The theorem on cutting a square into isometric triangles

Dividing a square into 6 equal triangles.

The theorem on cutting a square into equal triangles says that it is impossible to cut a square into an odd number of triangles of the same area ^[1] .

The theorem is famous for its unexpected proof using a 2-adic norm .

Content

History

The problem was posed by Fred Richman in the American Mathematical Monthly in 1965 and solved by Paul Monsky in 1970 ^[2] .

About proof

Using 2-adic numbers , a certain coloring of the points of the unit square in three colors is constructed.

The main properties of the coloring are as follows:

The area of any triangle with vertices of different colors cannot be expressed as a fraction with an odd numerator and denominator.
- In particular, if there were a partition of the square into an odd number of equal triangles, then none of the triangles would have vertices of all three colors.
Any line is painted exactly two colors.

This and some other properties of this coloring lead to a contradiction with Sperner's lemma .

Variations and generalizations

$N$ ${\ displaystyle N}$ -dimensional cube can be divided into simplexes of the same volume only if the number of simplexes is a multiple $N!$ ${\ displaystyle N!}$ ^[3] ^[4] .
The proof of the theorem also implies the existence of quadrangles that do not allow cutting into equal triangles.
For an integer $n\geqslant 5$ ${\ displaystyle n \ geqslant 5}$ , right $n$ ${\ displaystyle n}$ -gon can be cut into $m$ ${\ displaystyle m}$ equal triangles if and only if $m$ ${\ displaystyle m}$ divided by $n$ ${\ displaystyle n}$ ^[5] .

Notes

↑ Martin Aigner, Günter M. Ziegler. One square and an odd number of triangles // Proofs from The Book. - 4th. - Berlin, 2010 .-- S. 131–138. - ISBN 978-3-642-00856-6 . - DOI : 10.1007 / 978-3-642-00856-6_20 .
↑ P. Monsky. On Dividing a Square into Triangles // The American Mathematical Monthly : journal. - 1970. - Vol. 77 , no. 2 . - P. 161-164 . - DOI : 10.2307 / 2317329 . MR : 0252233 .
↑ Mead, David G. (September 1979), " Dissection of the hypercube into simplexes ", Proceedings of the American Mathematical Society T. 76: 302–304 , DOI 10.1090 / S0002-9939-1979-0537093-6
↑ Sperner's Lemma , Moor Xu
↑ EA Kasimatis, Dissections of regular polygons into triangles of equal areas, Discrete & Computational Geometry, August 1989, Volume 4, Issue 4, pp 375–381

Literature

B. Becker, S. Vostokov, Yu. Ionin. 2-adic numbers // Quantum . - 1979.- T. 2 . - S. 26-31 .

[1] Martin Aigner, Günter M. Ziegler. One square and an odd number of triangles // Proofs from The Book. - 4th. - Berlin, 2010 .-- S. 131–138. - ISBN 978-3-642-00856-6 . - DOI : 10.1007 / 978-3-642-00856-6_20 .

[2] P. Monsky. On Dividing a Square into Triangles // The American Mathematical Monthly : journal. - 1970. - Vol. 77 , no. 2 . - P. 161-164 . - DOI : 10.2307 / 2317329 . MR : 0252233 .

[3] Mead, David G. (September 1979), " Dissection of the hypercube into simplexes ", Proceedings of the American Mathematical Society T. 76: 302–304 , DOI 10.1090 / S0002-9939-1979-0537093-6

[xu-4] Sperner's Lemma , Moor Xu

[5] EA Kasimatis, Dissections of regular polygons into triangles of equal areas, Discrete & Computational Geometry, August 1989, Volume 4, Issue 4, pp 375–381