He applied the "Page 99 Test" to his recent book, In Pursuit of the Unknown: 17 Equations That Changed the World, and reported the following:
Page 99 manages to be about as unrepresentative of the rest of the book as it is possible to be. It is entirely devoted to a description and picture of the Klein bottle, a one-sided surface. In contrast, nearly all of the book is about the great equations of mathematics and physics, which have had huge effects on human history. Each chapter tells the story of one equation: who discovered it, what civilization was like at the time, how the equation changed everything, and why it remains important today.Learn more about the book at the author's website.
The chapter that contains page 99 is about Euler’s formula F+V-E = 2 where F is the number of faces, V the number of vertices, and E the number of edges of a solid. As the book states:
Why is that important? It distinguishes between solids with different topologies using the earliest example of a topological invariant. This paved the way to more general and more powerful techniques, creating a new branch of mathematics.I included Euler’s formula to show that important equations are not necessarily restricted to mathematical physics. Most of the other equations are about basic mathematics, like Pythagoras’s theorem, or they come from physics, like Newton’s law of gravity. The first led to trigonometry, surveying, and navigation; the second allowed us to understand how the planets move, and is used to this day to plan the trajectories of space probes and satellites. The two come together in the Global Positioning System, the basis of SatNav.
What did it lead to? One of the most important and powerful areas of pure mathematics: topology, which studies geometric properties that are unchanged by continuous deformations. Examples include surfaces, knots, and links. Most applications are indirect, but its influence behind the scenes is vital. It helps us understand how enzymes act on DNA in a cell, and why the motion of celestial bodies can be chaotic.
Euler’s formula and the Klein bottle led to topology, which revolutionised mathematics and can be found behind the scenes in many applied areas. The topology of the three-body problem in celestial mechanics was one of the discoveries that led to chaos theory, and the results are increasingly being used in space missions to design fuel-efficient trajectories. So in the end, page 99 isn’t such a bad test after all.