Euler's Gem: The Polyhedron Formula and the Birth of Topology

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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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• ISBN13: 9780691126777
• Condition: NEW
• Notes: Brand New from Publisher. No Remainder Mark.
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Product details:

Item number (ASIN): 0691126771
Author: David S. Richeson
Dewey Decimal Number: 514.09
Edition: 1
ISBN: 0691126771
Manufacturer: Princeton University Press
Number Of Items: 1
Number Of Pages: 332
Package Dimensions: 110 x 620 x 940 (hundredths-inches)
Publication Date: September 8, 2008
Publisher: Princeton University Press
Binding: Hardcover

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Rating: - Very Good, But Challenging
Euler's Gem is a fascinating & well written book. However, it is also a pretty challenging read, one can not really sit back & read it straight through. But this is also what mathematics & learning is all about, as you often have to stop, re-read, & think a bit about what is being said. The claim is made that someone with only high school mathematics could read the book, & while this is probably true, it would be a steep climb. Especially as one progresses further & further into the book, many references are made to calculus, differential equations, & other related ideas, which the author does a fantastic job of explaining the ideas to people that never had the courses, but in the end it really would help the reader to have that knowledge beforehand. What makes this a five star book is that it is so rich in knowledge. The average person won't be able to read it in a week, but if you're willing to put the time into the book, you'll get a lot of out it as it really is a great introduction to topology. Even if you can't pick up all the concepts, you're sure to be able to pick up many of the neat tricks the author points out, such as the wedding ring knot, coloring map problem, etc. Overall, one of the best books I've ever read, & one day I'll probably have to re-read it again because it's just so rich & packed with knowledge.

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Rating: - A Gem Indeed!
This is as good an introduction to topology as any for someone who isn't a professional mathematician. Even the professional can learn the history behind very familiar material.

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Rating: - 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.

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Rating: - 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.

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Rating: - 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.

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