Geometry and the imagination

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 R 3 R3. 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: π/4=1−1/3+1/5−1/7+−… π/4=1−1/3+1/5−1/7+−…. 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”.

What Hacker News says

adalaceloveDec 2024

Nice book. For another old but excellent math book I recommend Geometry and the Imagination by David Hilbert. No gutenberg remake I'm afraid, maybe because of the numerous (and incredibly high quality) illustrations.

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eimrineAug 2024

I was astonished that all the best and most important in methodical approach mathematical books are almost unknown to students here (and, seems to me, have not been translated into French). Among these are Numbers and figures by Rademacher and Töplitz, Geometry and the imagination by Hilbert and Cohn-Vossen, What is mathematics? by Courant and Robbins, How to solve it and Mathematics and plausible reasoning by Polya, Development of mathematics in the 19th century by F. Klein.

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enriqutoJan 2023

The classical stuff is great:

* Geometry and the imagination by Hilbert and Cohn-Vossen

* Methods of mathematical physics by Courant and Hilbert

* A comprehensive introduction to differential geometry by Spivak (and its little brothers Calculus and Calculus on manifolds)

* Fourier Analysis by Körner

* Arnold's books on ODE, PDE and mathematical physics are breathtakingly beautiful.

* The shape of space by Weeks

* Solid Shape by Koenderink

* Analyse fonctionnelle by Brézis

* Tristan Needhams "visual" books about complex analysis and differential forms

* Information theory, inference, and learning algorithms by MacKay (great book about probability, plus you can download the .tex source and read the funny comments of the author)

And finally, a very old website which is full of mathematical jewels with an incredibly fresh and clear treatment: https://mathpages.com/ ...I'm in love with the tone of these articles, serious and playful at the same time.

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chkhdJan 2023

* The Language of Mathematics: Utilizing Math in Practice by Baber https://www.amazon.com/Language-Mathematics-Utilizing-Math-P... really helped me "get it", as I always found programming natural but math hard. This one is written by a CS professor and it really makes all the difference.

* How to Solve it by Polya https://www.amazon.com/How-Solve-Mathematical-Princeton-Scie... and How to Prove it by Velleman https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/... helped strengthen that understanding.

* This year I am trying to master https://www.amazon.com/Methods-Mathematics-Calculus-Probabil... which focuses on how to "connect the dots".

* I am using Geometry and the Imagination by Hilbert https://www.amazon.com/Geometry-Imagination-AMS-Chelsea-Publ... as an attempt to "immerse" myself in Geometry. I just love this book.

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enriqutoJul 2020

The princeton companion is nice to have around, but you do not really read it end to end.

Spivak's calculus you bring to the beach and read it between swim and swim.

EDIT: Also, some books by Hilbert are breathtakingly beautiful: Geometry and the Imagination (just the chapter on synthetic differential geometry is worth more than 10 other great books), and the Methods of Mathematical Physics is also great. It begins by giving three proofs of cauchy-schwartz inequality, and then goes on to give several different definitions of the eigenvectors of a matrix. Both of those make great beach readings for this summer.

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enriqutoOct 2019

All the books cited here by Arnold, Spivak, and Lanczos are extraordinarily good.

Nobody has mentioned yet "Geometry and the Imagination" by Hilbert and Cohn-Vossen. If there is a Feynman equivalent in math it is certainly this book.

For elementary geometry, the Feynman equivalent is probably "Introduction to Geometry", by H.S.M.Coxeter. Beautifully written, figures on every page, covers all geometric topics (affine, projective, ordered, differential, ...)

For differential geometry, nothing beats "A Panoramic view of Differential Geometry" by Berger. It is a stunning comprehensive overview of the whole field, focused on the meaning and the applications of each part and, strangely for a math book, with no formal proofs. Only the main ideas of the proof and the relationships between them are given, but this allows to fit the whole subject into a single, manageable whole.

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