Decomposition into triangles of equal area

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Division of a square into six triangles of equal area

The division into triangles of equal area (also cross- sectioning ) is a problem of planar geometry . Among other things, it is investigated whether it is even possible to break down a given polygon into triangles of equal area .

Research into this problem began in the late 1960s with Monsky's theorem that a square cannot be broken down into an odd number of triangles of equal area. The proof uses valuation theory and is the only known proof for this theorem so far.

In fact, most polygons cannot be broken down into triangles of equal area. The question arises: Which polygons can be broken down into how many parts of the same area? Trapeze , dragon squares , regular polygons , point-symmetrical polygons and polyominos as well as the decomposition of hypercubes into simplices were examined in particular . In the case of regular, n-angled polygons with n ≥ 5, Elaine Kasimatis showed that these can only be broken down into m triangles of equal area if m is a multiple of n. For n = 3 or n = 4 this is obviously not correct: a square can be divided into two equal triangles and a triangle into any number.

Decompositions into triangles of equal area have only a few direct applications. But they are considered interesting because the results often contradict expectations at first glance and the theory for a geometrical problem with such a simple definition requires surprisingly sophisticated algebraic tools. Many results are based on the application of valuation theory to the real numbers and the coloring in graph theory based on Sperner's lemma .

Web links

Commons : Breakdown into triangles of equal area  - collection of images, videos and audio files

Individual evidence

  1. Victor Klee , Stan Wagon: Old and new unsolved problems in number theory and geometry of the plane , Birkhäuser, 1997, p. 37 (translation of Old and new unsolved problems in plane geometry and number theory , 1991, from the American by Manfred Stern )
  2. ^ Paul Monsky : On dividing a square into triangles , The American Mathematical Monthly 77 (2), February 1970, pp. 161-164, doi: 10.2307 / 2317329 (English; Zbl 0187.19701 ), reprinted as Paul Monsky: On dividing a square into triangles , Selected Papers on Algebra, Raymond W. Brink selected mathematical papers 3, Mathematical Association of America, July 1977, pp. 249-251, ISBN 0-88385-203-9 (English)
  3. ^ According to Monsky, the problem goes back to Fred Richman, John Thomas: Problem 5479, American Mathematical Monthly 74, 1967, 329. John Thomas: A dissection problem , Mathematics Magazine 41, 1968, pp. 187-190, proved it in a special case.
  4. Elaine A. Kasimatis, Sherman K. Stein: Equidissections of polygons , Discrete Mathematics 85 (3), December 1, 1990, pp. 281-294, doi: 10.1016 / 0012-365X (90) 90384-T (English; Zbl 0736.05028 )
  5. Sherman K. Stein: Cutting a polygon into triangles of equal areas , The Mathematical Intelligencer 26 (1), March 2004, pp. 17-21, doi: 10.1007 / BF02985395 (English; Zbl 1186.52015 )
  6. Elaine A. Kasimatis: Dissection of regular polygons into triangles of equal areas , Discrete & Computational Geometry 4, 1989, pp. 375-381 (English)
  7. Sherman K. Stein, Sándor Szabó: Tiling by triangles of equal areas , Chapter 5 in Algebra and tiling: homomorphisms in the service of geometry , The Carus Mathematical Monographs 25, Mathematical Association of America, 2008, pp. 107-134, ISBN 978-0-88385-041-1 (English; Zbl 0930.52003 )
  8. Sherman K. Stein: Cutting a polygon into triangles of equal areas , The Mathematical Intelligencer 26 (1), March 2004, pp. 17-21, doi: 10.1007 / BF02985395 (English; Zbl 1186.52015 )