I've been very positively impressed by Tristan Needham's book "Visual Complex Analysis", a very original and atypical mathematics book which is more oriented to helping intuition and insight than to rigorous formalization. I'm wondering if anybody knows of other nice math books which share this particular style of exposition.
This book explains some basic topological concepts using a lot of examples and it has quite a lot of pictures. In fact, it is rather hard to find a single page that has no pictures in it. Very good for intuition indeed. And also very cheap since it is a Dover reprint.
"Solid Shape" by Jan J. Koenderink, MIT Press
This is an older book, but it has some really nice approaches to thinking about differential geometry, and he encourages the reader to develop multiple views of the subject.
This, Indra's Pearls and Needham are my all time favorite mathematical tomes, and I return to them regularly.
He treats thoroughly the example of moduli of triangles which M. Artin claimed was all one needed to know to understand stacks.
I'm not sure it really qualifies as a textbook, but it deserves a mention.
Recently I am studying the book "An Introduction to Mathematical Billiards" written by prof. Utkir A. Rozikov.
I think the nature of the book and the good style of writing make it a candidate for this post.
He explains some basic topics that science students need to know. Excellent explanation, extremely intuitive and beautiful. Too elementary for most readers of this thread, but a good read for an advanced high school student / beginning undergrad.
Guide to distribution and fourier theory by Strichartz. This book doesn't have too much pictures, I suppose maybe because it deals with real analysis, but does indeed follow a similar style to that which Needham takes at the end, where he goes to Physics and heat equations to get intuitions of what he does mathematically, similar stuff going on here.
Another commonality is that this book author doesn't spend too much time on trivial details.
An example of the commonality is literally on the first page, where he introduces the theory of distribution by first starting with the physical example of how the error of a thermostat measuring temperature in can be accounted for.
Calculus: The Language Of Change (@005 1 ed) by Cohen, David W., Henle, James M.
It alike features many pictures in color that motivate and illustrate theorems and proofs.
There are some books said in this thread which I have skimmed through my self and can say are good, these are:
- Tristan Needham's VDG
- Naive Lie Theory
- Arnold's Differential equation book
- Robert Ghrist's Elementary applied Topology
One more book I found similar in the category of books above is Peter Salilev's Topology illustrated.
I highly recommend "A Topological Picturebook" by George Francis (1988). It contains extremely enlightening, very ably drawn sketches, of many quite mind-boggling topological phenomena. I do not know of any other source for such amazing pictures.
It starts with a very familiar contractible 2-dimensional CW complex: the "dunce cap", defined as the quotient space of triangle ABC after AB is identified to BC (so far, resulting in one nappe of a finite cone) ... but then the segment AB is identified with the circle BCB. (The resulting 2-complex is hard to grok, but turns out to be contractible.)
How Not to Be Wrong: the Power of Mathematical Thinking by Jordan Ellenberg; Penguin Press, 2014 is a wonderful, easily read exposition on subjects such as statistical analysis for a wide variety of examples. Also a lot of history. Surprising and clearly written.