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Euler's Gem: The Polyhedron Formula and the Birth of Topology Hardcover – September 28, 2008

by David S. Richeson (Author)

33 customer reviews

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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.

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

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Product Details

Hardcover: 336 pages
Publisher: Princeton University Press; 1 edition (September 28, 2008)
Language: English
ISBN-10: 0691126771
ISBN-13: 978-0691126777
Product Dimensions: 9.2 x 6.1 x 1.1 inches
Shipping Weight: 1.1 pounds (View shipping rates and policies)
Average Customer Review: 4.7 out of 5 stars See all reviews (33 customer reviews)
Amazon Best Sellers Rank: #165,926 in Books (See Top 100 in Books)
- #16 in Books > Science & Math > Mathematics > Geometry & Topology > Topology
- #118 in Books > Science & Math > Mathematics > History
- #385 in Books > Science & Math > Mathematics > Applied > Probability & Statistics

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Customer Reviews

4.7 out of 5 stars

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78 of 80 people found the following review helpful

More Evidence that Mathematics Is Beautiful

By Gene B. Chase on December 22, 2008

Format: Hardcover

If you want a popularized book-length treatment of string theory, you have two kinds of choices. Brian Greene uses no equations, save in an occasional endnote. Roger Penrose uses 1136 equation-filled pages to teach you all of mathematics you would need to know--although far too fast for anyone to learn it from Penrose alone. There is not much between Greene and Penrose.

If you wanted a popularized book-length treatment of topology before Dave Richeson's Euler's Gem: The Polyhderal Formula and the Birth of Topology, you had no choice at all.

This is a risky thing that Richeson attempts. Ian Stewart's 2007 book Why Beauty is Truth: The History of Symmetry cites the "conventional wisdom in science writing that every equation halves a book's sales." (34) On this basis, Richeson's book should have only
one ten billionth of the sales of other books popularizing science. Yet Richeson pulls it off with a well-written, nicely illustrated book surveying the history of topology from Plato to Poincaré to Perelman.

Richeson's book is accessible to an academically minded high school student, yet has something to offer the professional mathematican who happens not to be a topologist.

There are no typos in the book. There is a useful, although not comprehensive index. (Richeson mentions flexible polyhedra -- see mathworld.wolfram.com/FlexiblePolyhedron.html -- for example, but the index doesn't.) The only slight confusion that I encountered is at page 157, which says that we have seen V-E+F = 2-2g before. We have not. However, on page 148, we saw V-E+F = 2 - 2T + P + 2C, so let P = C = 0 and rename T as g, and all is clear.

Richeson's book ends on the theme of beauty, and well it should. It's a beautiful book! I bought three as Christmas presents for friends. You should buy one too.

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21 of 23 people found the following review helpful

Very Good, But Challenging

A Kid's Review on January 19, 2010

Format: Hardcover

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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27 of 31 people found the following review helpful

I thoroughly enjoyed this book.

By Dave on December 6, 2008

Format: Hardcover Verified Purchase

I really enjoyed this book. I found that the David Richeson's writing style made this topic very accessible. I thought that there was just the right balance of history and math. Having little experience with topology, learned a lot about it. I was really astounded at some of the unexpected connections between "Euler's Gem" and different branches of math.

Lots of fun!

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11 of 11 people found the following review helpful

A gem of mathematical results produced by one of the masters of mathematics

By Charles Ashbacher HALL OF FAMETOP 500 REVIEWERVINE VOICE on October 24, 2009

Format: Hardcover

The title of the book is derived from the formula V - E + F = 2 that holds for any convex 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.Read more ›

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Most Recent Customer Reviews

Five Stars

Wow!

Published 4 months ago by Moribund Alpaca

The finest example of popular math literature I have ever seen.

I often will flip through a new book reading short sections before starting from page 1. When I tried that with this book, I found I was so enthralled that I read each chapter... Read more

Published 4 months ago by Paul K. Zimmerman

An OK account of some simple stuff.

Published 4 months ago by Kenneth A. Perko, Jr.

Five Stars

Very good.

Published 6 months ago by Enrique

Interesting to those of us who like to read math books

Interesting to those of us who like to read math books. Informative and historical. Some medium to higher level math knowledge is necessary to get the most out of this book. Read more

Published 8 months ago by William L. Brunner

Euler's Gem - A gem of a book

This is a wonderful book. Even for a professional mathematician, there is much to learn from this book. This book is an excellent example in how to write Mathematics. Read more

Published 8 months ago by Dr S Muralidharan

Analysis Situs for everyone

I ran into this book because my ex professor recommended it (he didn't have a strong topology background). Read more

Published 10 months ago by Troy learns to design

Five Stars

an excellent book,

Published 13 months ago by Rajesh R Parwani

Five Stars

Can learns lot of math just reading
One book

Published 14 months ago by Janet Schofield

One needn't be a mathematician to enjoy it,

"Euler's gem" is a story of mathematical passion, the kind of passion for abstract ideas that leads mathematicians to discover unexpected relationships that may exist only... Read more

Published 14 months ago by Jim

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