Math Without Numbers

shape

Mathematicians like to overthink things. It's sort of what we do. We take some concept that everyone understands on a basic level, like symmetry or equality, and pick it apart, trying to find a deeper meaning to it.

Take shape. We all know more or less what a shape is. You look at an object and you can easily tell if it's a circle or a rectangle or whatever else. But a mathematician would ask: What is a shape? What makes something the shape it is? When you identify an object by shape, you're ignoring its size, its color, what it's used for, how old it is, how heavy it is, who brought it here, and who's responsible for taking it home when we leave. What are you not ignoring? What is it that you're getting across when you say something is shaped like a circle?

These questions are, of course, pointless. For all practical uses, your intuitive understanding of shape is fine-no significant decision in your life will ever hinge on how exactly we define the word "shape." It's just an interesting thing to think about, if you have some extra time and you want to spend it thinking about shapes.

Let's say you do. Here's a question you might think to ask yourself: How many shapes are there?

It's a simple enough question, but it isn't easy to answer. A more precise and limited version of this question, called the generalized Poincaré Conjecture, has been around for well over a century and we still don't know of anyone who's been able to solve it. Lots of people have tried, and one professional mathematician recently won a million-dollar prize for finishing up a big chunk of the problem. But there are still many categories of shapes left uncounted, so we still don't know, as a global community, how many shapes there are.

Let's try to answer the question. How many shapes are there? For lack of a better idea, it seems like a useful thing to do to just start drawing shapes and see where that takes us.

Milo Beckman has been addicted to math since a young age. Born in Manhattan in 1995, he began taking math classes at Stuyvesant High School at age eight and was captain of the New York City Math Team by age thirteen. His diverse projects and independent research have been featured in the New York Times, FiveThirtyEight, Good Morning America, Salon, the Huffington Post, the Chronicle of Higher Education, Business Insider, the Boston Globe, Gothamist, the Economist, and others. He worked for three tech companies, two banks, and a US senator before retiring at age nineteen to teach math in New York, China, and Brazil, and to work on this book.

With charm, unwavering enthusiasm, and a lot of cartoons, Math Without Numbers waltzes the reader through a garden of higher mathematics.
Jordan Ellenberg, professor of mathematics, University of Wisconsin-Madison, author of How Not To Be Wrong

So delightful! Mathematics is playful, surprising, and enchanting, but those qualities are often obscured behind intimidating equations and formalism. Milo Beckman brings them out into the open for everyone to share.
Sean Carroll, author of Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime

Math Without Numbers explores deep mathematical topics and shows how mathematicians think in completely readable prose. The puzzles and games are bonuses. Very enjoyable.
Will Shortz, crossword editor, The New York Times

The book s accessible language and illustrations makes understanding some of the most complex (and possibly most intimidating) math concepts feel as effortless as breathing. Beckman s approachable writing and Erazo s delightful illustration combine to tell an insightful and entertaining story about math.
Giorgia Lupi and Stefanie Posavec, co-authors of Dear Data and Observe, Collect, Draw!

This is the book for you if you ve ever been curious about the wonderful ideas and concepts underlying modern math, but been too frightened to make a start. Milo Beckman gives us a friendly introduction to unfamiliar concepts and ideas that show why modern math is such a fascinating and rewarding branch of human thought.
Graham Farmelo, author of The Universe Speaks in Numbers

Math Without Numbers offers an accessible and whimsically illustrated glimpse of what pure mathematicians study, all while capturing the playful spirit with which they do it.
Grant Sanderson, creator of 3blue1brown

A cheerful, chatty, and charming trip through the world of mathematics and its relation