In 1956 the Dutch graphic artist Maurits Cornelis
Escher (1898–1972) made an unusual lithograph
with the title Prentententoonstelling. It 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. . . .
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