one triangle whose area has a 2-adic valuation greater than 1.
  7. If n is odd then the 2-adic valuation of 1/n is 1, so it is impossible to dissect the square into triangles all of which have area 1/n.&91;5&93;

Optimal dissections

{{main|Equidissection#Related problems}}

By Monsky's theorem it is necessary to have triangles with different areas to dissect a square into an odd number of triangles. Lower bounds for the area differences that must occur to dissect a square into an odd numbers of triangles and the optimal dissections have been studied.[6][7][8]

Generalizations

{{main|Equidissection}}

The theorem can be generalized to higher dimensions: an n-dimensional hypercube can only be divided into simplices of equal volume, if the number of simplices is a multiple of n!.[2]

References

1. ^{{cite book|title=Proofs from The Book|last1=Aigner|first1=Martin|last2=Ziegler|first2=Günter M.|publisher=Springer-Verlag|year=2010|isbn=978-3-642-00855-9|edition=4th|location=Berlin|pages=131–138|contribution=One square and an odd number of triangles|doi=10.1007/978-3-642-00856-6_20|author1-link=Martin Aigner|author2-link=Günter M. Ziegler}}
2. ^1 {{Cite techreport|url=https://math.berkeley.edu/~moorxu/oldsite/misc/equiareal.pdf|title=Sperner's Lemma|last=Xu|first=Moor|date=April 4, 2012|format=PDF|publisher=University of California, Berkeley}}
3. ^{{Cite journal|last1=Monsky|first1=P.|author-link=Paul Monsky|year=1970|title=On Dividing a Square into Triangles|jstor=2317329|journal=The American Mathematical Monthly|volume=77|issue=2|pages=161–164|doi=10.2307/2317329|mr=0252233}}
4. ^{{Cite journal|last1=Stein|first1=S.|year=2004|editor-last=Kleber|editor-first=M.|editor2-last=Vakil|editor2-first=R.|title=Cutting a Polygon into Triangles of Equal Areas|journal=The Mathematical Intelligencer|volume=26|pages=17–21|doi=10.1007/BF02985395}}
5. ^{{cite web|url=http://www.math.lsu.edu/~verrill/teaching/math7280/triangles.pdf|title=Dissecting a square into triangles|last=Verrill|first=H. A.|date=September 8, 2004|website=|publisher=Louisiana State University|format=PDF|archiveurl=https://web.archive.org/web/20100818142143/http://www.math.lsu.edu/~verrill/teaching/math7280/triangles.pdf|archivedate=August 18, 2010|deadurl=yes|accessdate=2010-08-18|df=}}
6. ^{{Citation|last=Mansow|first=K.|title=Ungerade Triangulierungen eines Quadrats von kleiner Diskrepanz (en. Odd triangulations of a square of small discrepancy)|year=2003|type=Diplomarbeit|location=Germany|publisher=TU Berlin}}
7. ^{{Cite journal|last=Schulze|first=Bernd|title=On the area discrepancy of triangulations of squares and trapezoids|date=1 July 2011|url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p137|journal=Electronic Journal of Combinatorics|volume=18|issue=1|page=#P137|zbl=1222.52017}}{{open access}}
8. ^{{Cite journal|last1=Labbé|first1=Jean-Philippe|last2=Rote|first2=Günter|last3=M. Ziegler|first3=Günter|author-link3=Günter M. Ziegler|year=2018|title=Area Difference Bounds for Dissections of a Square into an Odd Number of Triangles|url=|journal=Experimental Mathematics|volume=|pages=1–23|arxiv=1708.02891|doi=10.1080/10586458.2018.1459961|via=}}

3 : Euclidean plane geometry|Theorems in discrete geometry|Geometric dissection
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