# 99 Variations on a Proof

###### Phillip Ording
Publisher:
Princeton University Press
Publication Date:
2019
Number of Pages:
272
Format:
Hardcover
Price:
24.95
ISBN:
9780691158839
Category:
General
BLL Rating:

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Geoffrey D. Dietz
, on
04/6/2019
]
Whenever I pick up a book to review for the MAA, the primary question in my mind is, "Who is the target audience for this book?" These answers can vary anywhere between general audience" through "experts interested in the latest results in the theory of gargantuan Macaulay-Culkin algebras" (That one may not quite be a real thing.)

When I picked up 99 Variations on a Proof, I found the title intriguing but immediately wondered who would really enjoy reading 99 different proofs of the same result? Thankfully, this book is not 99 Proofs of Quadratic Reciprocity (Each More Fiendishly Clever Than the Last.) Instead, 99 Variations on a Proof is a mathematical homage to the 1947 book Exercises in Style by Raymond Queneau of the literary group Oulipo. In that book, Queneau presents ninety-nine versions of the same short story under different literary constraints. If (like me) you are unfamiliar with Oulipo or the work of Queneau, your ignorance will not harm your enjoyment of Ording's book.

Ording presents ninety-nine proofs that a specific cubic equation has two real roots. The theorem itself is fairly uninteresting, but the proofs are the stars and each of them seeks to show a different aspect of the theory, history, or culture of mathematics. The variety aims to demonstrate the many different ways that mathematicians can view a problem and the plethora of tools at our disposal to solve an equation, including algebra, geometry, trigonometry, calculus, complex analysis, models, graphs, statistics, and even origami. History plays a big role throughout the book with frequent appearances from heroes of the past such as Cardano, Euclid, Euler, and Khayyam. The culture and philosophy of mathematics are also major factors. While some of the proofs inspired by mathematical culture are not strictly proofs," such as Social Media (a Twitter post of the result) or Preprint (an arXiv post of the paper), but their inclusion is a valuable reminder of the importance of modern tools which my be unfamiliar to many younger (or older) readers. Each proof is accompanied by a brief piece of commentary that adds extra explanation about the mathematics used or about the cultural or historical context.

I enjoyed reading the proofs and commentaries and was impressed by the diversity within the ninety-nine. I learned something new from many of the more creative proofs such as Wordless (slicing and augmenting a solid box), Electrostatic (using null points of an electric field), and Geometric (Eduard Lill's graphical method for solving polynomials). I laughed over several, including Research Seminar (using overpowered machinery and losing 90\% of the audience in the first two lines) and Tea (an informal chat with colleagues while worrying about where all of the snacks went). I was inspired to search out one of the references after reading Refereed (a parody referee report which recommended against publication). I even felt sentimental reading the commentary about Euler in First Person (a brief story illustrating how mathematicians often have ideas while performing other life activities). Ording asks, "How do you do mathematics with five [children]?" I thought back to typing code for a simulation while holding my sleeping newborn fifth child and a very productive sabbatical working at home with five home-schooled kids going about their daily routines while I proved some new results.

I return to my original question, Who is the potential audience for this book?" Here are (not quite ninety-nine) questions that may gauge your likelihood to enjoy this book:

• Have you ever read or heard a mathematical proof?
• Have you ever taken a mathematics course that included proofs during lecture?
• Have you ever taken a mathematics course where you had to write proofs?
• Have you ever taught a mathematics course involving proofs?
• Have you ever graded a class where students wrote proofs?
• Have you ever written proofs of original results, say, in a paper or dissertation?
• Have you ever been to a talk at a mathematics conference or research seminar?
• Have you ever fallen asleep at a research seminar?
• Have you ever had someone fall asleep while you gave a talk?
• Have you ever refereed a paper?
• Have you ever supervised a research student?
• Have you ever felt like Vanna White while trying to illustrate a concept during a lecture?
• Have you ever been horribly confused while reading a badly written proof of an otherwise interesting result?
• Have you ever discovered a shorter or more elegant proof for a result compared to the author of a book or paper you were reading?
• Are you fascinated by the history of mathematics and its personalities?
• Do you find the modern culture of mathematics both charming and absurd?
• Have you ever challenged yourself to deliver a lecture using only monosyllabic words?
The more of these questions to which you can say, yes," the more likely that you will find the proofs and discussions in this book enlightening, amusing, or even laugh-out-loud funny.

Geoffrey Dietz is a Professor of Mathematics at Gannon University in Erie, PA. He is married, has six children, and has published papers on topics including big Cohen-Macaulay algebras.