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Funding for this project generously provided by Overdeck Family Foundation

around 1900–1600 BCE

Susa Mathematical Tablets

Babylonian approximation to π based on a regular hexagon

In 1933, a total of 26 mathematical tablets were excavated some two hundred miles from Babylon in the ancient Elamite city of Susa. The tablets are believed to date from between 1900 and 1600 BCE and contain a number of interesting geometric figures and mathematical constants arising from geometric computations. Partial translations of the tablets were published in 1950, but the first detailed analysis of three particularly interesting tablets did not appear until 1961.

Susa Mathematical Tablets

The tablets referred to as TMS 2 and TMS 3 are of particular interest. TMS 2 is relatively straightforward, containing a diagram of a regular hexagon and markings that give an approximation to the area of an equilateral triangle. TMS 3, on the other hand, contains a tabular list of geometric constants corresponding to many shapes, including polygons but also featuring many more complicated geometric shapes, including the semicircle (termed a crescent) and lens-shaped figures known to the Babylonians as the ox-eye and grain figure (or grain-field, or barleycorn). The tabulation includes 68 constants spread over the 71 lines. On line 30, the perimeter-to-circumcircle circumference ratio for a regular hexagon is given as 24/25, which implies perhaps the world's oldest recorded approximation to π with value 3 + 1/8 = 3.125.

Original artifact location

Susa (historical name), Shush, Iran (current name)

Current artifact location

The Louvre, Paris


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Interactive Content

Computational Explanation

Other Resources

Additional Reading

  • Beckmann, P. A History of Pi. New York: Barnes & Noble, pp. 21–22, 1993.
  • Bruins, E. M. "Quelques textes mathématiques de la Mission de Suse." Proc. Roy. Dutch Acad. Sci., Vol. 53, pp. 1025–1033, 1950.
  • Bruins E. M. and Rutten, M. Mémoires de la Mission Archéologique en Iran, Tome XXXIV: Textes mathématiques de Suse. Paris: Librairie Orientaliste Paul Geuthner, pp. 26 and 33 and Pls. IV and 4, 1961.
  • Friberg, J. A Remarkable Collection of Babylonian Mathematical Texts: Manuscripts in the Schøyen Collection: Cuneiform Texts I. New York: Springer, pp. 217–218, 2007.
  • Friberg, J. and Al-Rawi, F. N. H. New Mathematical Cuneiform Texts. Springer, pp. 256–257, 325, and 396, 2016.
  • Harper, P. O.; Aruz, J. and Tallon, F. (Eds.). The Royal City of Susa: Ancient Near Eastern Treasures in the Louvre. New York: The Metropolitan Museum of Art, pp. 276–278, 1992.
  • Nemet-Nejat, K. R. Cuneiform Mathematical Texts as a Reflection of Everyday Life in Mesopotamia. New Haven, CT: American Oriental Society, p. 274, 1993.
  • Neugebauer, O. The Exact Sciences in Antiquity, 2nd ed. New York: Dover, pp. 46–48, 1969.
  • Park, J. "Cultural and Mathematical Meanings of Regular Octagons in Mesopotamia: Examining Islamic Art Designs." Journal of History Culture and Art Research, Vol. 7, No. 1, pp. 301–318, 2018.
  • Robson, E. Mesopotamian Mathematics, 2100-1600 BC: Technical Constants in Bureaucracy and Education. Oxford, England: Clarendon Press, pp. 19–21, 48–50 and 199–201, 1999.
  • Sarton, G. Ancient Science Through the Golden Age of Greece. New York: Dover, p. 74, 1993.