''Euler's Gem: The Polyhedron Formula and the Birth of Topology'' is a book on the formula

V-E+F=2

for the Euler characteristic of convex polyhedra and its connections to the history of

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

. It was written by David Richeson and published in 2008 by the

Princeton University Press Princeton University Press is an independent Academic publishing, publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, ...

, with a paperback edition in 2012. It won the 2010 Euler Book Prize of the Mathematical Association of America.

Topics

The book is organized historically, and reviewer Robert Bradley divides the topics of the book into three parts. The first part discusses the earlier history of polyhedra, including the works of

Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politic ...

Thales Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regard ...

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ...

, and Johannes Kepler, and the discovery by

René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathe ...

of a polyhedral version of the Gauss–Bonnet theorem (later seen to be equivalent to Euler's formula). It surveys the life of Euler, his discovery in the early 1750s that the Euler characteristic

V-E+F

is equal to two for all convex polyhedra, and his flawed attempts at a proof, and concludes with the first rigorous proof of this identity in 1794 by

Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are nam ...

, based on Girard's theorem relating the angular excess of triangles in spherical trigonometry to their area. Although polyhedra are geometric objects, ''Euler's Gem'' argues that Euler discovered his formula by being the first to view them topologically (as abstract incidence patterns of vertices, faces, and edges), rather than through their geometric distances and angles. (However, this argument is undermined by the book's discussion of similar ideas in the earlier works of Kepler and Descartes.) The birth of topology is conventionally marked by an earlier contribution of Euler, his 1736 work on the Seven Bridges of Königsberg, and the middle part of the book connects these two works through the theory of graphs. It proves Euler's formula in a topological rather than geometric form, for planar graphs, and discusses its uses in proving that these graphs have vertices of low

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...

, a key component in proofs of the four color theorem. It even makes connections to combinatorial game theory through the graph-based games of Sprouts and Brussels Sprouts and their analysis using Euler's formula. In the third part of the book, Bradley moves on from the topology of the plane and the sphere to arbitrary topological surfaces. For any surface, the Euler characteristics of all subdivisions of the surface are equal, but they depend on the surface rather than always being 2. Here, the book describes the work of Bernhard Riemann, Max Dehn, and Poul Heegaard on the classification of manifolds, in which it was shown that the two-dimensional topological surfaces can be completely described by their Euler characteristics and their orientability. Other topics discussed in this part include knot theory and the Euler characteristic of Seifert surfaces, the Poincaré–Hopf theorem, the Brouwer fixed point theorem, Betti numbers, and

Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...

's proof of the

Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured b ...

. An appendix includes instructions for creating paper and soap-bubble models of some of the examples from the book.

Audience and reception

''Euler's Gem'' is aimed at a general audience interested in mathematical topics, with biographical sketches and portraits of the mathematicians it discusses, many diagrams and visual reasoning in place of rigorous proofs, and only a few simple equations. With no exercises, it is not a textbook. However, the later parts of the book may be heavy going for amateurs, requiring at least an undergraduate-level understanding of

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

and differential geometry. Reviewer Dustin L. Jones also suggests that teachers would find its examples, intuitive explanations, and historical background material useful in the classroom. Although reviewer Jeremy L. Martin complains that "the book's generalizations about mathematical history and aesthetics are a bit simplistic or even one-sided", points out a significant mathematical error in the book's conflation of polar duality with Poincaré duality, and views the book's attitude towards computer-assisted proof as "unnecessarily dismissive", he nevertheless concludes that the book's mathematical content "outweighs these occasional flaws". Dustin Jones evaluates the book as "a unique blend of history and mathematics ... engaging and enjoyable", and reviewer Bruce Roth calls it "well written and full of interesting ideas". Reviewer Janine Daems writes, "It was a pleasure reading this book, and I recommend it to everyone who is not afraid of mathematical arguments".

References

{{reflist, refs= {{citation, url=https://www.maa.org/programs-and-communities/member-communities/maa-awards/writing-awards/euler-book-prize, title=Euler Book Prize, publisher= Mathematical Association of America, accessdate=2020-02-25 {{citation , last = Bradley , first = Robert , date = January 8, 2009 , newspaper = Times Higher Education , title = Review of ''Euler's Gem'' , url = https://www.timeshighereducation.com/books/eulers-gem-the-polyhedron-formula-and-the-birth-of-topology/404921.article {{citation , last = Bultheel , first = Adhemar , author-link = Adhemar Bultheel , date = January 2020 , journal = EMS Reviews , publisher =

European Mathematical Society The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current ...

, title = Review of ''Euler's Gem'' , url = https://euro-math-soc.eu/review/eulers-gem-0 {{citation , last = Ciesielski , first = Krzysztof , journal =

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also ...

, mr = 2963735 , title = Review of ''Euler's Gem'' {{citation , last = Daems , first = Jeanine , date = December 2009 , doi = 10.1007/s00283-009-9116-0 , issue = 3 , journal = The Mathematical Intelligencer , pages = 56–57 , title = Review of ''Euler's Gem'' , volume = 32, doi-access = free {{citation , last = Jones , first = Dustin L. , date = August 2009 , issue = 1 , journal = The Mathematics Teacher , publisher = National Council of Teachers of Mathematics , jstor = 20876528 , page = 87 , title = Review of ''Euler's Gem'' , volume = 103 {{citation , last = Karpenkov , first = Oleg , journal = zbMATH , title = none , zbl = 1153.55001 {{citation , last = Martin , first = Jeremy , date = December 2010 , issue = 11 , journal = Notices of the American Mathematical Society , pages = 1448–1450 , title = Review of ''Euler's Gem'' , url = https://www.ams.org/notices/201011/rtx101101448p.pdf , volume = 57 {{citation , last = Roth , first = Bruce , date = March 2010 , issue = 529 , journal = The Mathematical Gazette , jstor = 27821912 , pages = 176–177 , title = Review of ''Euler's Gem'' , volume = 94, doi = 10.1017/S0025557200007397 {{citation , last = Satzer , first = William J. , date = October 2008 , journal = MAA Reviews , publisher = Mathematical Association of America , title = Review of ''Euler's Gem'' , url = https://www.maa.org/press/maa-reviews/eulers-gem-the-polyhedron-formula-and-the-birth-of-topology {{citation , last = Wagner , first = Clifford , date = February 2010 , doi = 10.4169/loci003291 , journal = Convergence , publisher = Mathematical Association of America , title = Review of ''Euler's Gem'' Polyhedral combinatorics Topological graph theory Books about the history of mathematics 2008 non-fiction books

Topics

Audience and reception

See also

References