Square trisection

In geometry, a square trisection consists of cutting a square into pieces that can be rearranged to form three identical squares.

Square trisection using 6 pieces of same area (2010).

Problem History

The dissection of a square in three congruent partitions is a geometrical problem that dates back to the Islamic Golden Age. Craftsman who mastered the art of zellige need innovative techniques to achieve their fabulous mosaics with complex geometric figures. Thus, the first solution to this problem has been proposed in the 10th century AD by the Persian mathematician Abu'l-Wafa' (940-998) in his treatise "On the geometric constructions necessary for the artisan".[1] Abu'l-Wafa' used also his dissection to demonstrate the Pythagoras' theorem.[2] This geometrical proof of Pythagoras' theorem will be rediscovered in the years 1835 - 1840 [3] by Henry Perigal and published in 1875.[4]

Search of optimality

The beauty of a dissection depends on several parameters. However, it is usual to search for solutions with the minimum of parts. Far from being minimal, the square trisection proposed by Abu'l-Wafa' uses 9 pieces. In the 14th century Abu Bakr al-Khalil gives two solutions, one of which uses 8 pieces.[5] In the late 17th century Jacques Ozanam comes back to this issue [6] and in the 19th century, solutions using 8 and 7 pieces are found, including one given by the mathematician Édouard Lucas.[7] It's in 1891 that Henry Perigal publish the first known solution with only 6 pieces [8] (see illustration below). Nowadays, new dissections are still found [9] (see illustration above) and the conjecture that 6 is the minimal number of necessary pieces remains unproved.

See also

Bibliography

References

  1. ↑ Alpay Özdural (1995). Omar Khayyam, Mathematicians, and “conversazioni” with Artisans. Journal of the Society of Architectural Vol. 54, No. 1, Mar., 1995
  2. ↑ Reza Sarhangi, Slavik Jablan (2006). Elementary Constructions of Persian Mosaics. Towson University and The Mathematical Institute. online
  3. ↑ See appendix of L. J. Rogers (1897). Biography of Henry Perigal: On certain Regular Polygons in Modular Network. Proceedings London Mathematical Society. Volume s1-29, Appendix pp. 732-735.
  4. ↑ Henry Perigal (1875). On Geometric Dissections and Transformations, Messenger of Mathematics, No 19, 1875.
  5. ↑ Alpay Özdural (2000). Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World, Historia Mathematica, Volume 27, Issue 2, May 2000, Pages 171-201.
  6. ↑ (fr) Jean-Etienne Montucla (1778), completed and re-edited by Jacques Ozanam (1640-1717) Récréations mathématiques, Tome 1 (1694), p. 297 Pl.15.
  7. ↑ (fr) Edouard Lucas (1883). Récréations Mathématiques, Volume 2. Paris, Gauthier-Villars. Second of four volumes. Second edition (1893) reprinted by Blanchard in 1960. See pp. 151 and 152 in Volume 2 of this edition. online (pp. 145-147).
  8. ↑ Henry Perigal (1891). Geometric Dissections and Transpositions, Association for the Improvement of Geometrical Teaching. wikisource
  9. ↑ Christian Blanvillain, János Pach (2010). Square Trisection. Bulletin d'Informatique Approfondie et Applications N°86 - Juin 2010 also at EPFL: oai:infoscience.epfl.ch:161493.

External links