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

4.29  ·  Rating details ·  293 ratings  ·  27 reviews
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 r
Hardcover, 317 pages
Published September 28th 2008 by Princeton University Press (first published 2008)
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4.29  · 
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 ·  293 ratings  ·  27 reviews

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I've been dreaming of higher dimensions lately, and this book on topology just enthralled this fascination even more. "Euler's Gem" is really a look at one of the most famous equations you've never heard:
V-E+F=2, also known as Euler's Formula. This formula originally described a relationship between the faces, edges, and vertices of the 5 platonic Solids, but actually has a deeper significance as an equation connecting the vast subtopics of topology, including graph theory, knot theory, the 4-co
Bob Gustafson
Nov 19, 2013 rated it really liked it
Richeson's "Euler's Gem" is an excellent book. It gives the historical background, going back to ancient Greece, for this equation regarding faces, edges and vertices of polyhedra. It tells us about Euler (as well as more than a dozen other mathematical scholars) and the relationship. It goes on to tell us about various proofs and then extensions of and enhancements to the equation.

If this book is excellent, why four rather than five stars? It's because of the medium. This book was brought to my
Dec 17, 2008 is currently reading it
I am more or less in the middle of this book and I'm loving it. It explains, with simple words, the relevance of the polyhedron formula; filled up with history and proofs. Thumbs up!
Feb 17, 2013 rated it it was amazing
Shelves: mathematics, general
Nice topology and geometry book. The proof part is especially good. Would recommend this to go along with Weeks' The Shape of Space, or vice versa.
Ben Orlin
Mar 21, 2019 rated it it was amazing
A mathematician friend recently asked me, "Do you think pop math books could be a good way for me to learn about other fields?" (He studies partial differential equations.)

"Sure!" I said, and handed him this book. "Here's algebraic topology!"

"Great," he said. "Any others?"

I looked at my shelf, saw that the answer was no, and realized in that moment how unusual Richeson's achievement is. This is an engaging, readable, historical tour, covering a large swath of interesting math, pitched at an appr
Aug 07, 2019 rated it really liked it
Shelves: viii-pretty-good
This is quite an ambitious book for the author, and he does a great job showing the evolution of topology from its roots in geometry, graph theory, and knot theory. The math gets decently high-level by the end, but the author makes a great effort at trying to reach those without a formal understanding of the topics. This book isn’t for people who aren’t interested in mathematics, but for those who are amateurs to probably grad students this seems to be a fulfilling read.
Richard Holmes
Oct 26, 2017 rated it it was ok
Shelves: stopped-reading
This book wasn't what I was expecting; rather than a book centered on Euler's formula, it's an introduction to topology with the formula as a recurring theme. As such large parts of the book serve to introduce topics having little to do with the formula and which I was already familiar with. I didn't find Richeson's pedestrian exposition shed much in the way of interesting new light on these topics for the most part. I ended up skipping a lot, because it just wasn't what I was looking for.
Hyung Mook Kang
Feb 23, 2019 rated it it was amazing
Excellent book for the overall history of geometry and topology, as well as its motivations behind. The motivations behind the rigorous definitions of modern topology we learn (such as in Munkres) aren't in the book.
Matthew Dambro
Aug 21, 2017 rated it it was amazing
Excellent introduction to topology. Much of the math was over my head but the story of how mathematics is done is superb. The step by step accumulation of knowledge with the occasional flash of brilliance shown by a Euler or a Poincare is breathtakingly beautiful.
Jul 04, 2015 rated it it was amazing
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 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 somewh
Koen Crolla
Oct 11, 2013 rated it really liked it  ·  review of another edition
Shelves: mathematics
Euler has arguably produced a lot of gems, but the one Richeson is talking about is the observation that for convex polyhedra, the number of vertices minus the number of edges plus the number of faces is always equal to 2. That this is not true for less sensible polyhedra (it's 0 for a reasonable polyhedral torus, for example) is one of the foundational observations of the field of topology, which is indeed what the book is about.

Euler's Gem is more or less what popular mathematics should be: it
Gabriel Daleson
Mar 18, 2013 rated it really liked it
The first two thirds of this book is heavier on the history and geometry, and it's the better part. The back third gets into topology, and, while it's certainly interesting, it's also not for the squeamish. I'm really not sure how one could easily provide a casual treatment of any high-dimensional mathematics, even as casual as Richeson does here, without losing people. I appreciate the effort, certainly, but I can make a good argument that the chapters on the higher-dimensional analog to the Eu ...more
I read this book on the history of topology to my elementary-school-aged son. I cannot really remember when we finished it, but it was some time in the summer of 2011. The idea of a book for a popular audience about topology is absurd. Once I owned a book called Learn Gujrati in 30 Days, and this was something like that. My kid was too young to realize that it was hard to understand and just had a lot of fun. We read about a chapter a night for weeks and weeks. At the very end of the book was a ...more
Sep 04, 2015 rated it it was amazing
This turned out to be an excellent introduction to topology via Euler's second most famous formula, V - E + F = 2. The first 30 pages were a history lesson, and after that it kept ramping up the math, ending at about the same place where my brain stopped working in grad school (homology groups). The proofs are mostly of the "general overview without the nasty tricky details" variety, which is about right for a book like this. Recommended for those who want a pop-math introduction to topology or ...more
Aleksandr Jermakov
"The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living."

These are Poincaré's words author quotes at the end of the book. And he very well manages to show the nature's beauty in his book. More of biography/history of science than actual Mathematical technicalities so if you are
Mohammed Hashem
Aug 12, 2012 rated it really liked it
Great recap of the history of the relatively new mathematical field of Topology.

A quote I liked...

“Scientists and engineers have used computers to solve countless problems, but mathematicians have not. Computers are good at making speedy calculations, but not at the kind of precise and subtle arguments that are required in mathematical proof. Like philosophy and art, mathematics has always been a human endeavor, one that cannot be automated.” David S. Richeson, Euler’s Gem
Matthewmartinmurray murray
Jan 17, 2012 rated it really liked it
This was really fun to read at first but then goes heavily into topology and left me in the dust a little. I have a better understanding of some topological ideas but could have used a better explanation of most. I am much more comfortable visualizing the projective plan now which I am glad for. I also was interested in seeing some knot theory tie in. Its a good insight to one of the weirder sides of mathematics. So it was fairly good.
Trinity School Summer Reading
Leonhard Euler was an incredibly prolific mathematician who lived in the 18th Century. Among his many fabulous creations, his polyhedron formula is one that describes the structure of 3D shapes from soccer balls to salt crystals. In this book, Richeson describes the development of this indispensable mathematical idea and its connection to the modern fields of graph theory and topology.
Jul 25, 2018 rated it liked it
Very nice book talking about the history of Euler's polyhedron formula and it's applications.

Easily accessible with only a bit of mathematical background.

My favorite part was learning about Pick's theorem, a beautiful theorem that I didn't know anything about before this.
Nov 17, 2012 rated it really liked it
Yes math nuts, this book is for you. Euler's Gem is his mathematical statement that the sum of the vertices and faces of a polygon is equal to the sum of the edges and 2. It shows the relationship to modern topology.
Steve Schlutow
This was an okay book.. I did remind me of Euler's famous formula V-E+F=2 that I used once or twice in one of my math classes back in school (college).. I was interested in the connection of polyhedra and topology, which did not explain anything new.. Oh well.. Not a bad book--pretty easy read..
Jul 12, 2014 rated it it was amazing
Great book! Before I knew next to nothing on topology, now I know basic knot theory and a little graph theory.
Ralph Stoever
Jan 01, 2012 rated it it was amazing
Shelves: reference-books
A great, clear and still detailed introduction to topology.
Jan 03, 2011 rated it really liked it
Shelves: math, 2011
More interested in geometry than topology but this book is accessible, made up of 23 reasonably digestible chapters.
Apr 14, 2010 rated it it was amazing
I've seen explanations of a number of these topics before, but these are the best. The historical context makes it easier to remember the math and the diagrams are top-notch.
Jan 24, 2016 rated it it was amazing  ·  review of another edition
An interesting journey through the history of topology, along with the conceptual basics of topology presented in a way that is easy to understand.
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Jan 15, 2015
Jon Gould
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General comment 1 5 Dec 14, 2013 10:19AM  

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David Richeson is a professor of mathematics at Dickinson College and the editor of Math Horizons, the undergraduate magazine of the Mathematical Association of America. He received his undergraduate degree from Hamilton College and his masters and PhD from Northwestern University. He lives with his wife and two children in Carlisle, Pennsylvania.
“In his life of seventy-six years, Euler created enough mathematics to fill seventy-four substantial volumes, the most total pages of any mathematician. By the time all of his work had been published (and new material continued to appear for seventy-nine years after his death) it amounted to a staggering 866 items, including articles and books on the most cutting-edge topics, elementary textbooks, books for the nonscientist, and technical manuals. These figures do not account for the projected fifteen volumes of correspondence and notebooks that are still being compiled.” 2 likes
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