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

by David S. Richeson

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.

Genres: Mathematics, Science, Nonfiction, History, Stem, Physics, Biography

336 pages, Hardcover

First published January 1, 2008

About the author

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.

Reviews

[Ed Erwin · Rating 3 out of 5]

Basically the history of Euler's equation F + V − E = 2, how it was discovered and proved, and the mathematics that developed from these investigations, such as topology and knot theory. There are more equations in the book than in most science books aimed at a wide audience, but they mostly involve nothing more complicated than simple addition and subtraction. The math does get a bit complicated in the final chapters.

As in many such books, there are brief biographies of the mathematicians involved. I could have done without that, though a few of the stories were interesting. There were even a few that I haven't seen 10 times before.

The illustrations are mostly drawn by the author. Computer-generated art with proper perspective and lighting would have helped in some cases.

Quality: 4 stars. My enjoyment: 3 stars.

[Upom · Rating 4 out of 5]

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-colors problem, dynamic systems, and homology. Richeson uses Euler's Formula as a way to present the history of toplogy, as well as a grand tour of this little-known topic. Speckled with nice illustrations, pop-culture references, and some really crisp writing, Dickeson reveals a subject that radiates with wonder. The proofs are at times a bit hard to follow, but with a little patience and periodic page-flipping, they are easy enough to follow. The book also contains a nice little appendix with cutouts that you can use to make your own platonic solids, toruses (torii?), Klein bottle, and projective planes. The book could have done with a glossary, and I found myself reading proofs again and again to make sense of them. However, if you give it a little time and effort, "Euler's Gem" is a book that will blow your mind. Just as the book reminds us in the last line with Poincare's words, "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."

[Bob Gustafson · Rating 4 out of 5]

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 attention by "The Shape of Nature" a lecture series by Professor Devados on DVD from The Great Courses. Drawings in two-space can only do so much. Dynamic demonstrations in three-space, even though the computer screen is flat, makes things so much clearer.

An excellent presentation on DVD is simply better than an excellent book.

[Erickson · Rating 5 out of 5]

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.

[Kam Yung Soh · Rating 3 out of 5]

An interesting book that looks at Euler's Formula, namely for a polyhedron, the number of Faces plus the number of Vertices (corner points) minus the number of Edges always equals 2, or F + V − E = 2.

The book starts with a look at the life of Leonhard Euler, from his early mathematical education to his life in both the Russian and Prussian academies of science and his death. It then gives the history of polyhedra and some early findings about them, before presenting Euler's formula and some ways to prove it as devised by various mathematicians. It also provides a proof that there are only five regular polyhedra using the formula.

Other chapters show how the formula began to be used for other areas of mathematics. Graph theory is one area, as shown by the famous example of the Bridges of Königsberg problem. Attempts to provide a proper mathematical definition of a polyhedron using the formula would lead to the field of topology. Subsequent chapters show the application of the formula to various topological fields like knots, vectors on topological surfaces (via the Hairy Ball Theorem and the Brouwer Fixed Point Theorem).

Further applications of the theorem encompass other topics like the curvature of surfaces and how topology is now a field that has developed rigorous proofs, and its various parts combined into a unified whole.

The early chapters are light on mathematics, while some people may struggle with the later chapters on topology. But it should be possible to skip them and get the conclusion that Euler's formula that once defined a relationship of polyhedra has now been applied to the larger area of topology, leading to new mathematical findings.

[Anh Phan · Rating 5 out of 5]

An incredible math history book tracing Euler’s polyhedron formula V - E + F = 2 through advancements in geometry and topology with intuitive explanations and diverse illustrations, balancing technicality with proof outlines. Engaging and inspiring!

[Koen Crolla · Rating 4 out of 5]

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's gentle enough for a complete lay audience to follow from start to finish, but also thorough and not afraid to get technical. It covers some history, but doesn't repeat popular myths or devolve into crass gossip. The title isn't necessarily ideal—there are some other, largely unrelated observations at the basis of the field, which Richeson also covers and which fit the historical narrative, but not the title's conceit—and the cover art is awful enough that I was seriously considering covering it up, but on the whole, it's definitely one of the better general-audience books on mathematics I've read.

[Daniela]

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!

[Tom · Rating 4 out of 5]

For some decades now, the Princeton Science Library has reprinted paperbacks in mathematics and the sciences—from astronomy to zoology—written for general audiences possessed of a high school education, a sense of curiosity, and the patience work through the occasional tough spot. Mathematics was never my thang, and I declined to enroll in calculus my senior year in high school. A few years later, when I discovered Martin Gardner’s math games column in Scientific American, I realized how fascinating and—yes—cool mathematics can be in their implications, use, and elegance.

In Euler’s Gem, David S. Richeson, Professor Mathematics at Dickinson College, explains the implications and applications of just one of Leonhard Euler’s many theorems, this regarding just three properties of solid geometric shapes—their faces (surfaces), edges (along which the faces connect), and vertices (the point where two or more edges connect). Oddly enough, no other mathematician or philosopher before Euler had considered edges as a property of polyhedra. The simple recognition that shapes have edges allowed for the insight that V – E + F = 2 in the case of geometric solids with stiff surfaces but equals 0 in the case of such topographic solids as the donut-shaped torus. Richeson demonstrates—with plenty of examples that allow readers to prove to themselves the truth of the mathematical claims—why, how, and where Euler’s theorem works.

Euler’s Gem is more than a series of mathematical factoids. Richeson brings in historical and biographical background to the discoveries made by Euler and other mathematicians to illustrate the human personalities and motivations behind both the discoveries and the errors made over the centuries. (Negative results can present a lot to learn from.) We learn about the history of geometry, its discoveries, proofs, insights, and assumptions—all just by answering the “so what”? behind knowing that V – E + F = 2. These include implications for structures at the micro- and macroscopic levels—from the potential shapes of atomic molecules to graph theory; from the invention of topology (by Euler as a result of settling a debate within a town bisected by a river and island, across both of which were seven bridges, if it were possible to cross each bridge only once) to why the four-color conjecture for planar graphs is true (the conjecture stating that maps may be drawn of multiple countries sharing a border using no more than four colors to distinguish the countries from each other); from determining the area of oddly shaped plots of land, to “prime knots with 16 . . . crossings” (!); and more. Appendix A includes a variety of complex shapes that readers are encouraged to (photo)copy so they can demonstrate for themselves several principles covered in the book. Recommended.

For more of my reviews, please see https://www.thebookbeat.com/backroom/...

[Chayanika · Rating 5 out of 5]

I can't really think of anything to complain about. At the beginning of the book, Richeson says that he has written his book for a self selecting audience, neither too qualified, nor too new. This was a great book for me, as a person with no formal mathematical training beyond school and who loves mathematics. I never knew so many things... Had studied Euler's formula in school, but had never thought so deeply about it, never imagined it has topological implications, never really thought about the exact definition of a polyhedron! That it should lead to so many wonderful and fascinating concepts in topology, is amazing. This book is a mixture, and it worked very well for me; there is a lot of history, the story of the mathematicians who did work in this field, but thankfully it is not JUST those stories with handwavery when it comes to the actual concepts like a lot of other books. Richeson does explain and discuss the relevant concepts and even gives hints and let's the reader prove some of the things! And, unexpectedly, for a math book, there are even a few beautiful quotes in the book that left an impact on me,

"It is my belief that the audience for this book is self-selecting. Anyone who wants to read it should be able to read it. The book is not for everyone, but those who would not understand and appreciate the mathematics are precisely those who would never pick it up in the first place."

"The needs of scientists often do spur the creation of new mathematics, just as Kelvin’s vortex model of atomic theory inspired knot theory. But mathematicians do not relish being servants of science. Even when a mathematical theory is born from a practical application, it quickly takes on a life of its own and is advanced for its own inherent properties. Theoretical mathematicians are a stubborn bunch who as a whole are more interested in beauty, truth, elegance, and grandeur than in applicability."

'Let chaos storm!Let cloud shapes swarm!
I wait for form.

—Robert Frost, “Pertinax”1'

[Benjamin Linowitz · Rating 4 out of 5]

David Richeson is one of my favorite writers of popular mathematics. When I reviewed Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity for the Mathematical Association of America, I noted that the writing was “friendly but not chatty and scholarly but not dry." His book Euler's Gem: The Polyhedron Formula and the Birth of Topology deserves the same praise. In Euler's Gem Richeson gives a masterful historical introduction to Euler's famous V-E+F=2 formula for polyhedra and its subsequent generalizations. Beginning in Ancient Greece with Pythagoras, Euclid, and Archimedes, he traces the development of polyhedra and their study over the centuries, culminating in a discussion of the Euler–Poincaré characteristic of manifolds. Along the way the reader is treated to discussions on the birth of graph theory, topology, knot theory, and homology theory, and learns about the four color theorem, the Poincaré-Hopf theorem, the hairy ball theorem, and the Gauss-Bonnet theorem. As is evident, the book necessarily covers a wide swath of tough mathematics. Richeson's exposition of this mathematics is intuitive and non-technical, and for the most part effective. Towards the end of the text the mathematics begins to build upon itself very quickly, and I question whether many readers that had not already been exposed to Betti numbers and their connection to the homology of manifolds would get much out of Richeson's treatment. Regardless, I enjoyed Euler's Gem immensely, and have already begun to recommend it as supplementary reading in my geometry courses.

[Daniel Takemori · Rating 5 out of 5]

Euler's Gem is a book that talks about the birth and applications of the field of math called topology. Topology is a branch of geometry that focuses on properties of shapes like how many tunnels it has or having lines that connect points on a shape. Richeson talks about the polyhedron formula saying that the vertices of a polyhedron plus the number of faces minus the number of edges equals 2 for all closed convex 3D shapes. a moment of epiphany for a character was when mathematician Poincare discovered the reason for the seemingly arbitrary number 2 in Euler's formula that even Euler himself did not understand. He discovered that shapes with varying numbers of "tunnels" or holes in a polyhedron have different numbers so a shape with zero holes = 2 but a shape with 1 hole = 3. he discovered that constant that the formula equals is n+1 where n is the number of holes. This is the second most important discovery of the book aside from the original Euler's formula as this discovery carves the way for modern topology as one of the main focus's for topology is how many tunnels a shape has. Richeson tries to argue that changing the way we look at things is important. Poincare changes the way he sees Euler's formula and discovers something because of that. This is an important value to me because often people take for granted what they know and do not question it. I think that often when we are stuck on a problem it is beneficial to take a different view point to find a different way of solving similar to how Poincare did.

[Ben Orlin · Rating 5 out of 5]

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 appropriate level. It's not for a "general audience," per se, but anyone with a moderately technical undergraduate background should get a lot out of it (and there's no harm in skimming a few of the proofs sketched).

Bits of mathematics that I'd seen as disparate - graph theory, spherical geometry, Mobius strips, Platonic solids - are drawn together here, to create a pleasing and natural whole.

[Beau · Rating 5 out of 5]

The author praises Leonhard Euler as a great expositor, but the author himself must be just as great. This book made deep topological concepts accessible to the non-expert probably in ways no one has done before, and yet he did not shy away from proofs; there are some actual rigorous proofs and some proof sketches based on visual intuition. For example, I feel like I understand why the Poincare-Hopf theorem is true, even though I did not actually do the math. In addition to some serious mathematics supported by a collection of the best figures and pictures I've seen in such a work, this book is full of history, biography, and other interesting commentary. I highly recommend this book to anyone interested in mathematics; it probably should be required reading for students of mathematics taking their first course in topology.

[Wing · Rating 5 out of 5]

The relative locations and general arrangement of features determine the intrinsic and extrinsic properties of a structure. The most elemental building blocks have invariant basic features. This book is both a history and an exposition of this profound observation. Of the 270 pages of well-written text, the first 240 are easy to follow entirely. However, the last 30 pages, which deal with manifolds, are rather inscrutable. That doesn’t detract from the overall achievement of this very clear text. Overall, I love it. It demystifies many otherwise bewildering concepts. Five stars.

[Richard Holmes · Rating 2 out of 5]

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.

[Tyler · Rating 4 out of 5]

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.

[William Schram · Rating 5 out of 5]

Euler's Gem is a book referring to the Polyhedron Formula. It links the vertices, faces, and edges of any shape. Author David S. Richeson discusses the development of the formula and how mathematicians proved the theorem.

Along the way, we touch upon several aspects of mathematics. The book centers on topology but has chapters about graphs, projective geometry, and vector spaces.

Thanks for reading my review, and see you next time.

[John Golden · Rating 5 out of 5]

You'll be hard put to find a better written book of mathematics, nor a more interesting mathematical topic. Gets better the more math you know, but well addressed to a general audience. I recommend to every math major (literally, in my history of math course) and to anyone interested in one of the most beautiful pieces of mathematics.

[Rahul Banerjee · Rating 4 out of 5]

A really nice book that introduces several key ideas of modern geometry and topology, while tracing the genesis and development of these concepts through history, going back all the way to Plato and Pythagoras. In addition to the math, the text also includes a fair bit of biography of several relevant mathematicians.

[Matthew Dambro · Rating 5 out of 5]

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.

[Hyung Mook Kang · Rating 5 out of 5]

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 Emery · Rating 5 out of 5]

This is everything a math book aimed at a popular audience should aspire to be. The author tackles a challenging topic starting from the very beginning. There are plenty of interesting examples used and the bits of historical trivia liven up the more theoretical portions of the book.

[Stuart]

This is a tremendous book. The concepts are fascinating and the author does a great job organizing and clarifying them. The historical and biographical details give an interesting context while being a nice break from the concentration needed (at least by me) of the meat of the matter.

[Louis Meunier · Rating 5 out of 5]

An incredible introduction to topology, with a near-narrative approach. The intermixing of history and mathematical explanations provides just the right amount of context, with the inherent intertwining present this area of mathematics well reflected in the layout of this book.

[Benjamin]

Dear Santa...

[Kent Sibilev · Rating 5 out of 5]

There are not many really good popular math books. This is definitely one of them.

[Ivan Koma · Rating 4 out of 5]

if briefly, all 330 pages of the book about - "V − E + F = 2", but in mathematician way
it's just a recycling symbol ♲

[Kt · Rating 4 out of 5]

Difficult but very worthwhile read

[David Hosier · Rating 3 out of 5]

Started off interesting with biographical accounts of famous mathematicians, but then got way too into topology.