In 1957, the well known geometer H. S. M. Coxeter wrote to
Dutch graphic artist M. C. Escher to ask if he could use two
of Escher's symmetry drawings to illustrate his
paper *Crystal Symmetry and its
Generalizations*. Escher readily agreed. As a courtesy,
Coxeter sent Escher a reprint of his article. Escher was quite
excited about one of the figures in the paper. He wrote to
Coxeter, "some of the text-illustrations and especially figure
7, page 11, gave me quite a shock". The figure is just like
the computer redrawn version below.

This figure is a hyperbolic tiling with triangular tiles
which diminish in size and repeat infinitely within a
circle. Escher had been looking for a way to capture infinity
in a finite space for a long time. After figuring out the
geometry of this figure, he produced his *Circle Limit*
series,
e.g. Circle
Limit III. From his sketch we guess that he may draw those
figures with ruler and compass.

It is not a surprise that people tried to write programs to draw hyperbolic tilings after computer was invented. It can be done by reflecting an initial triangle hyperbolically. However this approach is wasteful and inaccurate because one needs to compare if two points with floating-point number coordinates are the same again and again. In 1990s, mathematicians found that the tiling's underlying group of symmetries is special. They developed the theory of automatic groups that can be used to draw hyperbolic tilings efficiently and accurately. I wrote a brief introduction about the hyperbolic geometry and group theory involved in drawing such hyperbolic tilings. I also summarized the key points in a large poster. Readers can zoom in this 60 MB PDF file to see the exquisite detail on the boundary of the big circle.

For interested readers who want to create their own tilings,
I suggest them to read James W. Anderson's *Hyperbolic
Geometry* to learn more about hyperbolic geometry. Basic
group theory concepts such as group, subgroup, coset and orbit
are also necessary. They can be found in any abstract algebra
textbooks. Then the readers can read Silvio Levy's
paper *Automatic Generation of Hyperbolic Tilings* to
know how to combine these things together. They may find my
GAP
program code
is also helpful.

Once the principles are grasped, the drawings are not limited to triangle tilings any more. There are infinitely many variants. I list one of the possibilities below. Readers are encouraged to generate their own artistic tilings.

by Shengyi Wang, 2023