Bolyai-Gerwien's Theorem

The square can be broken down into 7 triangles which, when put together differently, result in an equilateral triangle. (The single-colored squares can be broken down into 2 triangles)

The set of Bolyai Gerwien is a sentence from the geometry . He says that all flat polygons same acreage equidecomposable , many are so in last congruent triangles can be disassembled.

The sentence is sometimes also called Wallace-Bolyai-Gerwien's sentence. The Hungarian mathematician Wolfgang Bolyai and Paul Gerwien (then a lieutenant in a Prussian infantry regiment) proved the sentence, Gerwien in 1833. Wolfgang Bolyai published his studies in 1832/33 and also tried to include the case of any curvilinear area. The Scottish mathematician William Wallace is said to have found the solution earlier (1807).

Generalizations

The analogous statement for three- and higher-dimensional polyhedra does not apply. Polyhedra of the same volume with different strain invariants cannot be broken down into congruent simplices.

Others

At the end of the 19th century, the decomposition of polygons into other polygons of equal area was a frequent topic of popular puzzles.

literature

Paul Gerwien: Cutting up any number of the same straight-line figures into the same pieces , J. Pure Applied Math., Volume 10, 1833, pp. 228-234.
Ian Stewart From Here to Infinity , Oxford University Press 1996, pp. 169f
Max Zacharias elementary geometry , Encyclopaedia of Mathematical Sciences , Vol 3-1-2, p 917
Hugo Hadwiger, Paul Glur Decomposition Equality of Flat Polygons , Elements of Math, Vol. 6, 1951, pp. 97-106

Web links

Bolya-Gerwien theorem at cut the knot
Polygons - interactive demo of the Bolyai-Gerwien theorem

References and comments

↑ 22nd Prussian Infantry Regiment. Also teacher in the Royal Prussian Cadet Corps. He published another article in Crelles Journal (in the same volume, p. 235), in which he extends the sentence to include the sphere, and also, with H. von Holleben, task systems and collections from plane geometry for independent instruction in of analysis and prepared by law , 2 volumes, Berlin, Reimer 1831, 1832. H. von Holleben was also a lieutenant and teacher in the cadet corps.
↑ See Zacharias Elementarmathematik , Enzykl. Math. Wiss., P. 917
↑ Such a decomposition is shown in Ian Stewart From Here to Infinity , p. 170

[1] 22nd Prussian Infantry Regiment. Also teacher in the Royal Prussian Cadet Corps. He published another article in Crelles Journal (in the same volume, p. 235), in which he extends the sentence to include the sphere, and also, with H. von Holleben, task systems and collections from plane geometry for independent instruction in of analysis and prepared by law , 2 volumes, Berlin, Reimer 1831, 1832. H. von Holleben was also a lieutenant and teacher in the cadet corps.

[2] See Zacharias Elementarmathematik , Enzykl. Math. Wiss., P. 917

[3] Such a decomposition is shown in Ian Stewart From Here to Infinity , p. 170