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Review mashup: Ian Stewart’s latest book, “Visions of Infinity”

Visions of Infinity: The Great Mathematical Problems is Ian Stewart's latest book. Dr. Stewart is one of my heros. I interviewed him for my podcast series and I love his new book. But, I struggled to come up with the right words to write for a review. So, what I've done instead is to take snippets from a number of reviews of the book I found on the web that articulated my own impressions and turned those into a mashup review, followed by a very brief summary of my impressions.

Here's the publisher's description of the book:

For every problem mathematicians solve, another waits to perplex and galvanize them. Such challenges offer a tantalizing glimpse of the field’s unlimited potential, and keep mathematicians looking toward the horizons of intellectual possibility. In Visions of Infinity, celebrated mathematician Ian Stewart explains why these problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole. Stewart describes solved problems—like Fermat’s Last Theorem, proved three centuries after it was posited, and the Poincaré Conjecture, cracked by eccentric genius Grigori Perelman—as well as those like the P/NP problem, which could easily remain unproved for another hundred years. An approachable and illuminating history of mathematics as told through fourteen of its greatest problems, Visions of Infinity reveals how mathematicians the world over have risen to meet the challenges set by their predecessors—and how the enigmas of the past inevitably surrender to the powerful techniques of the present.

Below are excerpts from a half dozen reviews on the Web. Keep reading for my impression of the book as bullet points.


To be clear though, while this volume covers some of the most interesting problems in all of mathematics it is NOT a book to draw your non-mathematical friends into the math arena. Even the non-math person who wishes they could like math, and who may have enjoyed Steven Strogatz's "Joy of X," I think will find this particular book too heavy-going. But for the individual already enamored of the subject, and having some familiarity with math's deepest problems, this is a fantastic read. In fact, it's probably my favorite Stewart volume of all the ones I've read.


The four-color map problem can be understood by a bright fourth-grader (the question: whether four colors are enough to ensure that no two countries with a common border share a color). By junior high, most kids can grasp prime numbers and learn something about their properties and patterns. High school algebra students can comprehend what Fermat’s last theorem means. Yet these topics have for decades, or even centuries, occupied the world’s most sophisticated mathematicians.

Publishers Weekly

Capping the discussion is a quick chapter detailing some of the problems that may give mathematicians fits and nightmares into the next century, including quaintly named perfect cuboids, Langton’s Ant, and mysterious constructs called Thrackles. Once again, Stewart delivers an intriguing book that rewards random reading as much as dedicated study.


Stewart’s imaginative, often-witty anecdotes, analogies and diagrams succeed in illuminating many but not all of some very difficult ideas. It will enchant math enthusiasts as well as general readers who pay close attention.

New York Journal of Books

In the end chapters the mathematics gets more “pure” and explanations of the conjectures get more complicated. The reader has to pay more attention. CAUTION: Do not attempt these chapters without possessing full alertness. The chapter “Diophantine Dreams” addresses the search for the proof of the Birch-Swinnerton-Dyer conjecture, where the difficulty won’t just be in finding a proof. It will first be in understanding just what the conjecture is. Make this chapter your own personal Mount Everest.


Just as dynamic systems can "settle down" into chaos/fractals, strange attractors or an oscillation, the book, after taking us on a fascinating journey through the known and unknown, gives us a great, up to date feel for which problems are in which category of difficulty and likely vs. unlikely to be solved in our times. The "toughest" problems are the stuff of cryptanalysis and are "good" from the standpoint of providing security, but Ian also shows the many possible openings at the back of the tent in addition to the door, by suggesting possible "close enough" solutions and directions that are worth pursuing.

The reviews summed up my experience of the book.

  1. Ian Stewart is a great writer. He is able to weave together mathematics and story to keep the reader's attention and to provide context for why we care about solving these difficult problems.
  2. I'm not so sure that a motivated high school student can follow all of the ideas. I certainly couldn't. As Shecky points out in my excerpt from his MathTango review (above), "To be clear though, while this volume covers some of the most interesting problems in all of mathematics it is NOT a book to draw your non-mathematical friends into the math arena." Very well said.
  3. Having hammered in the last point, much of the material in the book is accessible to the motivated student who is willing to focus and persist.
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Review: Beautiful Mathematics

Beautiful Mathematics is a collection of interesting mathematical explorations published by the MAA. If you find the following questions (many of which are new to me) intriguing then you'll probably enjoy the book.

  1. Do you know the dimension of Sirpienski's Triangle?
  2. Have you ever explored squaring maps?
  3. What is the Riemann Sphere?
  4. Can you find a formula that associates Fibonacci numbers and Pi?
  5. Can a square be inscribed in any triangle?
  6. What are the first three digits of the millionth Fibonacci number?
  7. Do you know how to construct a regular heptagon using a straight edge, compass, and angle trisecting device?
  8. Can you prove Lagrange's Theorem, that every positive integer is the sum of four squares?
  9. How many triangles are there of perimeter n, where n is an integer, the sides are all integer lengths and the triangles are incongruent?

"Beautiful Mathematics" has nearly 100 challenging investigations, most with elegant solutions presented. Topics include words, images, formulas, theorems, proofs, solutions, and unsolved problems.

The book is available from the publisher in PDF format for half of the price of the hardback book.

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Review: Magical Mathematics

Deep Math meets great magic tricks. Magical Mathematics is an absolutely remarkable book. I don't say this lightly. Publishers send me plenty of books to review. Some I like more than others. "Magical Mathematics" is a fantastic book for someone who wants to explore the non-trivial math behind some impressive magic tricks.

While I enjoy purely recreational math puzzles that have no practical application I also love it when I discover challenges that are interesting and relevant. "Magical Mathematics" is chock full of fun (and deep) challenges that students (and adults) can sink their teeth into.

This great review by Sami Assaf at Amazon.com gets to the heart of what makes this book stand apart from many other recreational Math books:

The book is packed with fantastic card tricks that will surely dazzle friends and family (with enough practice), but goes beyond this by explaining the beautiful (often deep) mathematics behind the tricks. The book intersperses magic and mathematics in an engaging way that keeps the reader hooked. The book begins with a simple 4 card trick. Well, simple enough to perform; understanding is a different matter. The authors then explain what mathematical concepts (mostly involving combinations and permutations) are at the heart of the trick, and then generalize the principle involved into a truly impressive, more elaborate card trick. After that, it's back to math to see exactly how and why the magic works. Later chapters follow similar patterns, where the reader is drawn in by a beautiful card trick and the beauty is then heightened with a clear explanation of the underlying mathematics. Along the way, the authors give excellent advice on how often to rehearse the tricks before performing as well as entertaining tips to make for a more engaging performance. The mathematics is presented in digestible bites, with excellent examples and illuminating illustrations.

But be warned: this isn't your simple high school math! Many of these tricks employ sophisticated mathematics using Combinatorics, Group Theory, Graph Theory and more. Fortunately, the authors are adept at explaining these complicated concepts in a clear fashion, but the novice reader may have some trouble following some of the proofs. Hopefully, the reader will be so inspired by the beauty of the subject, that she'll see it as motivation to learn more mathematics! In fact, the authors' unapologetic goal with this text is to corrupt youngsters of all ages into pursuing mathematics in much the same way that the authors themselves were seduced by the subject. Here's hoping they succeed with you as they have with me!

The New York Times recently published a review of "Magical Mathematics." Another review is available at The Math Less Traveled. More information about the book is available at the Princeton University Press website.

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Review: Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction

Great stories. Interesting and challenging problems. Instructive MATLAB code. Lots of physics. That's my in-a-nutshell assessment of Princeton University Press's hot-off-the-press Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction.

Paul Nahin is a great story teller. Some of you might recall my review of an earlier book of Nahin's An Imaginary Tale where I noted Nahin's enjoyable writing style. Nahin has, in fact, written quite a number of books.

Nahin takes on the subject of using computers to solve difficult problems, many in physics, that couldn't be solved before computers. The publisher's page introduces some of the problems.

How do technicians repair broken communications cables at the bottom of the ocean without actually seeing them? What's the likelihood of plucking a needle out of a haystack the size of the Earth? And is it possible to use computers to create a universal library of everything ever written or every photo ever taken? These are just some of the intriguing questions that best-selling popular math writer Paul Nahin tackles in Number-Crunching. Through brilliant math ideas and entertaining stories, Nahin demonstrates how odd and unusual math problems can be solved by bringing together basic physics ideas and today's powerful computers. Some of the outcomes discussed are so counterintuitive they will leave readers astonished.


Review: The Man of Numbers: Fibonacci’s Arithmetic Revolution

Keith Devlin is a prolific writer. As The Math Guy on National Public Radio and author of some 40 books, Keith Devlin makes a tremendous contribution toward making math more accessible to the public.

The Man of Numbers: Fibonacci's Arithmetic Revolution is Devlin's latest book.

"The Man of Numbers," at 156 pages (plus notes, bibliography, and index) and ten chapters is a fairly quick read. Leonardo of Pisa, also known as Fibonacci, is mostly only known for the Fibonacci sequence. Devlin shows us that there was much more to Fibonacci's life and that, in fact, Fibonacci played a very key role in the marketing of arithmetic in 1202 to the world of commerce in Western Europe through his book, Liber Abacci (The book of calculation.)

I'm not going to review the book chapter by chapter as you can find that kind of information on the web. NPR has a nice review, an excerpt from the book, and an audio interview with Mr. Devlin. ScienceNews.org has a review and Amazon.com has several reviews. But, I will point out some items of particular interest.


Review: Charming Proofs: A Journey Into Elegant Mathematics

Charming Proofs: A Journey Into Elegant Mathematics is a delightful book, published by the Mathematical Association of America (MAA), that lives up to its name.

Given my joyful experiences of exploring challenging problems in middle school and in high school I have a soft spot for elegant problems that are accessible to motivated students who don't have any background in advanced mathematics. And, I have a soft spot for MAA books because they were among the first math books I devoured, specifically their MAA contest prep books.

Here's a brief description, from the publisher's page, of the structure of the book.

Charming Proofs is organized as follows. Following a short introduction about proofs and the process of creating proofs, the authors present, in twelve chapters, a wide and varied selection of proofs they consider charming, Topics include the integers, selected real numbers, points in the plane, triangles, squares, and other polygons, curves, inequalities, plane tilings, origami, colorful proofs, three-dimensional geometry, etc. At the end of each chapter are some challenges that will draw the reader into the process of creating charming proofs. There are over 130 such challenges.


Review: Math Dictionary for Kids

Math Dictionary for Kids: The Essential Guide to Math Terms, Strategies, and Tables is a great reference book that I think should be on every kid's bookshelf. (Disclosure: The publisher sent me a review copy.) New copies are available via Amazon for as low as $7.64 plus shipping in the U.S. making this book an inexpensive back to school gift for 4th to 9th graders.

When I was doing lots of computer programming I would always have reference books available (or online equivalents) to look up how to solve some particular kind of problem in some particular programming language or environment. "Math Dictionary for Kids" is that kind of book for kids who need to review (or learn) some mathematical concept and it's filled with tons of how-to's just like those good programming reference books.

I do think that "dictionary" is a misnomer. I typically use a dictionary to look up the meaning of a word (or the spelling, before spell-checkers were everywhere). This book is much more comprehensive than your typical dictionary.


Review: The Best Writing on Mathematics 2010

Imagine for a moment that you had a friend who was a voracious reader of Math journals and periodicals. And, imagine that this friend had a knack for finding articles that were of interest to mathematicians and non-mathematicians alike by well-known writers and by new talent. Would you be interested in reading a few dozen of these articles? Mircea Pitici, editor of The Best Writing on Mathematics 2010 is such a friend, even if you've never met him.

Publisher Princeton University Press has an interview with Pitici at their blog where he answers the question of how many articles he read to select the ones that got into the book.

It’s difficult to give an estimate, but let me try. I see several thousand articles in one year but obviously I discard most of them quickly (as far as this particular book series is concerned). Not necessarily because they are not worthy of my attention or do not deserve reading; I just know by reading the first paragraph or by a cursory look at the prose and the exposition that they wouldn’t fit in the book I envision. Perhaps I gave serious attention and read thoroughly in direct connection with this volume about four-five times more texts than I finally chose—which means 150 or so. That is a rough approximation.

The end result is 35 interesting and varied articles in six areas: Mathematica Alive, Mathematicians and the Practice of Mathematics, Mathematics and Its Applications, Mathematics Education, History and Philosophy of Mathematics, and Mathematics in the media. The contents are here.


Review: Sage Beginner’s Guide

Sage Beginner's Guide is a book from Packt Publishing that aims to help new users to break the ice and get comfortable and productive with Sage. Sage is an open-source Math software system that combines a large number of packages into a Python interface. Disclaimer: I am much more familiar with the commercial Math software system, Mathematica, which I blog about at Playing With Mathematica. I have dabbled with Sage but don't have enough experience with it to compare it to Mathematica. But, I'm curious enough about Sage that, when the publisher offered me a review copy, I accepted.

The Sage site has links to a number of types of documentation and support. You need to decide whether what's available for free meets your needs. Among other things, there's a tutorial, an installation guide, a book for newbies, and lots more, free for clicking and downloading.

Why might you want to buy a copy of the Sage Beginner's Guide? There are a number of reasons.

  1. The Guide combines a number of types of documentation into a single book. There's an overview chapter, an installation chapter, information helpful for getting your bearings, an introduction to Python, how-to's on plotting, information on symbolic and numerical computation, plus some advanced information on doing more with Sage, Python, LaTeX interactive applications and more. You can view the entire table of contents here.


Review: The Manga Guide to Relativity

I have a confession to make. I've never liked Physics a whole lot. As an undergrad at Stanford I had to take a number of basic Physics classes. Much of what we had to do was to apply formulas to compute the masses of tiny objects or to compute tiny forces. The way Physics was taught frustrated me because I developed no grounding in the subject. I could be off by many orders of magnitude and not have a clue. Was the mass of that tiny thing 10^-20 grams or 10^-30 grams? Beats me. My Math education on the other hand was much better, especially before my college years. I developed an intuitive ability to manipulate symbols and to work with abstract concepts which I never developed in Physics.

When the nice folks at No Starch Press offered me a review copy of The Manga Guide to Relativity (Manga Guide Series) I was reluctant to accept it. I won't review books I don't like although I'm certainly willing to report issues with books I do generally like.

I like Manga books. I've reviewed the Manga Guide to Calculus and the Manga Guide to Statistics. I love the idea of turning difficult and detailed ideas into a story. I recently reviewed Keith Devlin's new book, Mathematics Education for a New Era: Video games as a Medium for Learning. In that book I learned the importance of creating an environment that engages students in learning. Video games, if they're designed with the right principles, can do that in the context of computers. Manga books, in my judgment, create an excellent space for learning on the printed page.