Math, as mathemagician Vi Hart’s stop-motion doodles, photographer Robin Moore’s string portraits, and artist Anatolii Fomenko’s stunning black-and-white illustrations have previously shown us, can be the conduit of great fun and great beauty. In Measurement (public library), mathematician Paul Lockhart invites us to “make patterns of shape and motion, and then [try] to understand our patterns and measure them.” (Because, lest we forget, we have a natural penchant for patterns.) What results as we step away from physical reality and immerse ourselves in the imaginary — and imaginative — world of mathematical reality is a thing of infinite beauty and infinite fascination.

Lockhart writes in the introduction:

Physical reality is a disaster. It’s way too complicated, and nothing is at all what it appears to be. Objects expand and contract with temperature, atoms fly on and off. In particular, nothing can truly be measured. A blade of grass has no actual length. Any measurement made in the universe is necessarily a rough approximation. It’s not bad; it’s just the nature of the place. The smallest speck is not a point, and the thinnest wire is not a line.

Mathematical reality, on the other hand, is imaginary. It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I can never hold a circle in my hand, but I can hold one in my mind.

[…]

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

Lockhart underpins this excitement with a fair warning:

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.

And yet, he’s quick to reassure that the gold standard of math, not unlike that of science, isn’t the answer but the driver of curiosity:

What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for simple beauty.

But what makes Lockhart particularly compelling is his ability to relate mathematics to parallel concepts from disciplines and aspects of life that are more familiar, more comfortable, more ingrained in our everyday understanding of the world — for instance, in comparing math to storytelling:

A mathematical argument [is] otherwise known as a proof. A proof is simply a story. The characters are the elements of the problem, and the plot is up to you. 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.

In a nod to the value of “useless” knowledge and figuring things out, Lockhart argues:

People don’t do mathematics because it’s useful. They do it because it’s interesting … The point of a measurement problem is not what the measurement is; it’s how to figure out what it is.

He ultimately offers several pieces of advice on engaging with math:

1. The best problems are your own. Mathematical reality is yours — it’s in your head for you to explore any time you feel like it… Don’t be afraid that you can’t answer your own questions — that’s the natural state of the mathematician.
2. Collaborate. Work together and share the joys and frustrations. It’s a lot like playing music together.
3. Improve your proofs. Just because you have an explanation doesn’t mean it’s the best explanation. Can you eliminate any unnecessary clutter or complexity? Can you find an entirely different approach that gives you deeper insight? Prove, prove, and prove again. Painters, sculptors, and poets do the same thing.
4. Let a problem take you where it takes you. If you come across a river in the jungle, follow it!
5. Critique your work. Subject your arguments to scathing criticism by yourself and others. That’s what all artists do, especially mathematicians… For a piece of mathematics to fully qualify as such, it has to stand up to two very different kids of criticism: it must be logically sound and convincing as a rational argument, and it must also be elegant, revelatory, and emotionally satisfying. [But don’t] worry about trying to hold yourself to some impossibly high standard of aesthetic excellence.

Lockhart elaborates on the latter point with a poignant reflection that applies to math just as much as it does to life itself:

Part of the problem is that we are so concerned with our ideas being simple and beautiful that when we do have a pretty idea, we want so much to believe it. We want it to be true so badly that we don’t always give it the careful scrutiny that we should read. It’s the mathematical version of ‘rapture of the deep.’ Divers see beautiful sights that they forget to come up for air. Well, logic is our air, and careful reasoning is how we breathe.

Indeed, much of math sounds an awful lot like the art of living itself: Take for instance, that same old fear of failure that often stands in the way of creativity, which also holds us back from immersing ourselves in the art of figuring things out:

The important thing is not to be afraid. So you try some crazy idea, and it doesn’t work. That puts you in some pretty good company! 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, mostly failing. And then every once in a while, you succeed… And that feeling of unlocking an eternal mystery is what keeps you going back to the jungle to get scratched up all over again.

Here’s a little teaser for the whimsical jungle of mathematical reality and the logical aesthetic of math:

Measurement, from Harvard University Press, comes seven years after Lockhart’s exquisite critique of math’s tragic fate in contemporary education, A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.