The thing is, physical reality is a disaster. It’s way too complicated, and nothing is at all what it appears to be.

Now, there are lots of reasons people get interested in physical reality. Astronomers, biologists, chemists, and all the rest are trying to figure out how it works, to describe it. I want to describe mathematical reality. To make patterns. To figure out how they work. That’s what mathematicians like me try to do.

The point is I get to have them both—physical reality and mathematical reality. Both are beautiful and interesting (and somewhat frightening). The former is important to me because I am in it, the latter because it is in me.

My idea with this book is that we will design patterns. We’ll make patterns of shape and motion, and then we will try to understand our patterns and measure them. And we will see beautiful things!

Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively. The truth is, I don’t know of any human activity as demanding of one’s imagination, intuition, and ingenuity. But I do it anyway. I do it because I love it and because I can’t help it. Once you’ve been to the jungle, you can never really leave. It haunts your waking dreams.

What is a math problem? To a mathematician, a problem is a probe—a test of mathematical reality to see how it behaves. It is our way of “poking it with a stick” and seeing what happens. We have a piece of mathematical reality, which may be a configuration of shapes, a number pattern, or what have you, and we want to understand what makes it tick: What does it do and why does it do it? So we poke it—only not with our hands and not with a stick. We have to poke it with our minds.

What we’ve stumbled onto is a conspiracy. Apparently, there is some underlying (and as yet unknown) structural interplay going on that is making this happen. I think that is marvelous and also a little scary. What do triangles know that we don’t? Sometimes it makes me a little queasy to think about all the beautiful and profound truths out there waiting to be discovered and connected together.

Now here’s where the art comes in. In order to explain we have to create something. Namely, we need to somehow construct an argument—a piece of reasoning that will satisfy our curiosity as to why this behavior is happening. This is a very tall order. For one thing, it’s not enough to draw or build a bunch of physical triangles and see that it more or less works for them. That is not an explanation; it’s more of an “approximate verification.” Ours is a much more serious philosophical issue.

Is this such an extraordinarily difficult thing to do? Yes, it is. Is there some recipe or method to follow? No, there isn’t. This is abstract art, pure and simple. And art is always a struggle. There is no systematic way to create beautiful and meaningful paintings or sculptures, and there is also no method for producing beautiful and meaningful mathematical arguments. Sorry. Math is the hardest thing there is, and that’s one of the reasons I love it.

Don’t be afraid that you can’t answer your own questions—that’s the natural state of the mathematician.

So let it be hard. Try not to get discouraged or to take your failures too personally. It’s not only you that is having trouble understanding mathematical reality; it’s all of us. Don’t worry that you have no experience or that you’re not “qualified.” What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for simple beauty.

The goal, as in any literary fiction, is to write a story that is compelling as a narrative. In the case of mathematics, this means that the plot not only has to make logical sense but also be simple and elegant. No one likes a meandering, complicated quagmire of a proof. We want to follow along rationally to be sure, but we also want to be charmed and swept off our feet aesthetically. A proof should be lovely as well as logical.

Archimedes, Gauss, you and I—we’re all groping our way through mathematical reality, trying to understand what is going on, making guesses, trying out ideas, and mostly failing. And then every once in a while, you succeed. (Perhaps more frequently if you are Archimedes or Gauss.)

The real difference between you and more experienced mathematicians is that we’ve seen a lot more ways that we can fool ourselves. So we have more nagging doubts and therefore insist on a much higher standard of logical rigor. We learn to play the devil’s advocate.

Numbers like this that cannot be expressed as fractions are called irrational (meaning “not a ratio”).

It’s certainly not for any practical purpose. In fact, these imaginary shapes are actually harder to measure than real ones. Measuring the diagonal of a rectangle requires insight and ingenuity; measuring the diagonal of a piece of paper is easy—just get out a ruler.

The solution to a math problem is not a number; it’s an argument, a proof. We’re trying to create these little poems of pure reason.

Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying.

The point of doing algebra is not to solve equations; it’s to allow us to move back and forth between several equivalent representations, depending on the situation at hand and depending on our taste. In this sense, all algebraic manipulation is psychological. The numbers are making themselves known to us in various ways, and each different representation has its own feel to it and can give us ideas that might not occur to us otherwise.

Where does such an idea come from? How can you invent something like that? I don’t know. Mathematics is an art, and creative genius a mystery. Of course, technique helps—good painters understand light and shadow, good musicians have a thorough knowledge of functional harmony, and good mathematicians can untangle algebraic information—but a beautiful piece of mathematics is just as hard to make as a beautiful portrait or sonata.

Transcendental numbers—and there are lots of them—are simply beyond the power of algebra to describe. Lindemann proved that pi is transcendental in 1882. It is an amazing thing that we are able to know something like that.

If we look at the horizontal cross-sections, we can see that they are all the same—the same square as the base. It’s as if the different cross-sections have merely slid into new positions with their shape unchanged. The Cavalieri principle tells us that these two boxes have the same volume.

Nevertheless, we need some way to indicate precisely which polygon we’re talking about. We’re not going to be able to make measurements or communicate ideas about a shape that is described only as “that thing that looks sort of like a hat.”

The study of triangles is called trigonometry (Greek for “triangle measurement”).