Customer Reviews

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

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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.

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!

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.

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.

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.

I just finished reading this book, and it only took me a couple of days. Admittedly, I have some exposition to math, having taken linear algebra, calculus, and differential equations, all of which are very useful in understanding this little book. But even a lay person with only some basic knowledge of geometry and algebra can grasp the content fairly easily. Most of the proofs are visual, and all of them are extremely elegant and simple. If you didn't take much math before but you are interested in the subject, get this book, and don't be scared of the apparent difficulty.

It is very rare that a math book is both so simple and so insightful. The topic is quite advanced, and the concepts, especially in the later chapters, are quite complex. And yet the author explains them in great simplicity. He doesn't go into some details I would have liked to see as someone with a math background, but that makes the book much more clear. And it's very well written. The author is very involved and obviously loves the subject. He also introduces other, related branches of mathematics like graph theory and knot theory, which could have made his book too complicated. And yet he deals with them so simply you might think that all math is like that. He presents all the beauty and elegance with little of the complexity which can make math seem so ugly and incomprehensible.

I really, really recommend this book, especially to the lay audience, and high school/undergraduate students in particular. Many of my friends like math but think that they are not smart enough. This book can show you that this is not the case, you just didn't think about it in the right way before. It's important that people stop thinking of math as something out of their reach, and all that is needed for that is a good teacher. And Richeson is certainly a good teacher.

That said, even a more advanced reader can enjoy this book, both for the incredible presentation, including many illustrations, and the elegant proofs and their sketches, which one can carry out to completion during leisure hours. The historical background is fascinating and the book reads almost like a novel.

The author did a really good job on this book, if only there were more books like it.

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.

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 manifolds 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.

I chanced upon this new arrival book in my local National Library book shelf 2 weeks ago. I congratulate the librarian who put this book at the public loanable section. Truely as the author claims, this Topology is for anyone with or without advanced math backgroud.
Topology has been the 'scarest' subject in the University where my 'sadistic' Math professors used it to 'kill' (to fail) students.
This book tells you Topology is fun and intuitive, it is a 'Rubber-sheet' Geometry as opposed to rigid objects in the axiomatic Euclidean Geometry.
Starting from the Euler formula "V-E+R = 2" (replace F = Face equivalently by R=Region,'V-E+R=2" easier remember as "VERsion 2" with a hyphen '-' before E), the central topic of Topology begins from Descartes / Euler to Riemann, Poincare...
The Epilog on "The Million-Dollar Question" (Poincare Conjecture) details the legendary Russian mathematician Perelman, who refused Fields Medal and in June 2010 rejected the Clay US$ 1 million prize.
Appendix A "Build your own Polyhedra and Surfaces", useful for students to make the 5 Plato Polyhedra on paper.
All in all, this book is excellent as a starter of Topology.

The five Platonic solids--the tetrahedron, cube, octahedron, dodecahedron and icosahedron--are wonderful things. Forget their significance in math and science: their biggest impact had to be with Dungeons & Dragons, where dice came with a variety of sides, almost always based on these solids. How many trolls, mind flayers and cloud giants have been slain by the lucky roll of a twenty-sided die (or icosahedron)?

Surprisingly, D&D is not referred to in David S. Richeson's history of topology, Euler's Gem. Somehow, he does just fine without mentioning it. Of course, the first question for the lay reader might be, what is topology? It is the mathematical cousin of geometry, the study of the properties of objects that remain the same even when distorted. For example, in geometry, a sphere and a cube are different; in topology, they're the same: with some reshaping (but no tearing or cutting), one can be molded into the other. A sphere and torus, however, are different; no amount of reshaping will transform one to the other.

As much as a book like this can have a hero, it is Leonhard Euler, an eighteenth century mathematician who was perhaps the last great generalist: one who would work in all sorts of math fields, from algebra to geometry to calculus to number theory. Of course, Euler was preceded by other mathematical greats, from the Greeks onward. While Euclid would prove many properties of polyhedra, Euclid would come upon a key relationship between the vertexes, faces and edges that would be a constant for a given type of polyhedra. That formula, V-E+F=2, is considered one of the most beautiful formulas in math, second only to a different formula of Euler's, e^(pi)(i) + 1 = 0.

It's not always an easy concept to grasp, which is why it is usually taught at the graduate level. There are some things that topology has worked out that is easier to understand, such as the four color problem (the idea that no map requires more than four colors to illustrate) or the riddle of the seven bridges of Konigsberg (in which the walker attempts to cross all seven bridges without crossing any twice). Also, why are there only five Platonic solids/

As is common in these types of mathematics books, Richeson prefaces things by stating that no math higher than algebra or trigonometry is really needed. This is true, but it doesn't mean that the topics are always easy to grasp. (And my usual minor gripe with these sorts of books holds true for Euler's Gem: endnote references should not be superscripted numbers, where they can occasionally be confused with exponents.) Overall, Richeson does a good job introducing the reader to a sometimes mind-bending subject, but that's just fine: in topology, you can bend minds as long as you don't break them. This should be a good read for the mathematically inclined.