I'm a bit disappointed that Michael describes al-Nayrizi's and Perigal's (families of) simple perfect squarings of the (flat square) torus as different. They are, in fact, identical—not just “congruent” or “equivalent” or “isomorphic”, but actually indistinguishable. They only look different because al-Nayrizi and Perigal cut the torus into a square in two different ways.