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Ring of Four Almonds and the Omar Khayyam's Triangle in Islamic Art Design

이슬람 예술 디자인에서 회전하는 알몬드와 오마르 하얌의 삼각형

  • Received : 2019.07.04
  • Accepted : 2019.08.20
  • Published : 2019.08.31

Abstract

In this paper, we examine the brief history of the ring of four almonds regarding Mesopotamian mathematics, and present reasons why the Omar Khayyam's triangle, a special right triangle in a ring of four almonds, was essential for artisans due to its unique pattern. We presume that the ring of four almonds originated from a point symmetry figure given two concentric squares used in the proto-Sumerian Jemdet Nasr period (approximately 3000 B.C.) and a square halfway between two given concentric squares used during the time of the Old Akkadian period (2340-2200 B.C.) and the Old Babylonian age (2000-1600 B.C.). Artisans tried to create a new intricate pattern as almonds and 6-pointed stars by subdividing right triangles in the pattern of the popular altered Old Akkadian square band at the time. Therefore, artisans needed the Omar Khayyam's triangle, whose hypotenuse equals the sum of the short side and the perpendicular to the hypotenuse. We presume that artisans asked mathematicians how to construct the Omar Khayyam's triangle at a meeting between artisans and mathematicians in Isfahan. The construction of Omar Khayyam's triangle requires solving an irreducible cubic polynomial. Omar Khayyam was the first to classify equations of integer polynomials of degree up to three and then proceeded to solve all types of cubic equations by means of intersections of conic sections. Omar Khayyam's triangle gave practical meaning to the type of cubic equation $x^3+bx=cx^2+a$. The work of Omar Khayyam was completed by Descartes in the 17th century.

Acknowledgement

Supported by : Inha University

References

  1. A. AMIR-MOEZ, A paper of Omar Khayyam, Scripta Mathematica 26(4) (1962), 323-337.
  2. M. BERNAL, Black Athena: The afroasiatic roots of classical civilization, Vol. I, 10th ed. New Jersey: Rutgers University Press, 2003.
  3. M. BULATOV, Cosmos and architecture, Tashkent: SMI-ASIA, 2009.
  4. J. BONNER, Islamic geometric patterns: Their historical development and traditional methods of construction, New York: Springer, 2017.
  5. C. BOYER and U. MERZBACH, A history of mathematics, New Jersey: Wiley, 2011.
  6. Chinese Text Project: https://ctext.org.
  7. W. CHORBACHI, In the tower of babel: Beyond symmetry in Islamic design, Computers Math. Applic. 17(4-6) (1989), 751-789. https://doi.org/10.1016/0898-1221(89)90260-5
  8. K. CRITCHLOW, Islamic patterns: An analytical and cosmological approach, Vermont: Inner Traditions, 1976.
  9. P. CROMWELL and E. BELTRAMI, The whirling kites of Isfahan: Geometric variations on a theme, The Mathematical Intelligencer 33(3) (2011), 84-93. https://doi.org/10.1007/s00283-011-9225-4
  10. S. DEHKORDI, Iranian Seljuk architecture with an emphasis on decorative brickwork of the Qazvin Kharaqan towers, Journal of History Culture and Art Research 5(4) (2016), 384-394. https://doi.org/10.7596/taksad.v5i4.613
  11. H. EVES, An introduction to the history of mathematics, 6th ed. New York: Saunders College Publishing, 1990.
  12. I. EL-SAID and A. PARMAN, Geometric concepts in Islamic art, Kent: Westerham Press Ltd, 1976.
  13. J. FRIBERG, A remarkable collection of Babylonian mathematical text, New York: Springer, 2007.
  14. J. FRIBERG, Amazing traces of a Babylonian origin in Greek mathematics, New Jersey: World Scientific, 2007.
  15. T. HEATH(Ed), The works of Archimedes, New York: Dover, 2002.
  16. P. HUBER, Zu einem mathematischen Keilschrifttext (VAT 8512), Isis 46(2) (1955), 104-106. https://doi.org/10.1086/348404
  17. G. JAMES, Stolen legacy: The Egyptian origins of Western philosophy, CA: A Traffic Output Publication, 2015.
  18. V. KATZ, A history of mathematics, 3rd ed. New York: Addison-Wesley, 2009.
  19. M. KLINE, Mathematical thought: From ancient to modern times, Oxford: Oxford University Press, 1972.
  20. G. MARTIN, Transformation geometry: An introduction to symmetry, New York: Springer, 1982.
  21. G. NECIPOGLU(Ed.), The arts of ornamental geometry: A Persian compendium on similar and complementary interlocking figures, Leiden: Brill, 2017.
  22. A. OZDURAL, Omar KHAYYAM, Mathematicians, and conversazioni with Artisans, Journal of the Society of Architectural Historians 54(1) (1995), 54-71. https://doi.org/10.2307/991025
  23. A. OZDURALl, On Interlocking similar or corresponding figures and ornamental patterns of cubic equations, Muqarnas 13 (1996), 191-211. https://doi.org/10.1163/22118993-90000364
  24. A. OZDURALl, A mathematical sonata for architecture: Omar Khayyam and the Friday Mosque of Isfahan, Technology and Culture 39(4) (1998), 699-751. https://doi.org/10.1353/tech.1998.0099
  25. A. OZDURALl, Mathematics and art: Connections between theory and practice in the Medieval Islamic world, Historia Mathematica 27 (2000), 171-201. https://doi.org/10.1006/hmat.1999.2274
  26. J. PARK, Cultural and mathematical meanings of regular octagons in Mesopotamia: Examining Islamic art designs, Journal of History Culture and Art Research 7(1) (2018), 301-318. https://doi.org/10.7596/taksad.v7i1.1354
  27. R. RASHED, Classical mathematics from al-Khwarizmi to Descarte, London and New York: Routledge, 2015.
  28. R. RASHED and B. VAHABZADEH, Omar Khayyam, the Mathematician, New York: Bibliotheca Persica Press, 2000.
  29. J. ROTMAN, Advanced modern algebra, 2nd ed. Rhode Island: American Mathematical Society, 2002.
  30. G. TOOMER(Tr.), Ptolemy's Almagest, New Jersey: Princeton University Press, 1998.