Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else

After reading ellenberg’s other book (how not to be wrong) I right away ordered this one. He’s such a good writer and able to explain abstract and complicated math topics to common folk. His thought approach to everyday questions/problems is astounding. He’ll make you smarter.My only complaint about this author is that he’s only written 2 books. I’ll read whatever he writes.

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, 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  The Tao of Dating: The Smart Woman's Guide to Being Absolutely Irresistible , the highest-rated dating book on Amazon, and  Should I Go to Medical School?: An Irreverent Guide to the Pros and Cons of a Career in MedicineThe Tao of Dating: The Smart Woman's Guide to Being Absolutely IrresistibleShould I Go to Medical School?: An Irreverent Guide to the Pros and Cons of a Career in Medicine

A very interesting reading but very hard to follow. If you're not a mathematician you'll have some difficulties to extract all the richness of the text. In fact, I cannot imagine the time it would have taken me to develop any single issue in the book, to calculate, to get the correct mathematical expressions in order to make sense of everything. A month? Two months? A year? I don't know.Jordan Ellenberg is a mathematician but not like any mathematician (if there are...). His field of expertise is geometry and that is the topic of the book. "Geometry," he says at the beginning of the book, "is the cilantro of math." That's why you'll see lots of lines and triangles here and there, and in some moment you'll begin to realize that they are everywhere, I mean, not just in the pages of the book, but everywhere, inside and outside of you.Yes, we cannot see those lines (in a conversation, for instance), and here enters Ellenberg: he shows you exactly where there are and how mathematicians do in order to be able to work with them. Consider this: plain geometry is the part of geometry we see, so what we do in order to see it dynamically in three or four dimensions, or in curved spaces? Well, that's what they do (Ellenberg and Co.), and the ample variety of topics show you how present geometry is in everything, and I mean it.That said, I should add that the topics the author chose were very interesting and important. To me, the most important was the last one, "How math broke democracy (and might still save it)." It really tells you the things that you never thought about, for instance in the relationship between geometry and freedom or geometry and gerrymandering. Related? Yes.The book is full of quotes and anecdotes of great mathematicians. The hero of the book to me is Henri Poincaré. This quote is taken from him: "If a transcendent being knew the true nature of things... he could not find words to express it. Not only can we not divine the response, but if it were given to us, we could not understand nothing of it."So as you can see, even if the text is difficult, it gives you something valuable all the time. So read it slow and with some patience.In sum, a superb job, but to be enjoyed according to the expertise of every one. If you're a mathematician, I envy you. If not, well, something always remains, including of course the possibility of talking about it and recommending it.

Three things stood out to me: 1. A series of interesting explanations of how geometry is a much more far-ranging field than what we all learned in high school (analysis in higher dimensions, games as an application of geometry, differential equations as a way to express scientific laws). 2. A clever example of mathematical proof in Chapter 5, where Ellenberg points out how "proof" is a process that forces us to think more generally and to explain things clearly. 3. A wonderful tribute in Chapter 10 to the Princeton University mathematician, John Horton Conway, who was an amazingly creative and playful person... and who we lost to Covid in April 2020. (I had the pleasure of seeing Conway speak several times over the years.)I liked Ellenberg's previous book better ("How Not to be Wrong"), but this one is still fun. It's chock-full of interesting stories about mathematical personalities (especially Henri Poincaré). The section on the Covid pandemic was an excellent apolitical treatment of the mathematics for analyzing the spread. (The only semi-political comments were very restrained and only in the footnotes.)(Note that there is another review of this book where the reviewer expressed disappointment in the hand-drawn illustrations in the book. I actually found Ellenberg's hand-drawn pictures to be great, because they brought me back to my university days as a math major. As students, we had to present geometry or topology proofs on the blackboard, and our hand-drawn diagrams were really awful... but it didn't matter. The rest of the class knew that we were discussing the "abstraction behind the drawing," and a super-polished diagram could often be misleading.)[Update:] I just finished the book, and the last chapter ("How Math Broke Democracy") is absolutely amazing! Gerrymandering is a very complex topic, and it isn't easy to decide "what is the right thing to do?" Ellenberg bends over backwards to be as fair as possible to political line-drawers - we all know that it is impossible to keep politics out of the process. He gives an excellent summary of Prof. Moon Duchin's ReCom analysis, her brilliant way to assess the fairness versus sneakiness of redistricting maps.What makes this chapter special is that it doesn't stand alone. Ellenberg brings back examples from his earlier chapters (maps, complex polygons, random walks and transmission of diseases by mosquitos, graphs and trees and games like Nim, and poetry) to explain several alternative analysis methods. What math book has quotes from current Supreme Court justices? (By the way, Elena Kagan seems to be the most mathematically sophisticated justice.)

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.