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.

5$\begingroup$ Community Wiki? $\endgroup$ – Dan Ramras Jul 14 '10 at 18:54

6$\begingroup$ 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 Mathemagician Jul 14 '10 at 19:11

14$\begingroup$ 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). $\endgroup$ – danseetea Jul 14 '10 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 Mathemagician Jul 14 '10 at 22:32

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 Mathemagician Jul 15 '10 at 19:35
John Stillwell's recent book Naive Lie Theory is amazing and in a similar vein. It provides great geometrical intuition for many of the common matrix groups. What is particularly impressive about this book is how he motivates more complicated ideas, such as maximal torii in a very elementary fashion. It is perfect for undergrads looking for a good introduction.

7$\begingroup$ I would just like to second this opinion! There are several concepts I never entirely understood until a single sentence of Stillwell'sperfectly phrasedcleared years of fog. Understanding how Lie brackets connect to the conventional crossproduct is one such "Oh!" moment. $\endgroup$ – Joseph O'Rourke Jul 14 '10 at 20:48

1$\begingroup$ Agreed. The historical notes at the end of the book are particularly enlightening. $\endgroup$ – Qiaochu Yuan Jul 15 '10 at 1:22

1$\begingroup$ Absolutely agreed. I also had the sheer pleasure of being lectured by John Stillwell when he was here in Australia in the early to mid 1990s: see my answer on Stillwell's "Classical Topology and Combinatorial Group Theory”. $\endgroup$ – Selene Routley May 23 '11 at 7:13
I share your admiration for Needham's book!
One of my favorites is Geometry and the Imagination by David Hilbert and Stephan CohnVossen.
Some of their figures are stunning, almost works of art, and of course all drawn before computers!
Here they are explaining how one ellipsoid, one hyperboloid of one sheet, and one hyperboloid of two sheets, pass through any point in space:

2$\begingroup$ The original German edition of the book is true marvel to behold (I own a Russian translation and have used AMS translation, but they are not in the same league). $\endgroup$ – Victor Protsak Jul 15 '10 at 1:15

7$\begingroup$ What I find mind boggling about mathematics and physics of times gone by is the staggering ability of people to draw, graph and visualise WITHOUT COMPUTERS. They either did it with superb insight or with an almost soulcrushing slog of work  both to be admired. My day job is optical engineering and I am putting together a Lie theory text in my spare time  I truly believe that the only significant (but absolutely fundamental) way my jobs would differ without computers is that I would be lost without the ability to graph and visualise that they give .... $\endgroup$ – Selene Routley May 23 '11 at 6:42

6$\begingroup$ .... (Comment continued) Think of Abramowitz and Stegun  all those graphs drawn by hand. Or, there is a plot of the radiation pattern from a Mie scattering sphere in Born and Wolf, Principles of Optics. It has exquisitely fine structure in the plot, which was all done by hand calculation. I had to write a Mie scattering software function once, and I used Born and Wolf's hand calculations to debug my work  the hand calculations were right and I was wrong! It should have been the other way around!! $\endgroup$ – Selene Routley May 23 '11 at 6:45
I've just been reading "Visual Group Theory" by Nathan Carter. The similarity of title to Needham's may be coincidence, but the book has exactly the same effect: it SHOWS you WHY all these things are true, when conventional proofs so often just TELL you WHAT is true.
My favorite along with the Visual Complex Analysis by Tristan Needham
is Grad, Div, Curl and all that by H. M. Schey
NOTE: Alice's Adventures in Wonderland by Lewis Carroll is still the best mathematical book I have ever read.

2$\begingroup$ Re Alice's Adventures, you may want to take a look at bl.uk/onlinegallery/ttp/alice/accessible/introduction.html $\endgroup$ – Andrés E. Caicedo Jul 14 '10 at 19:53

$\begingroup$ @Andres Caicedo: Great link! Thanks! $\endgroup$ – Pratik Deoghare Jul 14 '10 at 20:41
"Mathematical Methods of Classical Mechanics" and "Ordinary differential equations" by late V.I. Arnold. In my opinion, these are THE books for anyone who wants to understand geometric theory of ODEs. I agree with Andrew's comment though, that the books might be a difficult read for an undergraduate (particularly, the first one).
My pick for Fourier analysis is, well, "Fourier analysis" by T.W. Körner. Very pedagogical, with lots of historical sections and nice illustrations. And it is probably more in the spirit of Needham's book than the books by Arnold.

1$\begingroup$ I totally agree,but these books are considerably harder then Needham's. But if you're serious about understanding the deeper aspects of the theory of ODE's,they are a necessity. Indeedall Arnold's book are a testament to one of the great mathematicians and teachers of all time and his awesome perspective unifying both mathematics and physical sciences through the shared language of geometry. $\endgroup$ – The Mathemagician Jul 14 '10 at 22:34

3$\begingroup$ "Ordinary Differential Equations" is the only one I have read, but it is among the best math books I've come across. I wouldn't have called it difficult, though  Arnold's style makes it a comfortable read for most graduate students, I'd say. (It actually makes quite nice "pleasure reading" when you want a break from more strenuous stuff.) $\endgroup$ – Tom Church Jul 15 '10 at 3:05

$\begingroup$ @Tom Yes,for GRADUATE students with very good command of linear algebra and basic analysis,yes,I agree. But an undergraduateunless they're superiorwill find them pretty tough.Especially the Classical Mechanics text,which is clearly a graduate level text by anyone's measure. $\endgroup$ – The Mathemagician Jul 15 '10 at 8:22


$\begingroup$ I believe that Arnold's textbook on ODE is oriented to undergrads. I read it when I first learn ODE when I was a sophomore. Arnold's lecture notes on PDE seem more demanding, which covers symplectic structures at the beginning. $\endgroup$ – user20948 Jan 17 '20 at 21:20
"Indra's Pearls: The Vision of Felix Klein" by David Mumford, Caroline Series and David Wright.
Most certainly visual, this book is not in the same category as most of the books mentioned so far. In fact, it defies categorization at all because it is a combination of an amazing ground level introduction to group theory and a monograph on Shottky groups, which grew out of desire to find mathematics to go along with stunning computer graphics. And it has outstanding cartoons by Larry Gonick.
[![alt text][1]][1]
1:

$\begingroup$ I wasn't able to find the picture of the front cover to embed (and David Wright's website seems to be down). If you know where to get one, please, put it in! $\endgroup$ – Victor Protsak Jul 15 '10 at 1:06



$\begingroup$ It certainly catches the eye and made me discover this great book ,thank you! $\endgroup$ – Jérôme JEANCHARLES Feb 18 '15 at 18:03
I haven't read Needham's book so I'm not totally sure what you mean, but it sounds like you might like "ThreeDimensional Geometry and Topology" by Bill Thurston and Silvio Levy.

$\begingroup$ I think it is far from didactic though trying to be clear but hard. Nothing to compare with the MO book. $\endgroup$ – Jérôme JEANCHARLES Jul 9 '20 at 11:47
John Stillwell's "Classical Topology and Combinatorial Group Theory (SpringerVerlag Graduate Texts in Mathematics)". See also the answer on his "Naive Lie Theory".
I also had the sheer pleasure of being lectured by John Stillwell when he was here in Australia in the early to mid 1990s. I took four of his courses in all, in general group and Galois theory as well two topics (topology and Riemann Surfaces) which were very much subtopics of the above book. I hope he wouldn't mind my saying that his gift for explanation did not appear magically: sheer hard work was evident in his lecture notes and he gave me the impression of someone never happy with an explanation as it was, he was always striving for a simpler and cleaner one for everything he lectured. Perhaps a mathematical analogue of Richard Feynman as a teacher. In his Galois theory lectures I and a few other students were lucky enough to join him as fellow learners: he was still getting his lectures straight and, in his honest way, warned us that this would be the case. So we "read" Emil Artin's "Galois Theory" together. Thus I got to see first hand the staggering amount of work he put into building his explanations.

3$\begingroup$ In his own words: "I read the books of Edwards, Tignol, Artin, Kaplansky, MacLane and Birkhoff and Lang, taught a course on Galois theory, and then discarded 90% of what I had learned."  from 'Galois Theory for Beginners'. $\endgroup$ – Marius Kempe Sep 1 '13 at 11:26

2$\begingroup$ I believe John Stillwell may be the single greatest author of mathematical textbooks that's currently active.He's going to leave behind a remarkable legacy of such textbooks.His topology and geometry textbooks alone would earn him this legacy,but he's written so many more such jewels. Let's hope he produces many more. $\endgroup$ – The Mathemagician Apr 22 '16 at 4:36

2$\begingroup$ @TheMathemagician Have you seen this: youtube.com/watch?v=9MV65airaPA. It's a wonderful talk in many ways, but one of the many things that really caught my attention was his beautiful one sentence description of induction to the lay person in the first few minutes. My daughter of ten grasped it in an instant after she had been reading about induction and had tried to learn more from me. His sentence drew the comment from her, "Wow, this guy is a way better stuffexplainer than you, Papa". $\endgroup$ – Selene Routley Apr 22 '16 at 23:22
Roger Penrose's The Road to Reality. Needham says in VCA that Penrose taught him what good style is.
A Panoramic View of Riemannian Geometry by Marcel Berger.
It gets into quite advanced and sometimes technical topics, but geometric intuition is always at the fore. Lots of great pictures! It must be impossible to read this book without getting passionately excited about differential geometry. Berger's other books on geometry are similarly outstanding, if more conventional.
Nonlinear dynamics and chaos by Steven Strogatz. Lots of pictures, intutive and clear explanations, interesting applications, great humor.

$\begingroup$ Yes! This is the book that to me comes closest to Visual Complex Analysis in spirit. $\endgroup$ – dfan Dec 8 '15 at 16:28
Tristan Needham has a new book coming out in 2021:
Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts. Princeton link.
A quite recent book is Advanced Calculus: A Geometric View by James Callahan. It is liberally illustrated and even contains a section on Morse's lemma in the chapter on critical points. Bear in mind, though, that the book is not intended for absolute novices to multivariable calculus. Familiarity with basic concepts such as partial derivatives is expected, as is some knowledge of linear algebra.
Edit: A review by William J. Satzer is available at http://www.maa.org/publications/maareviews/advancedcalculusageometricview.
Another book to try is Michio Kuga's Galois' Dream. It certainly has its own unique style (very playful) but I think its focus on intuition sets it apart from many other math books. Apparently it was quite a popculture hit in Japan!
If I may add my two cents, I would add two more books that are an integral part of my library, and which I have presently lent to a gifted middle school student. One is the 'shape of space' by Jeff Weeks, and the other is 'Symmetry of things' by John Conway
Jeff Week's book is an incredibly enjoyable account of the topology of 3manifolds. I came across someone mentioning the late Bill Thurston's book in this post. While Thurston's book is definitely more rigorous, I would say that Week's book is an overlooked classic. His invitation to experiment with intuition to extrapolate to the abstract, and tying in a theoretician's mental forays with cosmological measurements is quite an eyeopener.
John Conway's book, on the other hand, while it showcases some ideas of symmetry through the work of some artists like Bathsheba Grossman, is largely about abstraction. It is a major work, the latter part technical enough to challenge and inspire mathematicians on the forefront of their field (in his words, not mine!).
David Bressoud's "Proofs and Confirmations: The Story of the AlternatingSign Matrix Conjecture" is also wonderful. Emphasis on how the conjecture was proved, and its connections to many interesting areas of math.

3$\begingroup$ I'm one reputation point short of being allowed to edit the post, so here is a nitpicky comment instead: the subtitle should be "The Story of the Alternating Sign Matrix Conjecture". (And I agree that it's a lovely book!) $\endgroup$ – Hans Lundmark Jul 15 '10 at 14:30
It's not exactly as visual as Visual Complex Analysis, but Michael Spivak's A Comprehensive Introduction to Differential Geometry has a lot of the same appeal to intuition and conversational style. (Well, I've only read Volume 1, there's a total of 5, but if they're anything like other Spivak books I've read, this holds true of them as well).

4$\begingroup$ I like the artwork on the cover of these books! $\endgroup$ – Spice the Bird Jun 16 '11 at 21:32
A wonderful book which overviews a lot of these kinds of ideas is Glimpses of Algebra and Geometry by Gabor Toth. From the product description, "The purpose of Glimpses of Algebra and Geometry is to fill a gap between undergraduate and graduate mathematics studies. It is one of the few undergraduate texts to explore the subtle and sometimes puzzling connections between number theory, classical geometry and modern algebra in a clear and easily understandable style."
Roger Godement, Analysis, vols. I to IV (Springer). Contains many interesting historical, heuristic and motivational comments. Includes several details on Bourbaki ("bande militante") in Vol. III. Great mathematical content, plus some provocative thoughts.
The Shape of Algebra in The Mirrors of Mathematics by G. Katz and V. Nodelman
The Wild World of 4Manifolds by Alexandru Scorpan
Discrete Differential Geometry: An Applied Introduction by Keenan Crane
Calculus Blue Multivariable Volume 4: Fields by Robert Ghrist
A Gateway to Number Theory: Applying the Power of Algebraic Curves by Keith Kendig
Conics by Keith Kendig

$\begingroup$ As a complement to Needham, for superb visualizations and conceptual analysis: "Exploring Visualization Methods for Complex Variables" by Andrew J. Hanson and JiPing Sha. $\endgroup$ – Tom Copeland Jul 31 '20 at 15:20
Parallel Coordinates: Visual Multidimensional Geometry and its Applications by Alfred Inselberg
Has been praised by Stephen Hawking and others
https://www.amazon.com/ParallelCoordinatesMultidimensionalGeometryApplications/dp/0387215077
The barrier, imposed by our three dimensional habitation and perceptual experience, has been breached by this innovative and versatile methodology. There are beautiful visuals of multidimensional objects and insights into multidimensional problems: Air Traffic, Data Mining, Intelligent Process Control
David Bressoud's book Second Year Calculus: From Celestial Mechanics to Special Relativity is something like Needham's book. Both have an emphasis on history and applications.

2$\begingroup$ Speaking of Bressoud, the "history+applications" thread also runs through his other book "A Radical Approach to Real Analysis". It was truly a joy to peruse. $\endgroup$ – J. M. isn't a mathematician Aug 5 '10 at 10:35
Even more visual, even less formal, is "Dynamics, the Geometry of Behavior," by Ralph Abraham and Chris Shaw. I find the approach very useful for a difficult subject, however it needs to be supplemented with more rigorous material.
A digital edition can be purchased through Aerial Press http://www.aerialpress.com
I strongly recommend "The Essence of Chaos" by Edward N Lorenz. Not only considerable historical background, but a wonderful discussion of chaos, a unique and realistic model development and classic models. All without deep mathematics, but detailed so that one can program his model of a sled on a snow covered hill with moguls. A true classic that should be on every book shelf (after having read it in depth!)
I would recommend two books by David Bressoud
A Radical Approach to Real Analysis http://www.maa.org/press/books/aradicalapproachtorealanalysis
A Radical Approach to Lebesgue's Theory of Integration https://books.google.no/books?id=TxxMoGjXCwC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
They are not as visual as the book by Needham, but like the books by Stillwell, they combine history and insight.
By the way, Needham and Stillwell both work at Univ. of San Francisco. I visted there once. Great place!
Elementary Applied Topology by Robert Ghrist.
Book is full of interesting applications of ideas from topology/geometry. Good for adding some colour to a standard course.
if you are interested in dynamical systems/oscillators/differential equations, Pikovsky's Synchronization: A Universal Concept in Nonlinear Sciences is very wellwritten.
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 W Mar 17 '13 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.
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.