''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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...

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 ...

. It was written by

David Richeson David S. Richeson is an American mathematician whose interests include the topology of dynamical systems, recreational mathematics, and the history of mathematics. He is a professor of mathematics at Dickinson College, where he holds the John J. & ...

and published in 2008 by the

Princeton University Press Princeton University Press is an independent 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 the financial ...

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

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

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 poli ...

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, regarded ...

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 Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

, 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. Ma ...

of a polyhedral version of the

Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a t ...

(later seen to be equivalent to Euler's formula). It surveys the life of

Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...

, his discovery in the early 1750s that the

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 name ...

, based on Girard's theorem relating the angular excess of triangles in

spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...

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 The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in Prussia (n ...

, 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 graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...

s, and discusses its uses in proving that these graphs have vertices of low degree, a key component in proofs of the

four color theorem In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions sha ...

. It even makes connections to

combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the player ...

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 Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...

Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...

, and Poul Heegaard on the

classification of manifolds In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain. Main themes Overview * Low-dimensional manifolds are classified by geometric struc ...

, in which it was shown that the two-dimensional topological surfaces can be completely described by their Euler characteristics and their

orientability In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...

. Other topics discussed in this part include

knot theory In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...

and the Euler characteristic of

Seifert surface In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For example ...

s, the

Poincaré–Hopf theorem In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincar� ...

, the

Brouwer fixed point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest ...

Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplici ...

s, 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 ...

. 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 Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

. 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 In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...

, and views the book's attitude towards

computer-assisted proof A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a ...

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=

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Times Higher Education ''Times Higher Education'' (''THE''), formerly ''The Times Higher Education Supplement'' (''The Thes''), is a British magazine reporting specifically on news and issues related to higher education. Ownership TPG Capital acquired TSL 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 pu ...

, 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 ''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released qu ...

, 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 zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastruct ...

, title = none , zbl = 1153.55001 {{citation , last = Martin , first = Jeremy , date = December 2010 , issue = 11 , journal =

Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine sinc ...

, 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 ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

, 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 =

, 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 =

, 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