Robert Kaplans’s The Art of the Infinite.

The Art of the Infinite — Robert Kaplan

Straight up: This book is not for the faint of mathematical heart.

Robert Kaplan has made a surprising array of mathematics accessible in a conversational format in The Art of the Infinite. He includes some of the historical context and bits of trivia concerning certain discoveries as he develops the concepts and the proofs for these concepts in a way that gives the motivated reader the opportunity to work the answers out independently.

Most of the hard math proofs are placed in the appendix, leaving the reader the choice of tackling the proof while reading the text, or reading the appendix later as sort of hardcore review.


You have to be motivated to read this book, even then you might find you’ve reached a stage where you need to step back. You’ll either want to take a breath, get a drink, enroll in university, or just take the opportunity to consider what’s been covered to this point.

I’m guessing if you’ve read this far, you’re motivated to read the book. (I am not saying that just because you’ve read this far you’re motivated, just that if you’re motivated, you’ve gotten this far.) Based solely on the title you know that the majority of topics discussed end in infinity. Except there is no end in infinity, and that’s one of the more persistent concepts he presents.

If I may start in the middle (about Chapter Six, I believe), we start with a right triangle and by geometric proof inscribe a circle, using only a straight edge and a compass. Then we do the same to draw a circle enclosing the triangle. This signals the arrival of infinity. If we can have a triangle within a circle, can we not draw a square, using just compass and straight edge? What about a pentagon? A hexagon, a heptagon?

What can we and can we not draw inside a circle using only a straight edge and a compass? If you roll your eyes, and cough up blood at the next statement, you can safely conclude this book is not for you. Here’s your teaser. The answer to the previous question is based on the equation (2^2^k + 1). If you’re still with me, these are known as Fermat Numbers. Fermat thought all numbers of this form were primes. (To get your geek on go here:

By now, I’m pretty sure I’ve eliminated most of my initial audience. I’m going to speed on toward the end of the book. In the final chapter, Kaplan discusses the infiniteness of infinity. This is my phrase not his. If we have the infinite set of whole numbers, and there is an infinite set of real numbers between 0 and 1, do we not have a greater infinite number of real numbers than we do of whole numbers?  Then, if we have a greater number of real numbers, and we can plot those on a standard x-y graph, can we map this infinite set of real numbers 1 to 1 to the infinite set of whole numbers? (The answer to this is yes we can.) If we can map them 1 to 1, doesn’t that mean we have an equally infinite number of real and whole numbers?  So which is right?  Can it be that both right?

This is just a dip of the toe into an infinite pool of concepts that verge on, border against, run wild through the field of infinity in its infinite variety. If you’re fascinated, go see what else he has to say. If you aren’t fascinated, but you’ve gotten this far, thank you for listening while I gush over math.

Kaplan has authored several other books on mathematics, with illustrations provided by his wife, and is one of the primary contributors to a teaching organization The Math Circle. The Math Circle Summer Teacher Training Institute held its fifth Math Circle Summer Teacher Training Institute at Notre Dame in July 2012.