I’m going to try something new on this blog: a book review. For my younger readers, a book is an object made of a series of static screen images printed on cellulose fiber. Think of it as a collection of thousands of tweets, Snapchat screenshots, and Facebook status updates all related by a common “narrative”. Or just ask your parents.
Despite its silly title, The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life by Alex Bellos is a fascinating look at some of the most interesting developments in mathematics throughout history. Math books often come in one of two flavors. There are the hard-core textbook-style books that quickly get over my head, despite having words like “elementary” and “introduction to” in the title. Then there are the overly simplified and popularized books that lack sufficient depth or patronize readers by refusing to ever show an equation. For me, The Grapes of Math hits the sweet spot between these extremes and does an extraordinary job of providing clear explanations of some really complex and abstract math, while still challenging a numerate reader.
Grapes is divided into chapters each dedicated to a broad topic in mathematics, like number theory, power laws, trigonometry, imaginary and complex numbers, exponential functions, complex systems, etc. Each chapter covers some of the history of the ideas, some explanations of the ideas themselves, and some modern applications or research. It’s by no means a comprehensive review of the history of mathematics or even all the big ideas. But there is plenty of fascinating material, not to mention amusing anecdotes about history’s parade of quirky mathematicians and improbable discoveries.
Rather than try to summarize everything, I’ll just highlight a few of the most interesting bits (for me at least).
One of the my favorite chapters, about power laws, starts by introducing Benford’s law. This law is all around us, present in many real-world data sets, but it’s so unexpected and counterintuitive that it was only discovered a century ago. Benford’s law states that for many real-life datasets, the first digits of each number in the set are not equally distributed as you might expect. In fact, the small digits (e.g. 1,2,3) occur much more frequently than the large ones (7,8,9). In almost any dataset that varies over an order of magnitude (and meets a few other criteria), about 30% of the first digits are one, and less than five percent are nine. This is really weird! Here is the distribution of first digits under Benford’s law:
The law was discovered by observing that the books of logarithm tables were more worn on pages with tables of numbers starting with the smaller digits. The log books phenomenon was first noticed in 1880, but then rediscovered by Frank Benford in 1938. Benford found this distribution in all sorts of totally unrelated data sets, like the populations of US cities, areas of river basins, atomic weights of the elements, even baseball statistics.
It turns out that this phenomenon is so widespread that Benford’s Law is used by forensic accountants (yes, that’s really a thing) to look for falsified or manipulated data.
If you want an explanation for why Benford’s Law occurs, you’ll have to read the book.
Another of my favorite parts of the book is about the famous Mandelbrot set. Stunning computer-generated images of the Mandelbrot set, like the one below, are often used to illustrate fractal geometry and chaotic behavior in numeric systems. A fractal is an object (or a set of numbers) that looks similar no matter what the scale. In nature, examples of fractal geometries include the trace of a coastline, the drainage patterns in river basins, the topography of mountain ranges and geologic fault systems.
When you approach the edges of the Mandelbrot set you see amazing complex patterns that just keeping going (and changing) no matter how far you zoom. There’re really no way to explain with words. You need to see it:
But what is the Mandelbrot set? I must confess that before reading The Grapes of Math I didn’t really know, despite working on fractals as a graduate student in geology. Bellos gives a clear explanation of how the set is generated, which is at the same time incredibly simple and very counterintuitive. The Mandelbrot set consists of complex numbers generated by iterating over a quadratic equation (). The numbers that do not go to infinity upon iteration are members of the set, all others are not. The pictures are generated by projecting the set on the complex plane (and sometimes adding color to the edges). That’s all it takes to generate this image of extraordinary, indeed infinite, complexity!
Despite the simplicity of the algorithm, the discovery and implications of the Mandelbrot set were far from trivial. The work of Benoit Mandelbrot (who made the first computer image of the set) helped usher in an entirely new understanding of chaos in deterministic systems.
And if you don’t remember anything about imaginary or complex numbers from high school or college math (I needed a refresher), don’t worry. The Grapes of Math does a good job of walking the reader though it. In fact, the development of imaginary numbers is an extraordinary story in and of itself.
Speaking of high school or college math, this book introduced me to a beautiful equation that I can’t believe I never learned before:
This is Euler’s identity, discovered by the brilliant Swiss mathematician Leonhard Euler. It links five of the most basic numbers in mathematics: , e (the exponential constant), i (the square root of negative one), zero, and one. Why am I only finding out about this now?
Euler’s identity an example of math at its most elegant and mysterious. Mathematician Benjamin Pierce once said the Euler’s Identity is “absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”.
This is a fitting description for many of the mathematical concepts discussed in The Grapes of Math. The book shows how the history of math is a progression toward greater abstraction, from what we can physically see and count and measure, to concepts like Euler’s identify that cannot be intuitively understood, only discovered through applying mathematical logic.
I highly recommend this book. It’s perfect for summer beach reading, if you’re the kind of person who likes to draw curves and equations in the wet sand by the shore of a fractal coastline.