Escher’s Toroidal Print Gallery

If Art+Math brings one person to mind, it’s Escher. His tessellations present the best-known instance, but he did a lot more than that.

In April 2003, the mathematicians Bart de Smit and Hendrik Lenstra wrote a delightful article, Escher and the Droste effect, about Escher’s lithograph Prentententoonstelling. They pointed out that

We shall see that the lithograph can be viewed as drawn on a certain elliptic curve over the field of complex numbers…

de Smit and Lenstra recognized a toroidal structure implicit in Escher’s lithograph. Topologically, elliptic curves (over the complex numbers) are tori; to number theorists like our authors, probably the preeminent tori. (Elliptic curves have additional structure that I don’t find in Prentententoonstelling.)

If you haven’t already read the de Smit-Lenstra article, you’re in for a treat. Not much background required: just the exponential function in the complex plane, and the notion of topological identification (quotient spaces).

As is my wont, I wrote up my own notes on this. These amount to a marginal note on just one aspect of the de Smit-Lenstra article. But they do include a commutative diagram, illustrated as a medieval monk might have:

 

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