This book is based on notes I wrote when teaching an undergraduate
seminar on surfaces at Brown University in 2005. Each week I wrote
up notes on a different topic. Basically, I told the students about
many of the great things I have learned about surfaces over the years.
I tried to do things in as direct a fashion as possible, favoring concrete
results over a buildup of theory. Originally, I had written 14 chapters,
but later I added 9 more chapters so as to make a more substantial
Each chapter has its own set of exercises. The exercises are em-
bedded within the text. Most of the exercises are fairly routine, and
advance the arguments being developed, but I tried to put a few
challenging problems in each batch. If you are willing to accept some
results on faith, it should be possible for you to understand the mate-
rial without working the exercises. However, you will get much more
out of the book if you do the exercises.
The central object in the book is a surface. I discuss surfaces
from many points of view: as metric spaces, triangulated surfaces,
hyperbolic surfaces, and so on. The book has many classical results
about surfaces, both geometric and topological, and it also has some
extraneous stuff that I included because I like it. For instance, the
book contains proofs of the Pythagorean Theorem, Pick’s Theorem,
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