Geometry and the Imagination

by David Hilbert, Stefan Cohn-Vossen

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--after more than half a century! 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, or had 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 $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. 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.

Genres: Mathematics, Science, Geometry, Nonfiction, Textbooks, Stem, Physics

357 pages, Hardcover

First published January 1, 1932

About the author

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory).

Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century.

Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic.

Reviews

[Chris · Rating 5 out of 5]

One of the best mathematics books ever written. If it weren't reasonably difficult I would have selected it as my "should be in every high school" book. Still a strong 5/5, bordering on 6/5.



If you want to know what mathematics is, and if you're willing to work hard, read this.

[Roberto Rigolin F Lopes · Rating 5 out of 5]

We are in 1932, Hilbert is playing with geometry. You are about to understand why quantum mechanics books say that everything is happening at Hilbert’s space. Oras, seems that Hilbert dared to define all possible spaces. Was he a boundless thinker? In this book, he always starts with the simplest forms to end-up discussing how many x-order things there are. Using infinity power to some x to illustrate the algebraic nature of figures. Many figures. As a result, he will stretch your imagination to infinity and beyond (?).

[Ilib4kids]

513.8 HIL
Ref:The Process of Education by Bruner, Jerome S
David Hilbert, a German mathematician recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries

Quotes:
We must know. We will know.
David Hilbert liked to quote "an old French mathematician" saying "A mathematical theory should not be considered complete until you have made it so clear that you can explain it to the first man you meet on the street".

Review as not a academic approach to introduce math. Should be a good to self-reading book.

[Diana · Rating 4 out of 5]

Yesterday, I learned what an oblate spheroid is. Also a prolate spheroid (which I hadn't heard of before.) If you rotate an ellipse on its major (longer) axis you get a prolate spheroid (which looks like an egg). If you rotate an ellipse on its minor (shorter) axis you get an oblate spheroid (which looks like an onion).

[Woflmao · Rating 5 out of 5]

A magnificent and beautiful book. I recommend reading it to anyone wishing to get more deeply involved with geometry. Especially the introduction to topology from a geometrical point of view is a great motivation for the rather abstract approach that is commonly taught nowadays.

[Paul Mamani · Rating 5 out of 5]

"Geometrie und die Imagination"


Von David Hilbert und Stefan Cohn-Vossen:

Dieses Buch präsentiert eine Reihe von geometrischen Themen und Konzepten auf eine intuitive und anschauliche Weise. Hilbert und Cohn-Vossen zeigen die Schönheit und Eleganz der Geometrie durch zahlreiche Beispiele und Illustrationen. Sie decken sowohl klassische als auch moderne geometrische Probleme ab und betonen die Bedeutung der räumlichen Vorstellungskraft und der Visualisierung in der Mathematik.

Ein zentrales Thema ist die Darstellung geometrischer Objekte und Transformationen, die den Lesern ein tieferes Verständnis für die Struktur und Eigenschaften der Geometrie vermittelt. Das Buch richtet sich sowohl an Mathematiker als auch an Laien, die an den ästhetischen und theoretischen Aspekten der Geometrie interessiert sind.


Inti, aus Titicacasee Peru