Ebook Euler Gem The Polyhedron Formula and the Birth of Topology Princeton Science Library Book 64 Reprint David S Richeson
Download As PDF : Euler Gem The Polyhedron Formula and the Birth of Topology Princeton Science Library Book 64 Reprint David S Richeson
How a simple equation reshaped mathematics
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 theorem is so simple it can be explained to a child. 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. Using wonderful examples and numerous illustrations, David Richeson presents this mathematical idea’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. This paperback edition contains a new preface by the author.
Ebook Euler Gem The Polyhedron Formula and the Birth of Topology Princeton Science Library Book 64 Reprint David S Richeson
"I read this book after going through a series of books about mathematics aimed at the general public, including MacCormick's book about modern algorithms and a book about general mathematical logic and reasoning.
This book gave an excellent overview of proofs and theorems in topology. The bridges of Koenisberg and the Four Color problem being my favorites. Like any book about math, it can be a little dense sometimes, but I found it to be enjoyable more often than not."
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Euler Gem The Polyhedron Formula and the Birth of Topology Princeton Science Library Book 64 Reprint David S Richeson Reviews :
Euler Gem The Polyhedron Formula and the Birth of Topology Princeton Science Library Book 64 Reprint David S Richeson Reviews
- 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. Even though I knew the Euler's formula and several proofs, the proof by Legendre took me completely by surprise - what a beauty! Richeson gently leads you to the final result through carefully chosen exploratory path so that at the final destination (pulling out Euler's formula from area calculation on a sphere) leaves you speechless! By the time you finish the book, you would have fallen madly in love with Topology.
- Extremely well written book. And very well documented. It covers a very nice selection of material. My only reservation is that, while it may make the reader “wonder†and marvel, it doesn’t have problems or puzzles to test and extend his/her understanding; i.e., the reader is encouraged to wonder, but not to think.
- 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. - I read this book after going through a series of books about mathematics aimed at the general public, including MacCormick's book about modern algorithms and a book about general mathematical logic and reasoning.
This book gave an excellent overview of proofs and theorems in topology. The bridges of Koenisberg and the Four Color problem being my favorites. Like any book about math, it can be a little dense sometimes, but I found it to be enjoyable more often than not. - I knew about euler's formula from high school, but I'd never seen it connected with topology. This is such an easy way to approach topology and I wish my classes had started here, rather than with point sets and limits. It finally made it clear to me why people talk about rubber sheets when they talk about topology - it was completely not clear to me how this related to the stuff I'd learned about topology as a college student.
- 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 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. - I ran into this book because my ex professor recommended it (he didn't have a strong topology background). I already audited some courses in algebraic topology and graph theory before so this book to me is a page turner. But the book is so well written that even for layman, it should certainly be enjoyable, much more enjoyable than the notorious Love and Math...
- Fun read for math students
