The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else
By Jordan Ellenberg
Some of us have a certain “eat your vegetables” feeling at the mention of geometry; that it’s a necessary evil in order to get through school. “Shape” is Jordan Ellenberg’s 400-page counterargument. To Ellenberg, geometry is like poetry: able to mold and delight the human mind. But it is also deeply practical — poetry you can use to build houses. Art, then, as well as science: With geometry you can have your cake and cut it into symmetric pieces too.
Ellenberg’s previous book, “How Not to Be Wrong,” rested entirely on the thesis that we can use mathematical thinking to our advantage in everyday life. And the argument has been made many times over that math education is more about teaching thinking skills than mastering any specific probably-never-going-to-be-used math concept. But if that is the whole story, why not just force students to learn chess, or solve endless sudokus, or any other esoteric task that can train the mind?
Ellenberg claims that humans are innately geometric beings: “We teach geometry instead of any of those things because geometry is a formal system that’s not just a formal system. It’s built into the way we think about space, location and motion.”
In other words, it is the most appealing and human-friendly approach to math. This makes sense. We’ve evolved to survive in a world where the relative sizes and positions of things are important (particularly in reference to any animal located at a significantly different part of the food chain). Ellenberg’s skill as a storyteller, combined with a natural ability to spot otherwise obscure connections, enables him to capitalize on geometry as math’s gateway drug.
In the midst of recapping the standard ways you can move a shape about, Ellenberg throws in a new one: the “scronch.” It’s a more rigorous cousin to the lazy “squash” transformation used to cram a photo into an aspect ratio it has no business inhabiting. (I’ve heard rumors some people can watch a wide-screen movie squashed into a 4:3 screen without being constantly on edge, but I am not one of them.) The scronch allows a figure to be squashed in one direction, but it compensates with an equal expansion the other way.
This sounds as if it would make matters much worse, but cartoonists long ago discovered that the scronch (or “squash and stretch,” as it is known in the animation world) gives the zany distortions of a cartoon character the grounding in reality required to be parsed by the human brain. In other words: If a cartoon cat is hit with an oversize hammer and is compressed in one direction, it looks uncanny to us unless the cat also expands in an orthogonal direction. Real cats, I hasten to add, do not scronch. But what the animators discovered was that a scronch preserves the area of the (fictitious) cat’s underlying shape and this somehow gives it the substance and solidity craved by our brains.
It’s an interesting enough phenomenon in its own right, but Ellenberg returns to the scronch when talking about Lorentz contractions in the space-time of Einstein’s relativity. It ends up being the perfect purchase point to help grasp some otherwise very lofty and abstract concepts, which is one of the things that is so uncannily helpful about geometry (and you’ll have to believe me that it makes much more sense with Ellenberg’s full introduction).
And so the book progresses, from playing checkers to chasing mosquitoes, from machine learning to Abraham Lincoln’s unsuccessful efforts to “square the circle.” Be warned: Ellenberg uses a deliberately wide meaning of the word “geometry.” In the middle of learning about Fibonacci numbers in Sanskrit poetry you’ll lift your head to wonder how you got there. But in the spirit of Poincaré, who called mathematics “the art of giving the same name to different things,” Ellenberg gives his inner tour guide free rein and geometry becomes the shortest narrative path between any two seemingly disparate mathematical points. It makes for a deeply enjoyable and insightful book.