Adding to Valerio Talamanca's answer, there was a public lecture by Lenstra on Escher's "Paint gallery" at Princeton (April 3, 2007) which can be viewed and downloaded [here][1].

Also, my memory says that elliptic curves, rather than complex geometry, played the pivotal role in the explanation (that is why Lenstra explained it).

  [1]: https://www.princeton.edu/WebMedia/lectures/