Euler's Gem: The Polyhedron Formula and the Birth of Topology Paperback – April 15, 2012

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.

Perfect

This review is about the Kindle Version.I got this book as a way to understand some of the fundamental ideas of topology, but it reads as more of a guided tour through some historical geometry. Having said that, I definitely enjoyed reading it, and liked a lot of the little puzzles sprinkled throughout. Don't get this if you're looking for anything rigorous (I wasn't), but the author does a reasonable job of trying to build the intuition of why various theorems are true.

Euler's Gem uses a simple formula (V - E + F = 2) to relate topics in polyhedra, graphs, knots and topology. This is written for a non-professional audience, in an informal style and without assuming any particular mathematical background. I am not sure how well this works with a reader who really is completely new to all the topics covered in the book, but it should be a rewarding read for anyone sufficiently interested in the topic to crack open the book in the first place.In addition to a substantial section on Euler's life, most of the prominent mathematicians discussed are given at least a short biography before diving in to what they contributed to "Euler's Gem". The problems themselves are given a bit of a historical perspective (A started it, B fixed it, C extended it, D showed how it was related to another big topic, etc). This really helps emphasize the evolving nature of work in this area, starting around 2300 years ago, with substantial development 400 years ago, and continuing related work in the present day.Despite being familiar with much of the technical material at the advanced undergraduate or 1st year graduate level, I learned several results (e.g., Pick's theorem and the "five neighbors" theorem) and some techniques (e.g., Legrendre's proof of Euler's polyhedron formula). So a considerable breadth compensates for the lack of technical depth.The text cites ample references to the professional literature, if the reader wants to go there, but in a low-key way. There is certainly no pressure to go off and study a chapter of something else in preparation for the next topic in this book.

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 through as I turned to it. Richeson makes this easy by keeping each chapter almost entirely self contained and independent of other chapters. I will be reading this one cover to cover.Never before have I had an entire branch of mathematics explained to me by a single book and at just the right depth of coverage to both give me a good grasp of each topic, and to make me want to dig deeper and learn more. I took topology in college, only to learn that it was elementary point-set topology. Nothing could be more dry and disappointing. Absent from the material presented was every topic I had heard of that fits under the umbrella of 'topology'. Well here in this one volume are the platonic and Kepler polyhedra, the bridges of Konigsberg, graphy theory, knot theory, classification of surfaces, the 4 color theorem, the Poincare conjecture, Poincare-Hopf theorem, Brouwer fixed point theorem, and some algebraic topology. Amazingly, he ties it all together with the use of Euler's Formula. This is the book I should have had in college.Previously, I always looked for Paul Nahin's books. Now I will be looking for Richeson too.