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Euler's Gem: The Polyhedron Formula and the Birth of Topology
Binding: Hardcover
Author: David S. Richeson
Manufacturer: Princeton University Press
Availability: Usually ships in 24 hours
Features:
Average Rating: 5.0
Total Customer Reviews: 8
List Price: $27.95
Our Price: $18.45
Sales Rank: 37746
Product Description
Leonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child. Euler's Gem tells the illuminating story of this indispensable mathematical idea. From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V-E+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map. Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, Euler's Gem will fascinate every mathematics enthusiast.
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Users Product Reviews: |
Product Review Summary: A gem of mathematical results produced by one of the masters of mathematics The title of the book is derived from the formula V - E + F = 2 that holds for any polyhedron. V is the number of vertices, E the number of edges and F the number of faces. First demonstrated by Euler, the proof of this result is surprisingly simple. As is the case with most such formulas and their proofs, there is at least one near miss in the history of mathematics. Descartes was close; in retrospect it is somewhat surprising that he didn't reach the appropriate conclusion. Of course, we are considering the great master Euler here, a giant of mathematics who was able to see things in his mathematical sight that people with the physical vision that he lacked overlooked.
Topology is a relatively recent area of mathematics, one of the few that can be considered to have had a point of origin and a creator. Richison works through the historical mathematical preliminaries of the formula, the shapes it describes were well known to the ancient Greeks yet they were nowhere close to the formula. Some historical and mathematical background on Euler follows this and it includes some of his other accomplishments. The last chapters describe some of the results that follow from topology in general and Euler's gem in particular. One of the most interesting is the theorem of combing a sphere, where the conclusion is that there must always be at least one hair that stands straight up. This may seem like an absurd thing for mathematicians to be concerned about but it has a major conclusion, that at all times there must be at least one point on Earth where there is no wind. Even more significantly it means that there will always be a zero.
Richison uses a large number of diagrams and formulas when needed, which is to his credit. Mathematics is based on equations so when an author deliberately avoids them in an attempt to increase sales, it is hard to claim that they are actually writing mathematics. This is an excellent book about a great man and a timeless formula. Well within the reach of the intelligent layperson, it is also a good book to use as a resource for a course where the students are required to make presentations.
Published in Journal of Recreational Mathematics, reprinted with permission.
Product Review Summary: Start here. This is the best introduction to topology that I've found. While the book is very easy to read, I learned some very interesting mathematical concepts from it. The book partly history with some biographical information in almost every chapter. But the more important thing is the presentation of development of the field of topology. Euler is not the main character of this book, his equation is. There are a large number of illustrations which help you get the intuitive feel for the subject. After finishing with it, I'm ready to dig a little deeper.
Product Review Summary: outstanding read for anyone interested in math i was given this book as a gift after taking a course on algebraic topology. while only some of the material appeared in both the book and the text i used for the course, i developed a much deeper understanding of the subject after finishing richeson's outstanding presentation of a difficult subject.
the writing is efficient and enjoyable throughout. many of the chapters serve as interesting interludes or transitions to help clarify relationships between topics.
i have searched high and low for similar books in topology without luck. richeson seems to have a unique gift for "popularizing" topology without losing the interest of those of us who appreciate the depth and beauty of mathematics.
Product Review Summary: A Wonderful Book A rare book that has something for everyone from those with little experience with mathematics, to those with graduate level experience. There are insights for all here. There are, for example, any number of things I learned, beautiful ideas such as the projection onto a sphere proof, though I had seen the standard proof any number of times. The graph theory knot theory and topology discussions provided wonderful intuitions that I, though I had taken courses, had never seen. And there are deep issues about the philosophy of mathematics at work throughout the book. There is no question that once people start reading, they will be enthralled. Thus, the only question is how to get people to start reading it in the first place.
Buy it and read it. I very much look forward to Prof. Richeson's next book.
Product Review Summary: Sublime mix of popular and hardcore mathematics I may have never before read a book (and I have read hundreds of popular math and science books) that does such a fine job popularizing a potentially difficult math topic at such a deep level. My guess is that this book would even be a reasonable introduction to a college level course on topology. But with a bit of effort it also works for a lay person with an interest in math. It reviews the historical development of modern topology by working backwards and forwards from the development of Euler's famous theorem that forms the backbone of the book. It is astounding that such a simple formula could bridge the study and understanding of polyhedra and graph/network theory.
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