The Trigonometry of Escher’s Woodcut Circle Limit III

During the 1954 International Congress of Mathematicians in Amsterdam, my wife Hendrina introduced me to M. C. Escher, with whom we became close friends. On one of the occasions when he was visiting his son George in Nova Scotia, he gave an illustrated lecture in the Art Gallery of Ontario and spent a few nights with us in our Toronto house. He gave us four original prints; including one of Circle Limit III, which inspired this article, as well as an earlier one. In contrast to the other three Circle Limits (I, II and IV), this employs four colours in addition to black and white, and features arcs which are not orthogonal to the peripheral circle. In an earlier article [1], I used hyperbolic trigonometry for my analysis, but several years later I took up the challenge of using Euclidean trigonometry instead. My former student J. Chris Fisher kindly helped by reducing my expressions for the measurements to calculated numbers that could be compared with the actual print on my staircases wall. At first one of the six measurements seemed to be wrong by a few millimetres. (The diameter of the peripheral circle is 41 cm.) Rather than blame Escher, I asked Chris to check his computation again. When he admitted that the mistake was his, I realized that Escher’s intuition was completely justified. I still find it almost incredible that he, with no knowledge of algebra or trigonometry obtained accurately the centres and radii (r 1, r 2, r 3) of the three different circles to which the three different axes belong.