Thank you to the publisher and NetGalley for the opportunity to read and review this book. All of the opinions given are my own and have been given nothing for my review.

The 5-4 decision divided strictly along partisan lines. The majority opinion, written by Chief Justice John Roberts, ruled that gerrymandering was “nonjusticiable”—which is to say, beyond the scope of the court. In his new book, Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy and Everything Else, Jordan Ellenberg shows how wrong that is.

https://www.thedailybeast.com/gerrymandering-is-unfair-and-the-supreme-court-ignored-the-math-that-proves-it?ref=author

I loved Ellenberg's "How Not to Be Wrong" so I was excited to read this book. This book requires a good deal more mathematical knowledge than his other book. I was a math major, took non-Euclidean geometry in college, and have taught geometry so I have a pretty decent background in geometry and there were still some things that I struggled with. Perhaps it was a case of I know too much so I was reading too deeply? The chapters on COVID-19 and gerrymandering were excellent and would be great to use with higher level high school math classes. This book won't be able to reach as wide an audience as "How Not to Be Wrong" but for math people, there's a lot to take away from this book.

This book took me a little while to get through, but I enjoyed the process.

To start the book, Ellenberg describes the ways that geometry, and specifically Euclid, inspired writers like William Wordsworth and other historical figures like Jefferson and Lincoln. The first chapter then moves on to discuss some proofs, and the sometimes hidden complexity within them. Chapter two starts off discussing the “how many holes does a straw have” argument, as an introduction to topology. This is an approach that Ellenberg uses throughout the book; introducing a concept through a simple or well known example, and then expanding it to think more deeply about the underlying geometry. Ellenberg's descriptions and diagrams are helpful, but it still requires some thought to fully grasp the intricacies of these concepts; so unless you are very familiar with advanced geometry, it will probably take some time to read.

In later chapters Ellenberg tackles symmetry, polling, probability theory, decision trees, machine learning, and more; while inserting historical information and bit of humor along the way. His jokes, analogies, and encouragement help to keep the book from feeling too much like a textbook; and make it a little easier to keep reading. I sometimes found the historical backgrounds and biographical information a little distracting, as Ellenberg sometimes seemed to go off on a tangent (no geometry pun intended), but other people might enjoy these extra history lessons.

I liked this book, and I will probably read through it again soon now that I've spent some time pondering some of the ideas. Ellenberg really has done a great job of revealing how many things can be broken down into geometric concepts; but you might have to take your time with this book, as some parts will require some deep thought.

A glorious intellectual feast & mind-expanding brain workout, exuberantly presented

Once upon a time, back in college, there was a kid who was winning not just all the awards in mathematics (and by "awards" I mean "top prizes in worldwide competition"), but also in *writing*. Aren't there rules against someone being that good at both? That kid has since been teaching math as a college professor for 20+ years, and written a new book. Would you be interested in reading a book on math by the best combination mathematician-writer I know? I think you should, 'cause it's pretty damn good!

Note that "Shape" is probably not beach reading for most people. It's the mental equivalent of a high-intensity interval training workout: challenging to get through *and* rewarding in the end. I initially felt myself protesting when I started reading it because Jordan was actually making my brain do work. How dare you make me think! But gradually, I came to appreciate the magnitude of the intellectual journey he was taking us on, and merrily hopped on the train.

At its core, "Shape" is about the underlying patterns of how the world works, and the beauty of how they connect together. Take the shape of a tree, for example. It underlies not just the familiar family tree and company org chart, but also decision trees that allow you to craft winning strategies in real-life games and magically effective tools like artificial intelligence. (Also, trees.)

What I appreciated most about "Shape" was that it forced me to dust off the intellectual cobwebs in my brain and fill holes in my incomplete education. I didn't even realize that I had only half-assed notions of what squaring the circle, Markov chains, neural networks and eigenvalues meant — a case of the "it sounds familiar and I kind of studied it therefore I must know it" fallacy. Now, not only do I have a much better understanding of these concepts, but I also know whence they came and how they fit into the world.

That's because Ellenberg also provides the whole historical matrix of who came up with an idea, whose work motivated it in the first place, who else came up with the idea in another context, and how it's relevant today. This makes for a strangely satisfying intellectual tapestry, a squiggly visual representation of which Ellenberg is kind enough to provide (see photo). Other things I like about this book:
• Ellenberg is not afraid of going into the historical weeds. He leaves no stone unturned, probably digging up original programs from the 1904 St Louis World Fair, finding an obscure letter in a 1905 issue of "Nature", and scoring an interview with Euclid himself who's been dead 2000 years. He places a host of richly-drawn characters in their historical context, bringing their ideas to life: Ronald Ross, the cantankerously egotistical biostatistician; Paul Erdös, the supremely eccentric itinerant mathematician; Henri Poincaré, the math god who sparked a revolution in physics; Marion Tinsley, the nigh-invincible checkers player; Einstein, Claude Shannon, Emmy Noether, and (surprise!) geometry fanboy Abraham Lincoln.
• A sense of puckish humor permeates the book. While reading, I registered a steady rate of 3.618 cph (chuckles per hour), often in response to pop & literary references: Survivor, Talking Heads, wannabe poets, and most important, Akbar and Jeff. A skilled stylist, Ellenberg also threw in some well-crafted neologisms that cracked me up. Do NOT skip the footnotes; that's where 80% of the funny lives.

I will not attempt to summarize the book's content, because it is so dense with ideas as to be incompressible. Want to learn about machine learning, epidemics, biostatistics, game theory, Google search algorithms, Heisenberg's Uncertainty Principle, neural nets, cryptography, gerrymandering, and how they're all interconnected? It's all in there, explained in a way that felt like encountering the ideas for the first time, usually because I was.

The book's good at presenting ideas in a fresh way because Ellenberg is an educator at heart, and a very skilled one. In the spirit of his prior book, "How Not to Be Wrong", "Shape" is a paean to rigorous thinking: "The ultimate reason for teaching kids to write a proof is not that the world is full of proofs. It’s that the world is full of *non-proofs*, and grown-ups need to know the difference. It’s hard to settle for a non-proof once you’ve really familiarized yourself with the genuine article."

In a world overrun by disinformation and sloppy thinking, "Shape" is the antithesis and the genuine article. It joyously (and rigorously) guides you down paths of bullshit-proof reasoning with precise language, delivering you to the Land of A-ha. The book expanded my mind, giving me a whole new paradigm for looking at the world — geometry, baby! In the process, it awakened parts of my brain so dormant from disuse, I didn't even know they still existed -- thanks, personal trainer Jordan! Get into "Shape" to get your own brain in shape.
-- Ali Binazir, M.D., M.Phil., Happiness Engineer and author of [[ASIN:B003GAMXG0 The Tao of Dating: The Smart Woman's Guide to Being Absolutely Irresistible]], the highest-rated dating book on Amazon, and [[ASIN:B01N7U7RK3 Should I Go to Medical School?: An Irreverent Guide to the Pros and Cons of a Career in Medicine]]

Love Song To Geometry - And A Look At How It Is Truly Everywhere. This is a mathematician showing just how prevalent geometry is in our every day lives - and why modern math classes tend to ruin it for most people. As a mathematics oriented person myself (got one math-derived degree, very nearly got two others almost at the same time, former math teacher, current active software developer), this was fairly easy to follow - Ellenberg mentions some advanced concepts without actually *showing* many of them, though there *is* more actual equations in here than some might like in a "popsci" level book. Thanks to Ellenberg's explanations of said equations and concepts, this *should* be an easy enough follow for most anyone. And he really does do a great job of showing how even advanced ideas really do come down to the most basic principles - just applied in particularly interesting ways. Indeed, the only real critique I have here is that when Ellenberg gets off the math specifically and into more political and social commentary - even when ostensbily using the math as a shield - it gets much closer to "Your Mileage May Vary" level. Overall, those moments weren't quite pervasive enough nor did they stray far enough from the central premise to warrant dropping a star, and thus the book maintains the full five stars that all books start with for me. Very much recommended.