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Applying Mathematics to Escher’s Print Gallery

Prentententoonstelling—or Print Gallery—is a recursive M. C. Escher drawing. For Mathematics Awareness Month 2003, Escher and the Droste Effect delves into the mathematics behind one of Escher’s more intriguing pieces. The following from the published article.

[Prentententoonstelling] shows a young man standing in an exhibition gallery, viewing a print of a Mediterranean seaport. As his eyes follow the quayside buildings shown on the print from left to right and then down, he discovers among them the very same gallery in which he is standing. A circular white patch in the middle of the lithograph contains Escher’s monogram and signature.

What is the mathematics behind Prentententoonstelling? Is there a more satisfactory way of filling in the central white hole? We shall see that the lithograph can be viewed as drawn on a certain elliptic
curve over the field of complex numbers and deduce that an idealized version of the picture repeats itself in the middle. More precisely, it contains a copy of itself, rotated clockwise by 157.6255960832… degrees and scaled down by a factor of 22.5836845286….