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

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

  • Downloads:1396
  • Type:Epub+TxT+PDF+Mobi
  • Create Date:2021-08-14 08:51:56
  • Update Date:2025-09-06
  • Status:finish
  • Author:Jordan Ellenberg
  • ISBN:0593299736
  • Environment:PC/Android/iPhone/iPad/Kindle

Summary

From the New York Times-bestselling author of How Not to Be Wrong—himself a world-class geometer—a far-ranging exploration of the power of geometry, which turns out to help us think better about practically everything。

How should a democracy choose its representatives? How can you stop a pandemic from sweeping the world? How do computers learn to play Go, and why is learning Go so much easier for them than learning to read a sentence? Can ancient Greek proportions predict the stock market? (Sorry, no。) What should your kids learn in school if they really want to learn to think? All these are questions about geometry。 For real。

If you're like most people, geometry is a sterile and dimly remembered exercise you gladly left behind in the dust of ninth grade, along with your braces and active romantic interest in pop singers。 If you recall any of it, it's plodding through a series of miniscule steps only to prove some fact about triangles that was obvious to you in the first place。 That's not geometry。 Okay, it is geometry, but only a tiny part, which has as much to do with geometry in all its flush modern richness as conjugating a verb has to do with a great novel。

Shape reveals the geometry underneath some of the most important scientific, political, and philosophical problems we face。 Geometry asks: Where are things? Which things are near each other? How can you get from one thing to another thing? Those are important questions。 The word "geometry," from the Greek for "measuring the world。" If anything, that's an undersell。 Geometry doesn't just measure the world—it explains it。 Shape shows us how。

Download

Reviews

Adam

4。5 stars。 I think I liked "How not to be wrong" more, but this had some good stuff in it。 4。5 stars。 I think I liked "How not to be wrong" more, but this had some good stuff in it。 。。。more

Mike

Being an engineer means I took a lot of math in college。 I didn’t have patience for abstract math, as I wanted skills that I could directly apply to solving real world problems。 I didn’t like geometry, as it was presented as requiring rigorous proofs of, what seemed to my engineering mind, obvious qualities。 My wife, on the other hand, loves geometry, for this exact reason: supremely logical proofs。 So why read a book about geometry?It was the premise of actually using geometry to solve practica Being an engineer means I took a lot of math in college。 I didn’t have patience for abstract math, as I wanted skills that I could directly apply to solving real world problems。 I didn’t like geometry, as it was presented as requiring rigorous proofs of, what seemed to my engineering mind, obvious qualities。 My wife, on the other hand, loves geometry, for this exact reason: supremely logical proofs。 So why read a book about geometry?It was the premise of actually using geometry to solve practical problems。 Who know it could be also fun?!One of the themes throughout the book is forced learning in class doesn’t translate well into mastery。 A student needs to explore on their own a topic, which will then make it stick。 It is a different take on deliberate practice required for mastery of any subject or skill。 I took all those courses as I wanted to be an engineer, as using technology to solve problems was fun。 Here, the author uses several examples to illustrate this very fact。 So cool!The author is a joy to read。 The footnotes are even more fun, with snarky remarks on nearly every page。 He takes what could be dull writing about numbers to be humorous。 The book was written during the first months of COVID-19。 As such, he uses geometry and history to describe what the graphs we see everyday actually mean。 How he can use geometry and a bit of math to extrapolate where we could go based upon certain entrance parameters。If you are worried about there being a lot of math, there isn’t anything heavy or complicated。 A lot of it is simple, yet has the ability to be directly applicable to the real world。 By tying geometry in this way, I now have a better appreciation of the subject and why my wife thinks it is the best of all math。 。。。more

Matthew Emery

I had a lot of fun with Shape and I also learned a few things, too。 That’s really all one should expect from a popular mathematics book。 I especially like the chapter on Fermat's Little Theorem explained with strings of beads。 Another great chapter is on eigenvalues and random walks。 It has a good mix of introductory geometry, a bit of of mathematician gossip, and advanced concepts explained through analogy。 I think the last (possibly longest) chapter, on the geometry of gerrymandering, should h I had a lot of fun with Shape and I also learned a few things, too。 That’s really all one should expect from a popular mathematics book。 I especially like the chapter on Fermat's Little Theorem explained with strings of beads。 Another great chapter is on eigenvalues and random walks。 It has a good mix of introductory geometry, a bit of of mathematician gossip, and advanced concepts explained through analogy。 I think the last (possibly longest) chapter, on the geometry of gerrymandering, should have been its own book。 The tone is different from the rest of the book and it’s clear that this story is not over。 I eagerly await whatever Ellenberg decides to write next。 。。。more

C

The author brings up key ideas of geometry, ancient and modern, and illustrates how they are applied in various disciplines from pure math, to pandemic, to American elections。 The book is not about clever proofs as it is about clever applications。

Tim Dugan

Much of this you would know more formally from computer science math (discrete math)But the discussion about gerrymandering was pretty good

Nathan Cunningham

A bit disappointing after How Not To Be Wrong。 I was hoping for a book full of interesting ways that people solved problems using maths, with the complexities made simple by Jordan Ellenberg。 Instead, there's a lot more history on mathematical figures and developments, and a lot less really practical applications of complex maths。The book is still interesting, but just not as engrossing as How Not To Be Wrong。 A bit disappointing after How Not To Be Wrong。 I was hoping for a book full of interesting ways that people solved problems using maths, with the complexities made simple by Jordan Ellenberg。 Instead, there's a lot more history on mathematical figures and developments, and a lot less really practical applications of complex maths。The book is still interesting, but just not as engrossing as How Not To Be Wrong。 。。。more

Sagar Jethani

Shape? More like "Math。" Disappointed the book had so much actual calculating involved in it。 Shape? More like "Math。" Disappointed the book had so much actual calculating involved in it。 。。。more

Zachary Dial

A Wonderful Interlacing of Math and HistoryThis is a book for people who enjoy learning about a bunch of different people, ideas, concepts, views, and what I'd call "interesting tidbits" with a central theme throughout。 It was done really well, and I by the end I felt I had a greater appreciation for mathematics, particularly geometry。My only complaint is it gets a little dry/heavy at some parts。 I loved the beginning but each chapter afterwards felt harder to "get。" It's partially because the s A Wonderful Interlacing of Math and HistoryThis is a book for people who enjoy learning about a bunch of different people, ideas, concepts, views, and what I'd call "interesting tidbits" with a central theme throughout。 It was done really well, and I by the end I felt I had a greater appreciation for mathematics, particularly geometry。My only complaint is it gets a little dry/heavy at some parts。 I loved the beginning but each chapter afterwards felt harder to "get。" It's partially because the subject matter gets harder to be sucked into。 For example, learning about all of the ways math can illuminate discussions on gerrymandering for a whole chapter was very boring, and the math doesn't make it much more interesting。However, I still think this book was done well and is worth the money, especially because it might be one of those books you need to read a few times to soak it all in。 I imagine I'll like it better upon rereading。 。。。more

Roger Mattson

Ya gotta love mathematicians!Amazing insights from what we thought we learned in high school if the reader just sticks with the hard stuff up front。

Kalyan Turaga

Good book, well researched。 Honestly, it sounded like a math textbook and a political book another point。 All is well which ends well, Author stuck to his guns and drove the theme of the book to a logical。 Not for all, more for math or engineering background readers。 Will I read this book again no, did I enjoy it yes, do I want my kids read this book no。

Michael Finocchiaro

This is an eye-opening book about the prowess of geometry and how it impacts the real world。 The author successfully explains everything from graphs to game theory, from differential equations to high-level statistic in an easy to read manner with interesting and thought-provoking examples。 Even for non-math types, there is plenty to learn and enjoy here。

Kevin Crown

Really good balance of more simple concepts and more advanced ones。 I enjoyed the writing style and the wit the author possesses。 Overall great book, highly recommended for anyone who likes mathematics, or even logic/reasoning。

Mexscrabbler

Fabulous book by a mathematician which brings math concepts to bear on topical issues such as elections, democracy, covid, etc。 Frequently the book got way over my head at times, but I was always able to follow the gist of the discussion, and Ellenberg has a wonderful humourous style which is appreciated!Very very timely!

Richard Thompson

I don't know。 It was OK, but just didn't do it for me。 Mr。 Ellenberg explores ways in which various graphs and visualizations can be used in solving interesting easily understood, but sometimes complex problems。 Maybe it was his writing style。 Maybe I just wasn't in the right mood。 My one takeaway is that I'd like to do a bit more reading about Markov chains and Eigenvalues。 I don't know。 It was OK, but just didn't do it for me。 Mr。 Ellenberg explores ways in which various graphs and visualizations can be used in solving interesting easily understood, but sometimes complex problems。 Maybe it was his writing style。 Maybe I just wasn't in the right mood。 My one takeaway is that I'd like to do a bit more reading about Markov chains and Eigenvalues。 。。。more

Jim Willse

I can’t claim to have gotten all of this, or even most of it。 Despite Ellenberg’s chatty tone, it’s pretty tough going even for a devotee of popularized science writing。 So I picked my shots and took out what I could。

Kim Zinkowski

Enjoyable, humorous, oftentimes a bit tough to parse。

Ron

Mostly enjoyable, but like other reviewers I found that the title and subtitle of the book were not quite justified。 I did not find enough Geometry here, even if broadly conceived。

Bernard

Informative and wildly readableLots of great stuff in here touching on topics ranging from card shuffling to saving democracy, presented in a witty manner that teaches - easy without being facile。

April

Interesting, easy to follow, and very current and relevant。 I most enjoyed the sections about disease spread and gerrymandering。 I’d be interested to read more from the same author on other topics if he released them!

Michiel

"Shape" is the successor of Ellenbergs' popular work "How Not to Be Wrong"。 Whereas the earlier book showed how to use mathematical thinking in everyday life (and win the lottery), this book aims to show that geometry is absolutely everywhere。 I am no mathematician, but I have the feeling that Ellenberg considers absolutely everything geometry。 There is a chapter on Euclid, about Euclid, the topology of straws and pants, Gerrymandering but also about random walks, graphs game theory, the uncerta "Shape" is the successor of Ellenbergs' popular work "How Not to Be Wrong"。 Whereas the earlier book showed how to use mathematical thinking in everyday life (and win the lottery), this book aims to show that geometry is absolutely everywhere。 I am no mathematician, but I have the feeling that Ellenberg considers absolutely everything geometry。 There is a chapter on Euclid, about Euclid, the topology of straws and pants, Gerrymandering but also about random walks, graphs game theory, the uncertainty principle, etc。 Nothing much connects the chapters, so the whole style felt meandering。Ellenberg is a charismatic writer with a talent for explaining complicated math to the layperson。 I felt that after the success of this previous book, this has hardly been edited。 I have read several interviews with the author that geometry is his least favourite branch of mathematics。 It shows。 Most chapters feel like a collection of brainstormed, vaguely connected ideas weaved into a text on an evening。 For example, consider one chapter。 It jumps from: a series of numbers, an epidemiological model, Fibonacci numbers, rational numbers, stocks, eigenvalues, page rank, eigensequences and the uncertainty principle。 Compare with Strogatz' "Infinite Powers", which guides the reader over the wonders of calculus or Parker's "Things to Make in the Fourth Dimension", in which each chapter is a delightful self-contained story about a math topic。 These are more enjoyable math books for laypersons, at least in my opinion。 。。。more

Melody

Shape - Jordan Ellenberg (2021)2021 read #83Grade: ASynoposis: Jordan Ellenberg is a math professor at the University of Wisconsin - Madison。 In his book, Shape, he explores how geometry can be used to solve a variety of real-world problems, from board games to gerrymandering。 He describes the math behind a lot of the virus spread data we've been seeing on the news since 2020。 This book is written not just for professional mathematicians, but for everyone who desires to learn why geometry is imp Shape - Jordan Ellenberg (2021)2021 read #83Grade: ASynoposis: Jordan Ellenberg is a math professor at the University of Wisconsin - Madison。 In his book, Shape, he explores how geometry can be used to solve a variety of real-world problems, from board games to gerrymandering。 He describes the math behind a lot of the virus spread data we've been seeing on the news since 2020。 This book is written not just for professional mathematicians, but for everyone who desires to learn why geometry is important。Thoughts: This was a birthday gift from my boyfriend so I was so excited to read this! I am getting my PhD in Math so this book was right up my alley。 I've found with a lot of similar books, the material is either too basic for my level and I learn nothing, or it's too advanced and I feel like I'm reading for work instead of pleasure。 Ellenberg has found a happy medium。 What you won't do while reading Shape is crunch numbers in formulas, instead you'll learn the concepts and applications of geometry。 If you read a lot of nonfiction and want to learn a little math, then I highly recommend this book! 。。。more

Ann Hriciga

I wanted to like this book but found it like a random walk through space。 I was not expecting so much discussion about the personalities of various mathematicians。 The discussion of how to analyze the spread of a virus was very well done and easy to understand, so kudos for that。

Lance Johnson

This book has many excellent pieces but they are poorly linked together and surrounded by bloat。 In my opinion, it could be polished into another masterpiece (like his previous book "How Not to be Wrong") if the theme was more concrete and topics were more carefully strung together。 Before I go through my criticisms, there are several positives。 Generally, I like Ellenberg's writing style and his humor。 He also has a knack for choosing good examples and historical tidbits。 I also appreciate that This book has many excellent pieces but they are poorly linked together and surrounded by bloat。 In my opinion, it could be polished into another masterpiece (like his previous book "How Not to be Wrong") if the theme was more concrete and topics were more carefully strung together。 Before I go through my criticisms, there are several positives。 Generally, I like Ellenberg's writing style and his humor。 He also has a knack for choosing good examples and historical tidbits。 I also appreciate that he is not afraid to talk about advanced concepts and present them in a simple language。 There were many points in the book that caused me to think about something in a different light or see a connection I previously had not。 So if you are already interested in mathematics and are willing to wade through the book to find the gems, I would recommend it。 My first major criticism is that the overall theme is poorly defined。 Math is broad and densely interconnected but has several different fields: algebra, analysis, topology, probability, combinatorics, number theory, geometry, etc。 Most problems fall naturally into one or more fields but can benefit from the tools or perspectives of other fields as well。 The theme of this book is geometry but for many of the topics discussed the author does little to convince me that the topic is either naturally a geometric problem or that a particularly geometric lens is being applied to it。 He seems to lump anything that has a shape present or involves a distance under the umbrella of geometry, which is most of math。 He is not wrong exactly, but taking such a loose approach robs it of any particular significance。 It is a bit like classifying most of math as arithmetic since many areas involve the use of addition, subtraction, and multiplication。 The statement is not quite wrong, but it lacks any meaningful insights。 My second criticism is that the topics are poorly strung together。 Each chapter meanders from tangent to tangent until I had forgotten what the original point was。 Then suddenly it jumps back to a previous topic。 Many topics were split into pieces and scattered across several chapters。 For example, there is a particular subplot that is broken up across at least 5 chapters。 The division just makes it hard to follow。 My third criticism is there is a significant amount of bloat。 The book is over 400 pages long, but many pages felt redundant or tedious。 By the end, I really didn't want to keep reading。 Sometimes he would go out of his way to avoid a certain technical term or concept, but to avoid it would involve all these gymnastics that took longer and were more confusing than it would be to quickly introduce the term and explain it。 Other bits were fine, but did not feel like they had a good reason to be in the book。 Either they overlapped heavily with another section or seemed so tangential that they did not fit with the surrounding material。There are gems in here, you just have to dig for them。 Looking forward to his next book。 。。。more

Isaac

Pretty sure I could not have correctly defined "geometry" prior to reading this book。 I still can't, but at least now I know it's more than just angles and flat shapes。This book was kind of all over the place。 I probably use the expression "dizzying intellect" a bit too much, but it seems particularly apt to describe Jordan Ellenberg here。 He seems to start accessibly with a topic and often spends a while there but seems to inevitably spiral off on tangents of increasing complexity, occasionally Pretty sure I could not have correctly defined "geometry" prior to reading this book。 I still can't, but at least now I know it's more than just angles and flat shapes。This book was kind of all over the place。 I probably use the expression "dizzying intellect" a bit too much, but it seems particularly apt to describe Jordan Ellenberg here。 He seems to start accessibly with a topic and often spends a while there but seems to inevitably spiral off on tangents of increasing complexity, occasionally spiraling back out, occasionally just lurching back to the initial topic。That's all part of the charm though, his enthusiasm for the subject is so unmistakably genuine that even when I wasn't following the math at all, I was still enjoying the book。 He's also got a great sense of humor, the book is really funny in places。As a bit of a Lincoln nerd I really loved the first section about Lincoln and Euclid, and as a Wisconsonite I really loved the final section on our recent history of Gerrymandering。 There was some really interesting pandemic geometry in there, but it was very "last summer", almost dated given all that's transpired, and I think that will all be much better if it gets expanded and revised in a few years。 。。。more

Marcos

Like many of the reviews here, I also loved Ellenberg's earlier book! So maybe I was too hyped about this one。The problem is that the topics are so interesting but it's very hard to follow the common thread in the book。 We jump from subject to subject without any (at first) obvious connection。 This worked really well in his previous book, where concepts were beautifully woven together to make a coherent whole。Another issue I had with the book was that it felt like some concepts are dumbed down p Like many of the reviews here, I also loved Ellenberg's earlier book! So maybe I was too hyped about this one。The problem is that the topics are so interesting but it's very hard to follow the common thread in the book。 We jump from subject to subject without any (at first) obvious connection。 This worked really well in his previous book, where concepts were beautifully woven together to make a coherent whole。Another issue I had with the book was that it felt like some concepts are dumbed down past the what could be reasonably expected。 For example, when explaining loss functions in machine learning, Ellenberg writes "。。。we can assign our strategy a 'wrongness function。'" and in a footnote says "*among actual computer scientists, usually called an error or a loss" (p。 169)。 Why? Saying loss function is less confusing than saying wrongness function is also called loss or error function。There are quite a few tangents to tangents to tangents in the storyline。 For example, Simpson's paradox (p。 226), which I find fascinating and enjoyable to explore, is mentioned in passing while explaining something else (Covid-related)。 This was a shame because in that specific setting, I got no wiser about Simpson's paradox or about the Covid-example。It also felt like the book could have been much more concise。 Then again, I didn't write any book。 So, good job Jordan! It's so much easier for us to criticise a book than write one ourselves。 Looking forward to your reading your next book! 👏 。。。more

Kumar Ayush

Less technical than I expected it to be, but I enjoyed reading it very much。

Bowman Dickson

Enjoyed! Strangely wanted more geometry (he defines geometry loosely) but got a bunch of teaching ideas

Susan

This is the geometry book that you didn't know that you wanted to read。 Let's be truthful--it's not a fast or easy read, and despite the reviews saying that you'll follow the math, there are probably places where you won't, unless you're a lot better at math than I am。 But Ellenberg is an entertaining writer and knows his stuff cold, so you're sure to learn a lot anyway。 The long chapter on redistricting--Ellenberg was involved in the efforts in Wisconsin, where he lives--is enlightening, enragi This is the geometry book that you didn't know that you wanted to read。 Let's be truthful--it's not a fast or easy read, and despite the reviews saying that you'll follow the math, there are probably places where you won't, unless you're a lot better at math than I am。 But Ellenberg is an entertaining writer and knows his stuff cold, so you're sure to learn a lot anyway。 The long chapter on redistricting--Ellenberg was involved in the efforts in Wisconsin, where he lives--is enlightening, enraging, and discouraging, but there are laughs along the way。 。。。more

Keven Wang

Some parts of the book is quite entertaining。 A math major would probably enjoy it more。

Miguel

It’s really hard for a writer to make math, let alone geometry, a fun and inviting subject。 Ellenberg does about as good a job as possible for those not well versed in this area (even after taking courses up to Calc V and diff equations in college I am not familiar with most of his topics in any sort of depth)。 His style keeps the reader engaged, although some of the concepts still get a bit tough to meticulously follow such as the larger portion near the end on jerrymandering。 The charts and fi It’s really hard for a writer to make math, let alone geometry, a fun and inviting subject。 Ellenberg does about as good a job as possible for those not well versed in this area (even after taking courses up to Calc V and diff equations in college I am not familiar with most of his topics in any sort of depth)。 His style keeps the reader engaged, although some of the concepts still get a bit tough to meticulously follow such as the larger portion near the end on jerrymandering。 The charts and figures in the book do help somewhat and would have been difficult to follow the audiobook alone for this particular one。 。。。more