Symmetry in an Old Babylonian geometry problem

25 October 2025

The #Mathober 2025 day 25 prompt is ‘Wedge’, so I would like to point out a little historical curiosity involving ‘wedge’.

A detail from an Old Babylonian clay tablet. Above, a partial geometrical diagram shown (and described in the alt text of) the other attachment. Below, a few lines of cuneiform text.

The images show a detail from an Old Babylonian clay tablet of geometrical problems and a reconstruction of the diagram.

The cuneiform text reads:

‘The square-side is 1 cable. ⟨Inside it⟩ I drew 12 wedges and 4 squares. What are their areas?’ (trans. Robson, ‘Mesopotamian mathematics’, p.95)1

The term ‘wedge’ translates the Akkadian ‘santakkum’, which names any figure with three (possibly non-straight) sides. (1 ‘cable’ = approximately 360 metres)

A geometrical diagram reconstructed from the tablet shown in the other attachment. The diagram consists of a square A, with another square B set inside it at an angle of 45 degrees, so that the corners of B touch the mid-points of the sides of A. Inside B is another square C, at the same angle as A, with the corners of C touching the mid-points of the sides of B. The smallest square C is divided into four equal square quarters by lines joining the mid-points of opposite sides. The four parts of B outside C form four isosceles right-angled triangles. Lines run from the corners of C to the corners of A, dividing the four parts of A outside B into a total of eight isosceles right-angled triangles. There are thus, in total, twelve equal isosceles right-angled triangles.

The exact symmetry of the configuration is vital to the problem. Without symmetry, which is suggested by the (necessarily approximate) diagram, but which is not made explicit in the question, the ‘wedges’ could be (e.g.) non-isosceles triangles of different sizes, and the problem would be insoluble.2

The problems on the tablet comprise various geometric configurations in which symmetry is implicitly required for the solution.

Notes

  1. E. Robson. ‘Mesopotamian Mathematics’, p. 95. Ch. 2, pp. 57–186 in: V. Katz, ed. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, 2007. ISBN: 978-0-691-11485‑9 ↩

  2. A. J. Cain, Form & Number: A History of Mathematical Beauty (Lisbon, 2024), p. 34. ↩

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