By Ernest Davis
Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. By Jordan Ellenberg. Penguin Press, New York, NY, May 2021. 480 pages, $28.00.
Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. By Jordan Ellenberg. Courtesy of Penguin Press.
Seven years ago, Jordan Ellenberg’s
How Not to Be Wrong—which explained the importance of mathematics for a popular audience—was a runaway success and earned both critical praise (including a
review by me in SIAM News) and a place on
The New York Times’ Best-Seller list. His new book,
Shape, has raised the game; it seems to me even better than his last.
Ellenberg is an enviably skillful writer. His explanations of technical material are simultaneously crystal clear, patient, complete, and entertaining. He is also an exceptional raconteur. Ellenberg’s many biographical and historical passages are skillfully integrated with the mathematical content, and he recounts them in a way that seems leisurely—as if he had all the time in the world to tell readers about a curious story or fun anecdote—but never drags. His inclusions of himself in the commentary are unfailingly tasteful — enough to make the narrative vivid and personal but never so much as to appear egotistical.
Ellenberg is witty yet balanced. His accounts of historical events and people are clear-eyed but sympathetic. His descriptions of contentious topics—such as artificial intelligence (AI) and gerrymandering—are scrupulously fair. Ellenberg loves math but is fully aware of the limitations of the mathematical viewpoint and the occupational vices of mathematicians. He writes that “Mathematicians are prone to an imperial tendency: We often see other people’s problems as consisting of a true mathematical core surrounded by an irritating amount of distracting domain-specific knowledge, which we impatiently tear away to get as quickly as possible to ‘the good stuff.’”
Incidentally, Ellenberg has a particular affinity for poetry. As such, he discusses both poets who engaged with mathematics—such as William Wordsworth and Rita Dove—and mathematicians who wrote poetry, including William Rowan Hamilton and James Joseph Sylvester.
What is actually in Shape? Well, the book contains a chapter on Euclidean proof, with a discussion of its impact on figures like Wordsworth, Abraham Lincoln, and Thomas Jefferson. A second chapter explores the question, “How many holes does a straw have?”—which was apparently a hot topic on the internet in 2014—with discussions of Henri Poincaré (a leading figure throughout the book) and Johann Benedict Listing, a lesser-known 19th-century savant. Yet another chapter addresses symmetries and geometric transformations in the context of Poincaré and Emmy Noether.
Ellenberg then moves on to the theory of random walks. He begins with Ronald Ross’ analysis of mosquitoes’ transmission of malaria, passes through Karl Pearson, Poincaré, Albert Einstein, and Andrey Markov, and concludes with Claude Shannon’s use of Markov chains to generate English text. Three chapters focus on AI, game playing, and machine learning, with an excursion into cryptography. A chapter on metric spaces ends up mostly examining the vector representation of words that are used in AI programs. Two chapters detail prediction and include a long discussion of the attempts to predict the course of the COVID-19 pandemic. One chapter describes nearly geometric sequences and eigenvalues, while another delves into graph connectivity.
Next comes a true masterpiece: Ellenberg’s chapter on districting and gerrymandering. He covers the implications, history, law, and relevant algorithmics. He also addresses the difficulty of determining what is fair and reasonable; the various suggested mathematical objectives for districting and criteria for gerrymandering detection; and the strengths and flaws of each approach. This chapter is very long—almost twice the length of all the others—but exceptionally clear, absorbing, and of course important.
I did find one aspect of the book disappointing: there is very little geometry after the introduction and first three chapters, which comprise only 63 of 425 total pages of content. Shape includes a lot of math of all kinds, but game trees, Markov chains, cryptographic codes, integer sequences, graphs, and so forth do not constitute geometry. Even the geometric considerations in the section on gerrymandering—such as district requirements for spatial compactness—are largely peripheral. There are very few shapes, other than the shapes of Congressional districts: some triangles, rectangles, circles, and occasional passing references to ellipses, hyperbolas, Alexander’s horned sphere, and such.
This seeming uninterest in spatial and visual aspects of math manifests itself in other ways as well. There is a really beautiful proof of Fermat’s little theorem \(a^p = a \; \textrm{mod} \; p\) if \(p\) is prime1 that I had never seen before, but the book has no proofs of geometric theorems beyond the pons asinorum and a dissection proof of the Pythagorean theorem. There are 100 pages on AI—mostly as applied to language—but barely a reference to computer graphics, computational geometry, or solid modeling. As I mentioned, the text contains a lot of poetry, but the only work of visual art that Ellenberg acknowledges is Salvador Dali’s Crucifixion: Corpus Hypercubus. The subtitle promises “the hidden geometry of…biology,” but Shape fails to comment on the marvelous geometric forms that living organisms create and only remarks on the flight of mosquitoes, spread of disease, and network structure of the circulatory system. Many years ago, my father Philip Davis decried the disappearance of the visual and spatial sense from mathematics [1]; it is dismaying to see this tendency in a book by an eminent geometer that is entitled Shape.
Nevertheless, this is at most an objection to the title. Shape is a wonderful book; it is superlatively well written and full of history, mathematics, and much else. Readers that range from professional mathematicians to people who disliked geometry in high school can enjoy and learn from the material.
1The book gives the proof only for the case \(a=2\), but the generalization is immediate.
References
[1] Davis, P.J. (1974). Visual geometry, computer graphics, and theorems of perceived type. In The influence of computing on mathematical research and education. Proceedings of symposia in applied mathematics (Vol. 20, pp. 113-128). Providence, RI: American Mathematical Society.
Ernest Davis is a professor of computer science at New York University’s Courant Institute of Mathematical Sciences.