This book is a wonderful introduction to differential geometry. Most books on the subject get right into the algebraic side of it, which is good and all, but makes it hard to build a physical intuition. This book works pictorially and gets one thinking in terms of movement and geometry rather than formula. Instead of dealing with the technical issues of limits, this book uses Newton's idea of things being eventually equal as they get smaller. and reinforces this with a series of geometric demonstrations of proofs that would have been algebraically complex but are simple to visualize.<br><br>I haven't taken a math course in forever, so I didn't expect to get very far in this book, but the narrative was seductive. My freshman calculus returned slowly as I was reintroduced to curvature and finally got a visual understanding of how positive and negative curvature work. The historical approach was encouraging. If it took the likes of Gauss and Euler decades to figure something out, I had no reason for shame in having to reread a section a few times and spend a few days making sure I understood it.<br><br>The narrative carried me along when the book moved from those surfaces familiar from algebra to more general topological surfaces. It always seemed that there was so much freedom in the geometries of generalized surfaces that the mathematics must be inscrutable, but it turns out that curvature has all kinds of constraints. It isn't just that a donut can be deformed into a coffee cup, it's that the sum total of curvature of both surfaces are the same. Every inward bit of bending has to be balanced by the same bending out. Even better, there were multiple proofs, all geometric and well illustrated, of a number of important theorems. If one didn't suit my learning style, another would. Visualizing bananas dripping honey and oddly shaped squashes embedded in three dimensional space really helped.<br><br>Then the narrative got deeper as the viewpoint shifted from the space outside the surface into the surface itself. If you've read the classic Flatland, you'd appreciate how much can be learned by changing one's point of view. With diagrams and clear descriptions, the book explained how that now somewhat familiar curvature appeared to a creature living within the surface. Moving within a curved surface yields strange motions and peculiar rotations. As odd as they may seem, they all make sense, and they are much more constrained than they first appear. An accountant would appreciate so much of this book. Everything has to add up.<br><br>Then comes a bigger jump, from two dimensional curved surfaces to three dimensional curved surfaces, and few of us have a geometric intuition here. Having laid the groundwork, however, the book discusses the strange motions and peculiar rotations that we can experience living in three dimensional space. It is harder to visualize curved three dimensional space, but the book had readied us for this jump. I had to read the section on parallel transport three times before I felt I understood it properly, but in the end I did. Again, more illustrations, more theorems, more demonstrations. There is a bit more algebra here. Two dimensional curvature can be described with a single number, but higher dimensions require more numbers best arranged in matrices.<br><br>I really appreciated the section on general relativity and the way it treats gravity not as a force but as a property of curved space. Do the geodesics, the paths that appear to be straight to those dwelling within within the space, converge or diverge? That can be seen as a geometric question or as a question of attractive or repulsive force. The book had a really nice explanation of this and added a lot of depth to the idea of gravity as geometry.<br><br>I was exhausted by the end of Act IV, but the journey had been worth it. I didn't read the fifth act with the chapters on forms. Forms are a more modern, more abstract way of understanding those strange motions and peculiar rotations, but for now they are beyond me. Perhaps some day I will return to Visual Differential Geometry and finish with Forms, but my brain needs to recover a bit.<br><br>This book is definitely not for everyone. It assumes that you are at least familiar with vector calculus and some linear algebra. If you aren't you will be lost, clear explanations and useful illustrations notwithstanding.<br><br>Also, a few quibbles, the book has some typographic and proof reading problems. The captions in a few of the illustrations seem to have been scrambled. There are an awful lot of variables distinguished by font or by markings above like carets and dashes and squiggles. Some of the block letter characters look all too similar to my aging eyes, and I would have appreciated a guide to the conventions of those markings, especially since they aren't completely consistent throughout the chapters. A particular problem for me was the use of a comma that looked a lot like the prime marker indicating differentiation. More than once I had to stop and go over a derivation before realizing that an apparent derivative was just a comma.<br><br>All told, a great book. If you are differential geometry curious, think you have sufficient background and finally want to make sense of curvature, this is an excellent introduction.