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

David S. Richeson
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Book Description

15 April 2012

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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Product details

  • Paperback: 336 pages
  • Publisher: Princeton University Press; Reprint edition (15 April 2012)
  • Language: English
  • ISBN-10: 0691154570
  • ISBN-13: 978-0691154572
  • Product Dimensions: 23.1 x 15.2 x 2.3 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 154,707 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Product Description


Winner of the 2010 Euler Book Prize, Mathematical Association of America

One of Choice's Outstanding Academic Titles for 2009

"The author has achieved a remarkable feat, introducing a nave reader to a rich history without compromising the insights and without leaving out a delicious detail. Furthermore, he describes the development of topology from a suggestion by Gottfried Leibniz to its algebraic formulation by Emmy Noether, relating all to Euler's formula. This book will be valuable to every library with patrons looking for an awe-inspiring experience."--Choice

"This is an excellent book about a great man and a timeless formula."--Charles Ashbacher, Journal of Recreational Mathematics

"I liked Richeson's style of writing. He is enthusiastic and humorous. It was a pleasure reading this book, and I recommend it to everyone who is not afraid of mathematical arguments and has ever wondered what this field of 'rubbersheet geometry' is about. You will not be disappointed."--Jeanine Daems, Mathematical Intelligencer

"The book is a pleasure to read for professional mathematicians, students of mathematicians or anyone with a general interest in mathematics."--European Mathematical Society Newsletter

"I found much more to like than to criticize in Euler's Gem. At its best, the book succeeds at showing the reader a lot of attractive mathematics with a well-chosen level of technical detail. I recommend it both to professional mathematicians and to their seatmates."--Jeremy L. Martin, Notices of the AMS

"I highly recommend this book for teachers interested in geometry or topology, particularly for university faculty. The examples, proofs, and historical anecdotes are interesting, informative, and useful for encouraging classroom discussions. Advanced students will also glimpse the broad horizons of mathematics by reading (and working through) the book."--Dustin L. Jones, Mathematics Teacher

"The book should interest non-mathematicians as well as mathematicians. It is written in a lively way, mathematical properties are explained well and several biographical details are included."--Krzysztof Ciesielski, Mathematical Reviews

From the Inside Flap

"Euler's Gem is a thoroughly satisfying meditation on one of mathematics' loveliest formulas. The author begins with Euler's act of seeing what no one previously had, and returns repeatedly to the resulting formula with ever more careful emendations and ever-widening points of view. This highly nuanced narrative sweeps the reader into the cascade of interlocking ideas which undergird modern topology and lend it its power and beauty."--Donal O'Shea, author of The Poincaré Conjecture: In Search of the Shape of the Universe

"Beginning with Euler's famous polyhedron formula, continuing to modern concepts of 'rubber geometry,' and advancing all the way to the proof of Poincaré's Conjecture, Richeson's well-written and well-illustrated book is a gentle tour de force of topology."--George G. Szpiro, author of Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles

"A fascinating and accessible excursion through two thousand years of mathematics. From Plato's Academy, via the bridges of Königsberg, to the world of knots, soccer balls, and geodesic domes, the author's enthusiasm shines through. This attractive introduction to the origins of topology deserves to be widely read."--Robin Wilson, author of Four Colors Suffice: How the Map Problem Was Solved

"Appealing and accessible to a general audience, this well-organized, well-supported, and well-written book contains vast amounts of information not found elsewhere. Euler's Gem is a significant and timely contribution to the field."--Edward Sandifer, Western Connecticut State University

"Euler's Gem is a very good book. It succeeds in explaining complicated concepts in engaging layman's terms. Richeson is keenly aware of where the difficult twists and turns are located, and he covers them to satisfaction. This book is engaging and a joy to read."--Alejandro Lpez-Ortiz, University of Waterloo

--This text refers to the Hardcover edition.

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Most Helpful Customer Reviews
3 of 3 people found the following review helpful
5.0 out of 5 stars A gem of a book for a gem of a formula. 5 Oct 2012
Format:Hardcover|Verified Purchase
This is a lovely book. The 18th Century mathematician Leonhard Euler discovered many mathematical formulae - almost every discipline I know has its "Euler's formula". His polyhedron formula, that is the subject of this book, marked the beginning of a mathematical revolution that today underpins everything from the difference between a baseball and a donut to the ten-dimensional world of string theory in theoretical physics.

Richeson takes the beginner on a gentle but irresistible walk through the history of the formula (the "gem" as he calls it) and its mathematical adulthood, the study of shapes in space nowadays known as topology. From Euler's birth in Switzerland to the Frenchman Poincaré's explosive elaborations a hundred years later, we are led step by step on the path of discovery.

There are few mathematical formulae in the book besides the gem itself, and the lay reader may cheerfully skip these without losing the gist of the story. That would be a shame though, as the beauty and elegance of mathematics shines through them.

Learned, clearly and accurately written, thoughtfully referenced and carefully and profusely illustrated, this book is almost as much a gem as the formula itself.
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Most Helpful Customer Reviews on (beta) 4.8 out of 5 stars  24 reviews
72 of 74 people found the following review helpful
5.0 out of 5 stars More Evidence that Mathematics Is Beautiful 22 Dec 2008
By Gene B. Chase - Published on
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 -- 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.
24 of 28 people found the following review helpful
5.0 out of 5 stars I thoroughly enjoyed this book. 7 Dec 2008
By Dave - Published on
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!
17 of 19 people found the following review helpful
5.0 out of 5 stars Very Good, But Challenging 20 Jan 2010
A Kid's Review - Published on
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.
9 of 9 people found the following review helpful
5.0 out of 5 stars A gem of mathematical results produced by one of the masters of mathematics 24 Oct 2009
By Charles Ashbacher - Published on
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.
14 of 16 people found the following review helpful
5.0 out of 5 stars legendre's proof alone is worth the price of this book 10 Oct 2011
By muddy glass - Published on
Format:Hardcover|Verified Purchase
it is hard not to fall in love with topology after reading "euler's gem." this book is the epitome of outstanding mathematical exposition, presenting the history and consequences of euler's humble looking polyhedron formula with extraordinary clarity. richeson takes the reader on a leisurely journey of mathematical exploration to get to the land of algebraic topology, while visiting along the way the surrounding territories of graph theory, knot theory, and classical and differential geometry. by the end, the reader should have realized that the various branches of mathematics are intimately intertwined and the journey itself was of significant value. the reader will see mathematical truth and beauty in the process of creation, as well as in its results.

euler's polyhedron formula is: v - e + f = 2, where v is the number of vertices, e is the number of edges, and f is the number of faces. such a simple formula, and yet so deep! if by some chance you've never plugged this formula before, try it now with a cube. draw a cube and start counting the number of vertices, edges and faces. you will get: v = 8, e = 12, f = 6, and so 8 - 12 + 6 = 2. incidentally, euler was a highly "experimental" mathematician in the sense that he was not afraid of calculations and would crunch things out to see if a pattern emerges. that was how euler found this formula in the first place, even wondering how such a simple observation could have escaped other mathematicians before him.

euler's original proof of his formula was combinatorial in nature and somewhat interesting, but it was legendre's proof that completely blew me away. legendre's proof made me utter the words, "so beautiful!!!" (actually, i also used an f-word in there, but amazon is a family website.) legendre's ingenious idea was to consider the images of the vertices, edges and faces as projected onto a sphere encompassing the convex polyhedron. the projection is with respect to a point light source inside the polyhedron. the problem then transforms into a counting problem of areas on the sphere, completely out of left field! everyone who has an interest in mathematics should see the details of this proof before leaving this world. legendre's contribution to uncovering the truly topological aspect foreshadows some of the later consequences of euler's polyhedron formula. we see here an entrance to the road leading to triangulations of surfaces and the results that followed that development.

while richeson's book is suitable for a large readership, its potential is perhaps greatest among high school students who show promise in mathematics. this book expounds the history of the polyhedron formula, offers biographical sketches of great mathematicians, goes through different proofs, explores connections and cross-fertilization in the mathematical empire, and gives the reader a sense of the art of mathematical thinking. it is almost certain that not everything in "euler's gem" will be fully understood by a student at the high school level, but that's perfectly ok. it is good for the mind to see glimpses of where mathematics is heading in future courses so that math doesn't feel like meaningless memorization without any direction. i hope "euler's gem" will gain popularity among high school faculty members so that they will recommend it to their brightest students; i hope this book will be used to stoke the fires in the minds of those who will later walk the path of math and science.

in writing "euler's gem," richeson has done the mathematical community a tremendous service. topology has never before been so lucidly explained to so wide an audience. well done.
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