Musings

Modular Escher

There’s a wonderful application of modular transformations and conformal mappings in Escher’s The Print Gallery. In it, one sees, through a row of arched windows, a young man looking at a painting of a harbour scene. The painting mushrooms out of its frame, and on the waterfront one sees the museum, with its row of arched windows.

What’s going on mathematically in the picture was figured out this past summer by some Dutch mathematicians. Their site is filled with all kinds of wonderful stuff, including alternate renderings of The Print Gallery using different modular transformations.

Consider an ordinary image on the plane. View it as a function of w, the complex coordinate on the plane. It is, of course, periodic under 2π rotations of the plane.

w ~ e^{2π i} w

A “Droste effect” picture (after the Dutch Chocolate company, whose boxes have a picture of a young woman holding a box which has a picture of a young woman …) has a second periodicity under constant rescalings

w ~ r w

for some r > 0 (r = 256 is the factor relevant to The Print Gallery).

This, as you know, defines a torus. And we can conformally map it via

z = log(w)

where now the periodicities are

z/(2πi) ~ z/(2πi) + 1 ~ z/(2πi) + τ

with

τ = i log(r)/(2π)

Next perform a modular transformation of the torus.

z’ = z/(cτ+d)
τ’ = (aτ+b)/(cτ+d)

where a,b,c,d are integers satisfying ad-bc = 1. Finally, conformally map back

w’ = e ^z’

The picture in the w’ plane is Escher’s picture, if the original “Droste” picture was the w plane.

You can see this all worked through in detail, with wonderful illustrations here.