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.
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.
I share your admiration for Needham's book!
One of my favorites is Geometry and the Imagination by David Hilbert and Stephan Cohn-Vossen.
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:
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.
"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.
"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.
John Stillwell's "Classical Topology and Combinatorial Group Theory (Springer-Verlag 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.
Tristan Needham has a new book coming out in 2021:
Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts. Princeton link.
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.
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 pop-culture hit in Japan!
Nonlinear dynamics and chaos by Steven Strogatz. Lots of pictures, intutive and clear explanations, interesting applications, great humor.
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/maa-reviews/advanced-calculus-a-geometric-view.
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 3-manifolds. 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 eye-opener.
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!).
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).
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."
The Shape of Algebra in The Mirrors of Mathematics by G. Katz and V. Nodelman
The Wild World of 4-Manifolds 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
How Surfaces Intersect in Space: An Introduction to Topology by J. Scott Carter
As a complement to Needham, for superb visualizations and conceptual analysis: "Exploring Visualization Methods for Complex Variables" by Andrew J. Hanson and Ji-Ping Sha.
Parallel Coordinates: Visual Multidimensional Geometry and its Applications by Alfred Inselberg
Has been praised by Stephen Hawking and others
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.
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
A Singular Mathematical Promenade by Étienne Ghys is a perfect example in my opinion. The word singular in the title refers primarily to singularity theory, but the book is also singular in that its style is unique. Here is how the book begins.
In March 2009, I attended an administrative meeting and the colleague sitting next to me was even more bored than I was. Obviously Maxim Kontsevich had something else in his mind. Suddenly, he passed me a Parisian métro ticket containing a scribble and a single word: “impossible”. That was the new theorem he wanted to share with me! It took me a few minutes and some whispering before I could guess the statement of the theorem and a few more minutes to find the proof. Here is the statement.
Theorem. Four polynomials P1, P2, P3, P4 of a real variable x cannot satisfy
• P1(x) < P2(x) < P3(x) < P4(x) for small x < 0,
• P2(x) < P4(x) < P1(x) < P3(x) for small x > 0.
The relative position of the graphs of four real polynomials is subject to some constraints. I was fascinated: a new elementary result on four polynomials in 2009!
In the book, Ghys takes the reader on a leisurely tour through an astonishingly wide swath of mathematics, nominally with the goal of proving a generalization of the above theorem, but actually using that as an excuse to showcase many beautiful gems, including the resolution of singularities, the Hopf fibration, permutation pattern avoidance, the associahedron, the fundamental theorem of algebra, operads, Kontsevich's universal invariant for knots, and much more. The book is lavishly illustrated with diagrams in full color, and best of all, the PDF can be freely downloaded completely legally.
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/a-radical-approach-to-real-analysis
A Radical Approach to Lebesgue's Theory of Integration https://books.google.no/books?id=TxxMoGjXC-wC&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!
if you are interested in dynamical systems/oscillators/differential equations, Pikovsky's Synchronization: A Universal Concept in Nonlinear Sciences is very well-written.
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.