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.

8$\begingroup$ You SHOULD be impressed,Marco. It's a truly remarkable book that should be read by anyone seriously interested in analysis or physics,particularly for it's historical insight. $\endgroup$– The MathemagicianJul 14, 2010 at 19:11

18$\begingroup$ On one hand I want to +1 because that's indeed a great book. On the other hand I should probably 1 because it's a bad question (well, at least as long as it's phrased the way it is and since it's not communitywiki). So I'll do neither for now. I'll just leave this comment: while many books come to mind, one particular author who has a style similar to that of Needham's (that is, his books are full of intuition and historical insight) is John Stillwell. I'm not going to recommend a particular book because most of them are great (I say 'most' and not 'all' because I haven't read them all). $\endgroup$– danseeteaJul 14, 2010 at 19:22

7$\begingroup$ @danseetea The similarity between the texts of Stillwell and Needham's masterwork is indeed striking and is not an accident:Professor Stillwell is acknowledged in the introduction to Needham's book as a strong influence on the style of the text. We can all learn a great deal from Professor Stillwell's texts,emphasizing the historical development of the great edifice of mathematics. $\endgroup$– The MathemagicianJul 14, 2010 at 22:32

$\begingroup$ @Andrew L: Does that mean that Needham's text is useful for the student who has not yet understood the rigorous formalism of complex analysis and wants to understand it? Or should a student who already understands complex analysis reasonable well still read Needham's book? $\endgroup$– David CorwinJul 15, 2010 at 10:36

5$\begingroup$ @Davidac897 Both,ideally.A student struggling with a more rigorous presentationsuch as Alfhors or Narishamwill benefit tremendously from the geometric and historical presentation of the basics and drawing connections between them. A student who has mastered a rigorous presentation will gain much deeper insight into what the abstract construction "means". $\endgroup$– The MathemagicianJul 15, 2010 at 19:35
42 Answers
From Geometry To Topology by H. Graham Flegg
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.

$\begingroup$ You could see this book as a gentle warmup to topology, which purpose it fulfils admirably. By the way, I was amused to read in Chapter 8 that the four colour theorem had never been proved. Of course, since the book was originally published in 1974. $\endgroup$– J WMar 17, 2013 at 14:35
"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.
https://books.google.com.au/books/about/Solid_Shape.html?id=pIyNQwAACAAJ
This, Indra's Pearls and Needham are my all time favorite mathematical tomes, and I return to them regularly.
Behrend's Intro to Algebraic Stacks or here is so nicely explicated and illustrated that even doofi like myself have a chance of grasping stacks.
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.

$\begingroup$ That's a beautiful exposition  thanks for sharing! $\endgroup$– pinakiApr 26, 2022 at 16:53
Sheaf Theory through examples by Daniel Rosiak, and also, Conceptual Mathematics: A First Introduction to Categories by Lawvrere
Recently I am studying the book "An Introduction to Mathematical Billiards" written by prof. Utkir A. Rozikov.
https://www.worldscientific.com/worldscibooks/10.1142/11162
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.
Galois' Dream: Group Theory and Differential Equations
Explanation of poles
This book has so many pictures that it apparently got the professor in trouble for keeping too many pictures!
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 mindboggling topological phenomena. I do not know of any other source for such amazing pictures.
It starts with a very familiar contractible 2dimensional 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 2complex 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.

3$\begingroup$ I'm sure it's a good book, but that is not what this question is asking for $\endgroup$ Aug 26, 2014 at 21:55

2$\begingroup$ "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 is just what Ellenberg's book is about. $\endgroup$ Aug 26, 2014 at 22:00

2$\begingroup$ I recall trying to read that, and couldn't continue. The attitude struck me as conceited and vaguely reminiscent of Harris's "Cows, Pigs, Wars, and Witches." Much as I would like to believe that we mathematicians really are that smart, time and again I see no clear evidence for it once we venture outside of our field. $\endgroup$ Jan 25, 2018 at 11:54