# Geometry and the Imagination

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*D. Hilbert; S. Cohn-Vossen*

AMS Chelsea Publishing: An Imprint of the American Mathematical Society

This remarkable book has endured as a true masterpiece of
mathematical exposition. There are few mathematics books that are
still so widely read and continue to have so much to offer—even
after more than half a century has passed! The book is overflowing
with mathematical ideas, which are always explained clearly and
elegantly, and above all, with penetrating insight. It is a joy to
read, both for beginners and experienced mathematicians.

“Hilbert and Cohn-Vossen” is full of interesting facts,
many of which you wish you had known before. It's also likely that
you have heard those facts before, but surely wondered where
they could be found. The book begins with examples of the simplest
curves and surfaces, including thread constructions of certain
quadrics and other surfaces. The chapter on regular systems of points
leads to the crystallographic groups and the regular polyhedra in
\(\mathbb{R}^3\). In this chapter, they also discuss plane
lattices. By considering unit lattices, and throwing in a small
amount of number theory when necessary, they effortlessly derive
Leibniz's series: \(\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots\).
In the section on lattices in three and more dimensions, the authors
consider sphere-packing problems, including the famous Kepler problem.

One of the most remarkable chapters is “Projective
Configurations”. In a short introductory section, Hilbert and
Cohn-Vossen give perhaps the most concise and lucid description of
why a general geometer would care about projective geometry
and why such an ostensibly plain setup is truly rich in structure and
ideas. Here, we see regular polyhedra again, from a different
perspective. One of the high points of the chapter is the discussion
of Schlafli's Double-Six, which leads to the description of the 27
lines on the general smooth cubic surface. As is true throughout the
book, the magnificent drawings in this chapter immeasurably help the
reader.

A particularly intriguing section in the chapter on differential
geometry is Eleven Properties of the Sphere. Which eleven
properties of such a ubiquitous mathematical object caught their
discerning eye and why? Many mathematicians are familiar with the
plaster models of surfaces found in many mathematics departments. The
book includes pictures of some of the models that are found in the
Göttingen collection. Furthermore, the mysterious lines that mark
these surfaces are finally explained!

The chapter on kinematics includes a nice discussion of linkages
and the geometry of configurations of points and rods that are
connected and, perhaps, constrained in some way. This topic in
geometry has become increasingly important in recent times, especially
in applications to robotics. This is another example of a simple
situation that leads to a rich geometry.

It would be hard to overestimate the continuing influence
Hilbert-Cohn-Vossen's book has had on mathematicians of this century.
It surely belongs in the “pantheon” of great mathematics
books.

#### Readership

Advanced undergraduates, graduate students and research mathematicians; mathematical historians.

#### Reviews & Endorsements

This book is a masterpiece — a delightful classic that should never go out of print.

-- MAA Reviews

[This] superb introduction to modern geometry was co-authored by David Hilbert, one of the greatest mathematicians of the 20th century.

-- Steven Strogatz, Cornell University

A fascinating tour of the 20th century mathematical zoo … Anyone who would like to see proof of the fact that a sphere with a hole can always be bent (no matter how small the hole), learn the theorems about Klein's bottle—a bottle with no edges, no inside, and no outside—and meet other strange creatures of modern geometry, will be delighted with Hilbert and Cohn-Vossen's book.

-- Scientific American

Should provide stimulus and inspiration to every student and teacher of geometry.

-- Nature

Students, particularly, would benefit very much by reading this book … they will experience the sensation of being taken into the friendly confidence of a great mathematician and being shown the real significance of things.

-- Science Progress

A person with a minimum of formal training can follow the reasoning … an important [book].

-- The Mathematics Teacher