Customer Reviews

Experiencing Geometry (3rd Edition)

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The first five chapters or so are an interesting and intuitive introduction to surface geometry. We "experience" the geometry of a sphere by stretching rubber bands on tennis balls or rolling them across freshly painted floors etc., and make observations like "The water strider is very sensitive to motion and vibration on the water's surface, but it can be approached from above or below without its knowledge. Hungry birds and fish take advantage of this fact." Then we study cylinders and cones. Since they have zero curvature their straight lines can be be understood as straight lines on their covering surfaces (straightness is preserved when a cylinder unfolds like a roll of toilet paper, etc.). Then hyperbolic geometry. The plane of hyperbolic geometry can be built by pasting together thin, semi-circular annuli. It can also be crocheted by the same principle. The pseudosphere (a hyperbolic cylinder) is easier to build: just pile up increasingly pointy cones. It can also be successfully crocheted. The annular plane is essentially a pseudosphere cut lengthwise. Thus it is a hyperbolic piece of paper rather than a hyperbolic cylinder, which should be a good thing. But unfortunately the annular plane cannot be extended very far without scrunching up. This defeats its purpose. It would be very difficult to develop any intuition for Euclidean geometry by working on a piece of paper scrunched up to a ball; a good, clean cylinder would be much more useful, provided of course that its imperfections are kept in mind. So I think we might as well stick to the pseudosphere. Over the next few chapters the "experiencing" approach deteriorates slightly, and necessarily so I think, as we study triangle congruencies and parallelism. The fruits of our efforts are limited. We make a start at the problem of determining the area of hyperbolic triangles, but only later will we be able to complete it using double integrals. And the relation between hyperbolic geometry and the parallel postulate is not done justice: it appears to be incidental rather than a major reason for studying hyperbolic geometry in the first place, and besides it is not very clear since lines to us are stitches in the scrunchy waltz-dress model (the standard plane models for hyperbolic geometry will appear briefly only later). The second half of the book is quite useless. There are isolated chapters on all sorts of completely unconnected topics: isometries and patters, Euclidean circle geometry, 2-manifolds, geometric solution of quadratic and cubic equations, polyhedra, etc. There is not much "experiencing" going on, although we do experience that life is easy for our authors and hard for us: they introduce something new, draw a few pictures, and as soon as there is any work to be done they claim it as a problem for the reader and move on to something else.

This book takes an interesting approach to exploring geometry with a lot of drawings and instructions on how to create physical models. however a lot of the knowledge is placed on the user to explore the subject and think. a lot of theorems and lemmas are not given or proven in the book. i think this book is great along side a course in undergraduate geometry, but lacks examples, excersises, and solutions, too much so to be used for self study. so overall a great textbook but not a great book to read during your free time. for that reason i only give it 3 stars.