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Форма реальности Скрытая геометрия стратегии, информации, общества, биологии и всего остального
Эта книга изменит ваше представление о мире. Джордан Элленберг, профессор математики и автор бестселлера МИФа «Как не ошибаться», показывает всю силу геометрии — науки, которая только кажется теоретической.
Математику называют царицей наук, а ее часть — геометрия — лежит в основе понимания мира. Профессор математики в Висконсинском университете в Мэдисоне, научный сотрудник Американского математического общества Джордан Элленберг больше 15 лет популяризирует свою любимую дисциплину.
В этой книге с присущими ему легкостью и юмором он рассказывает, что геометрия не просто измеряет мир — она объясняет его. Она не где-то там, вне пространства и времени, а
Математику называют царицей наук, а ее часть — геометрия — лежит в основе понимания мира. Профессор математики в Висконсинском университете в Мэдисоне, научный сотрудник Американского математического общества Джордан Элленберг больше 15 лет популяризирует свою любимую дисциплину.
В этой книге с присущими ему легкостью и юмором он рассказывает, что геометрия не просто измеряет мир — она объясняет его. Она не где-то там, вне пространства и времени, а
723 pages, Kindle Edition
First published May 25, 2021
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About the author
Jordan Ellenberg
5 books417 followersJordan Ellenberg is the John D. MacArthur Professor of Mathematics at the University of Wisconsin-Madison. His writing has appeared in Slate, the Wall Street Journal, the New York Times, the Washington Post, the Boston Globe, and the Believer.
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Displaying 1 - 30 of 285 reviews
May 30, 2021
I really enjoyed Jordan Ellenberg’s earlier book How Not to be Wrong, so looked forward to Shape with some anticipation. In principle what we have here is a book about geometry - but not seen from the direction of the (dare I say it) rather boring, Euclid-based geometry textbooks some of us suffered at school. Instead Ellenberg sets out to show how geometry underlies pretty much everything.
Along the way, we are given some nice turns of phrase. I enjoyed, for example, Ellenberg’s remark on the philosopher Thomas Hobbes, where Ellenberg remarks Hobbes was ‘a man whose confidence in his own mental powers is not fully captured by the prefix “over”’. Whether or not what we read about here is really all geometry is a matter of labelling (as is the ‘number of holes in a straw’ question that Ellenberg entertainingly covers). Arguably, for example, there is some material that is probability that can be looked at in a geometric fashion, rather than geometry that produces probabilistic results - I find the probability viewpoint a lot simpler and more interesting. In the end, despite his efforts, unless you are a mathematician, some aspects of geometry (and maths in general) feel laboured and uninteresting. But the marvel of this book is that he does make a surprising amount of it quite the opposite.
Unfortunately, Ellenberg can go into far too much detail sometimes (which may be why the book is a bit of a doorstop at 463 pages) - for example, a story that starts with the mosquito’s random walk seems to go on and on for ever and I rather lost the will to continue, in a topic that interests me a lot more than geometry does. In a different way, I struggled to get my head around the lengthy section on US electoral district gerrymandering which seemed only of interest to someone with in-depth enthusiasm for US politics, while some of the final meandering final chapter should certainly have been lost in the edit.
I do wonder if the success of the earlier book meant this one was not given the editorial scrutiny it needed. Although Ellenberg failed to convince me that geometry is the foundation of many of the topics he discusses, there is material to interest the reader here - and it’s certainly a far cry from those laborious proofs and QEDs.
Along the way, we are given some nice turns of phrase. I enjoyed, for example, Ellenberg’s remark on the philosopher Thomas Hobbes, where Ellenberg remarks Hobbes was ‘a man whose confidence in his own mental powers is not fully captured by the prefix “over”’. Whether or not what we read about here is really all geometry is a matter of labelling (as is the ‘number of holes in a straw’ question that Ellenberg entertainingly covers). Arguably, for example, there is some material that is probability that can be looked at in a geometric fashion, rather than geometry that produces probabilistic results - I find the probability viewpoint a lot simpler and more interesting. In the end, despite his efforts, unless you are a mathematician, some aspects of geometry (and maths in general) feel laboured and uninteresting. But the marvel of this book is that he does make a surprising amount of it quite the opposite.
Unfortunately, Ellenberg can go into far too much detail sometimes (which may be why the book is a bit of a doorstop at 463 pages) - for example, a story that starts with the mosquito’s random walk seems to go on and on for ever and I rather lost the will to continue, in a topic that interests me a lot more than geometry does. In a different way, I struggled to get my head around the lengthy section on US electoral district gerrymandering which seemed only of interest to someone with in-depth enthusiasm for US politics, while some of the final meandering final chapter should certainly have been lost in the edit.
I do wonder if the success of the earlier book meant this one was not given the editorial scrutiny it needed. Although Ellenberg failed to convince me that geometry is the foundation of many of the topics he discusses, there is material to interest the reader here - and it’s certainly a far cry from those laborious proofs and QEDs.
July 5, 2023
I loved Jordan Ellenberg's earlier book, How Not to Be Wrong: The Power of Mathematical Thinking, and it was a hard act to follow. But not for Ellenberg, as this book is also great. While the title, "Shape" implies that this book is about geometry--and it is--it is also about so much more. This book shows how mathematics is applicable to just about everything under the sun. And Ellenberg manages to make it all so fascinating! The sheer depth at which he covers an incredibly diverse range of topics is staggering. This is the type of book that I love! Ellenberg is a professor of mathematics, so he certainly knows what he is talking about!
I enjoyed his description of how computers have been taught to play chess. I enjoyed his description of the problem of gerrymandering in states. I had previously thought that, were it not for politics, it would be an easily solved issue. Ellenberg shows why politics is just one of the thorny issues. The geometry, and the issue of fairness is just as difficult! The book describes issues about pandemics, including Covid-19. How can you turn away from a book that talks about politics, eigenvectors, Poincare, neural networks, Euclid and ... and ... Wow! The book does tend to spend a bit too many pages on the subject of gerrymandering, but it is a big problem, and I have never seen such a treatment of the subject.
I enjoyed his description of how computers have been taught to play chess. I enjoyed his description of the problem of gerrymandering in states. I had previously thought that, were it not for politics, it would be an easily solved issue. Ellenberg shows why politics is just one of the thorny issues. The geometry, and the issue of fairness is just as difficult! The book describes issues about pandemics, including Covid-19. How can you turn away from a book that talks about politics, eigenvectors, Poincare, neural networks, Euclid and ... and ... Wow! The book does tend to spend a bit too many pages on the subject of gerrymandering, but it is a big problem, and I have never seen such a treatment of the subject.
July 27, 2021
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.
May 25, 2021
Shape is University of Wisconsin math professor and bestselling author Ellenberg’s 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? 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. It's an interesting, informative and accessible exploration of the way geometry touches our everyday lives without us giving it a second thought. I admit, I used to not be that fond of anything relating to maths in school many moons ago, but I found this easily digestible and rather compelling to read. Showing exactly how geometry applies to real-life situations, Ellenberg pens a fascinating book even to those of us who are rather maths-averse, which is quite the feat. Highly recommended.
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. It's an interesting, informative and accessible exploration of the way geometry touches our everyday lives without us giving it a second thought. I admit, I used to not be that fond of anything relating to maths in school many moons ago, but I found this easily digestible and rather compelling to read. Showing exactly how geometry applies to real-life situations, Ellenberg pens a fascinating book even to those of us who are rather maths-averse, which is quite the feat. Highly recommended.
May 26, 2021
I am not sure you can possibly understand how high my expectations were for this book. Jordan Ellenberg's How Not To Be Wrong is my favorite book of all time about mathematics and I have been waiting for him to write another since I read it. I have read How Not To Be Wrong at least three times and I have badgered everyone I know into reading it. "If you want to understand me," I tell my friends, "you need to read this book." Ellenberg gives people a short cut to getting a mathematician's view of the world. He's funny, he knows great stories. You'll laugh, you'll gasp. Read this book, see the world differently. Learn. Understand.
July 7, 2021
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.
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.
February 12, 2022
The 3 is because this is a clever and fun book for people who aren't me, probably a 5 for the number-savvy.
I struggled through the first half, reading every word, getting farther and farther behind in proofs and theorems that were increasingly out of my reach. This is a book for the true math adjacent. As the author says "MATH IS HARD".
Some truly commendable notes - there are women in math, forging new ideas. Emmy Noether in early 1900s modernizing the theory of holes was uplifting. I did enjoy coverage of Ronald Ross figuring out how far the mosquito could take malaria - very relevant. The next to last chapter is a VERY deep dive into gerrymandering, which I have been very interested in (living through it in North Carolina, so happy that our STATE supreme court threw out the districting - SO DISAPPOINTED IN OUR RIDICULOUS FEDERAL SUPREME COURT).
But I couldn't read most of it, I'm just not numberly enough to fully enjoy it.
I struggled through the first half, reading every word, getting farther and farther behind in proofs and theorems that were increasingly out of my reach. This is a book for the true math adjacent. As the author says "MATH IS HARD".
Some truly commendable notes - there are women in math, forging new ideas. Emmy Noether in early 1900s modernizing the theory of holes was uplifting. I did enjoy coverage of Ronald Ross figuring out how far the mosquito could take malaria - very relevant. The next to last chapter is a VERY deep dive into gerrymandering, which I have been very interested in (living through it in North Carolina, so happy that our STATE supreme court threw out the districting - SO DISAPPOINTED IN OUR RIDICULOUS FEDERAL SUPREME COURT).
But I couldn't read most of it, I'm just not numberly enough to fully enjoy it.
July 10, 2021
"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.
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.
January 30, 2022
One of the key mathematical stories early in this book is "the random walk". And the the book walks randomly over a bunch of material, staying too long on some parts, and not long enough on others.
One of the longest parts, too long for my taste, was about gerrymandering. The key question is can you prove that districts were created specifically to give a certain party the advantage? The simple fact that percent of representatives chosen is not proportional to the number of members of each party is not enough to prove it. For example, Massachusetts has some percentage of Republicans, but no Republican wins seats, and that is not Gerrymandering. People of all parties are randomly spread-out in Massachusetts, so that is just the way things turn out. But Wisconsin is a different story. Anyone who looks hard at that state can see that it was obviously rigged. And the people who rigged it pretty much admit that is what they did. But, still, can you prove it?
The author spends a lot of time on this because it is personal to him. He was one author on a brief to the Supreme Court examining this issue. This mathematical analysis shows that, yes, you can prove it with a high degree of probability. Elana Kagen clearly read and understood that analysis. Kavanaugh, and most others, clearly did not understand it. It is really frustrating to see them miss the point over and over. The court decided that since you can't prove intentional unfairness (though you CAN), the court can do nothing about it.
Anyway... still recommended to math nerds, like me. Early in the book there was an entertaining chapter on "how many holes does a straw have?" The question leads to some interesting answers. He correctly points out that mathematics cannot really answer the question because "holes" has a multiple, vague definitions in English. And even in math, it is more a question of choice: we pick a definition for our terms that we think turns out to be most useful. That is all great! But even while being very careful about that, he is very un-careful about defining whether he is talking about a real straw (where the edges have some volume, making it homeomorphic with a torus) or a mathematical abstraction (where the edges have no volume, making it homeomorphic with an annulus). He is in some sections clearly talking about a straw through which one can drink a milkshake, but switches to treating it mathematically without apparently noticing the difference.
One of the longest parts, too long for my taste, was about gerrymandering. The key question is can you prove that districts were created specifically to give a certain party the advantage? The simple fact that percent of representatives chosen is not proportional to the number of members of each party is not enough to prove it. For example, Massachusetts has some percentage of Republicans, but no Republican wins seats, and that is not Gerrymandering. People of all parties are randomly spread-out in Massachusetts, so that is just the way things turn out. But Wisconsin is a different story. Anyone who looks hard at that state can see that it was obviously rigged. And the people who rigged it pretty much admit that is what they did. But, still, can you prove it?
The author spends a lot of time on this because it is personal to him. He was one author on a brief to the Supreme Court examining this issue. This mathematical analysis shows that, yes, you can prove it with a high degree of probability. Elana Kagen clearly read and understood that analysis. Kavanaugh, and most others, clearly did not understand it. It is really frustrating to see them miss the point over and over. The court decided that since you can't prove intentional unfairness (though you CAN), the court can do nothing about it.
Anyway... still recommended to math nerds, like me. Early in the book there was an entertaining chapter on "how many holes does a straw have?" The question leads to some interesting answers. He correctly points out that mathematics cannot really answer the question because "holes" has a multiple, vague definitions in English. And even in math, it is more a question of choice: we pick a definition for our terms that we think turns out to be most useful. That is all great! But even while being very careful about that, he is very un-careful about defining whether he is talking about a real straw (where the edges have some volume, making it homeomorphic with a torus) or a mathematical abstraction (where the edges have no volume, making it homeomorphic with an annulus). He is in some sections clearly talking about a straw through which one can drink a milkshake, but switches to treating it mathematically without apparently noticing the difference.
January 11, 2022
Cool math stuff, but not really tied together by central theme. Only read if you want more math in your life.
August 4, 2024
this was great! but only if you like math. if you don’t like math (specifically some abstract stuff) i would not recommend at all. not to be exclusive but bc i just don’t know if you’d like it. i listened to the audiobook initially so i could listen at work and then bc i liked it. this book is pretty dense so i think the audiobook is why i was able to finish it but i felt like it was easier to zone out and miss stuff and he would often mention figures that u just wasn’t able to see. so honestly i would consider a physical reread. i thought he made the math super accessible and was still funny and engaging despite it being a book about math. i rlly enjoy seeing math in day to day life so i loved this!
July 28, 2023
[15 July 2023]
This guy really loves geometry. He thinks it's the solution to all the world's problems. Not really, but it almost seems that way. I admire his love for his field, but I don't think he was entirely successful if he was trying to convince others to join his geometry fan club. Maybe if you're already a math geek. I loved high school algebra, but I always thought geometry was sort of meh. And Ellenberg didn't really change my mind.
He focuses a lot on the history of geometry and that was mostly boring. Lots of names that I will never remember.
He also went off on frequent tangents that I thought didn't add to the book. For example he spent several pages discussing irrational numbers like pi and phi and how you could get to a fraction which was very close to the actual value of those numbers. But there was no point to it. I couldn't figure out why I should care. And I was mostly lost during his extended discussion of eigenvalues. I just didn't get it.
There were some parts of the book that I did enjoy, however, like the explanation of programming a computer to play games like checkers, chess, and go. That was interesting and mostly understandable. Also I enjoyed the discussion of gerrymandering.
I'm not sure I would recommend this book if you're not already interested in math, but I'm not sorry I read it, either. YMMD.
This guy really loves geometry. He thinks it's the solution to all the world's problems. Not really, but it almost seems that way. I admire his love for his field, but I don't think he was entirely successful if he was trying to convince others to join his geometry fan club. Maybe if you're already a math geek. I loved high school algebra, but I always thought geometry was sort of meh. And Ellenberg didn't really change my mind.
He focuses a lot on the history of geometry and that was mostly boring. Lots of names that I will never remember.
He also went off on frequent tangents that I thought didn't add to the book. For example he spent several pages discussing irrational numbers like pi and phi and how you could get to a fraction which was very close to the actual value of those numbers. But there was no point to it. I couldn't figure out why I should care. And I was mostly lost during his extended discussion of eigenvalues. I just didn't get it.
There were some parts of the book that I did enjoy, however, like the explanation of programming a computer to play games like checkers, chess, and go. That was interesting and mostly understandable. Also I enjoyed the discussion of gerrymandering.
I'm not sure I would recommend this book if you're not already interested in math, but I'm not sorry I read it, either. YMMD.
June 18, 2021
On the book jacket Bill Gates calls this book "accessible" which has proven once and for all that intellectually, accessible means two very different things to me & Bill Gates. That aside, it is a fun book if you enjoy semi-dense math reads, particularly applied math. If I dare to say so, Jordan the wickedly funny author, has managed to give me the guilty pleasure of watching something trashy like Jersey Shore through a book about geometry by dishing out colorful bits about revered people.
Did you know Einstein played violin on the street for extra cash? Or that Gauss was often only a few steps ahead of his debts? Or that Wordsworth (the poet) and Lincoln (the politician) were excellent mathematicians? Can you even imagine the last president read Euclid for fun!? Oh and my favorite bit -- Karl Pearson, the correlations guy, apparently looked like a Greek God. He also taught his class the law of large numbers by throwing 10,000 pennies on the floor and making students count the heads. I remember the dreary day I was taught that theorem. Yikes. Maybe this is how we should teach math!
Now that we've laughed at the gossip-y bits, the more serious topics covered include an examination of Euclid's geometry, encryption keys as bead bracelets, wall street stock movements as mosquito paths across bogs, AI as mountaineering, and finally gerrymandering. Some topics I was more lost than others, but some places -- boy was the read worth it. (Like finally truly intuitively understanding convex hull after 6years doing Machine Learning).
Couple of the quotes made me audibly laugh out loud at how true they were:
"Mathematicians have an imperial tendency - we often see other people's problems as consisting of a true mathematical core surrounded by an irritating amount of domain specific information"
"If my grandma had wheels, she'd be a wagon" -- comparing this to the hypothetical incorrect statistical practice of "if we only consider xyz then we find abc"
Some bits I'd like to remember:
- the distance a bond travels is about inverse of sqrt of time that has passed
- wrong answers are bad but wrong questions are worse
- a connected graph stays connected only if the connections at each point >=2, interesting to think what this says about friendships and networks
- we teach geometry not to prove the exterior angles of a polygon add to 360 but rather to learn deductive reasoning with which to discern non-proofs. Like young Lincoln breaking things into their first principles until they become simple enough for him to understand
- Markov chains were discovered because and atheist Markov wanted to disprove the free-will proof that his super religious peer Nekrasov had formulated (the proof hinged on the assumption that to have stable long term probability the agents in the system must be independent)
- the hilarious story of how the Electoral College came about as an exhausted compromise rather than the brilliant design we tout it to be
- through gerrymandering, understanding that the opposite of something can just be not-that-something (counterfactual), rather than an individual entity of its own. This is important when being asked "okay if not this then what? well, not-this"
- understanding through the sorites paradox (at how many grains of wheat do you get a pile), that even if you don't know when something bad starts, you can tell when it's VERY bad. Draw the line there, might be arbitrary but still useful
Finally a wonderful thought to leave with:
"complex computations can feel like blind groping until you discover the common mathematical understructure shared by the two, illuminating each in the light of the other"
Did you know Einstein played violin on the street for extra cash? Or that Gauss was often only a few steps ahead of his debts? Or that Wordsworth (the poet) and Lincoln (the politician) were excellent mathematicians? Can you even imagine the last president read Euclid for fun!? Oh and my favorite bit -- Karl Pearson, the correlations guy, apparently looked like a Greek God. He also taught his class the law of large numbers by throwing 10,000 pennies on the floor and making students count the heads. I remember the dreary day I was taught that theorem. Yikes. Maybe this is how we should teach math!
Now that we've laughed at the gossip-y bits, the more serious topics covered include an examination of Euclid's geometry, encryption keys as bead bracelets, wall street stock movements as mosquito paths across bogs, AI as mountaineering, and finally gerrymandering. Some topics I was more lost than others, but some places -- boy was the read worth it. (Like finally truly intuitively understanding convex hull after 6years doing Machine Learning).
Couple of the quotes made me audibly laugh out loud at how true they were:
"Mathematicians have an imperial tendency - we often see other people's problems as consisting of a true mathematical core surrounded by an irritating amount of domain specific information"
"If my grandma had wheels, she'd be a wagon" -- comparing this to the hypothetical incorrect statistical practice of "if we only consider xyz then we find abc"
Some bits I'd like to remember:
- the distance a bond travels is about inverse of sqrt of time that has passed
- wrong answers are bad but wrong questions are worse
- a connected graph stays connected only if the connections at each point >=2, interesting to think what this says about friendships and networks
- we teach geometry not to prove the exterior angles of a polygon add to 360 but rather to learn deductive reasoning with which to discern non-proofs. Like young Lincoln breaking things into their first principles until they become simple enough for him to understand
- Markov chains were discovered because and atheist Markov wanted to disprove the free-will proof that his super religious peer Nekrasov had formulated (the proof hinged on the assumption that to have stable long term probability the agents in the system must be independent)
- the hilarious story of how the Electoral College came about as an exhausted compromise rather than the brilliant design we tout it to be
- through gerrymandering, understanding that the opposite of something can just be not-that-something (counterfactual), rather than an individual entity of its own. This is important when being asked "okay if not this then what? well, not-this"
- understanding through the sorites paradox (at how many grains of wheat do you get a pile), that even if you don't know when something bad starts, you can tell when it's VERY bad. Draw the line there, might be arbitrary but still useful
Finally a wonderful thought to leave with:
"complex computations can feel like blind groping until you discover the common mathematical understructure shared by the two, illuminating each in the light of the other"
August 22, 2021
Mathematics is not my forte. It was a problem for me in school and for a long time, I was convinced that I just wasn't a math person. Of course, when I grew older, a lot older, some really smart mathematicians online tried to convince me that anyone can become good at math. Not excellent or exceptional, but good is possible. As a humanities major, however, I think I approach the whole problem the wrong way because instead of actually trying to do some maths (Who has time for that?), I prefer to read books about it. "Shape" is one such book.
The description reads:
"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."
Perfect, I thought. The first chapter was excellent, the second was also good, and after that I was lost. I was relieved to read that many other readers had the same problem. Jordan Ellenberg writes about some topics, such as gerrymandering in the USA, for example, in great detail, but only touches upon others. He also jumps from one topic to the next and it was difficult to follow him sometimes, especially since I have an audiobook. Nevertheless, I will be re-listening to this one, I think it deserves my full attention, and I will be reading his praised "How Not to Be Wrong: The Power of Mathematical Thinking" soon.
The description reads:
"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."
Perfect, I thought. The first chapter was excellent, the second was also good, and after that I was lost. I was relieved to read that many other readers had the same problem. Jordan Ellenberg writes about some topics, such as gerrymandering in the USA, for example, in great detail, but only touches upon others. He also jumps from one topic to the next and it was difficult to follow him sometimes, especially since I have an audiobook. Nevertheless, I will be re-listening to this one, I think it deserves my full attention, and I will be reading his praised "How Not to Be Wrong: The Power of Mathematical Thinking" soon.
March 27, 2022
Spoiler: this book is about math because everything is about math. And this is a former math-hater saying this.
I’d say 3.5 stars in truth. I’m not a person whose brain grasps and wrestles around mathematics easily, but because I’ve learned to make sense of enough mathematics to enjoy it I did enjoy this book. I’m a firm believer that math is more about experience and less about ability ( also conceding that some have a more natural tendency towards understanding than others in the same way that people all have strengths in various areas of life).
Confession: I glossed over some parts when my energy at the time wasn’t into full grasping of the topic in that section.
I was unable to stop collecting quotes from various sections of this book. I will share them because many people will not read a book on geometry but may read a review and I think these thoughts are worth sharing with more people. Stay with me here:
Mathematics is a fundamentally imaginative enterprise, which draws on every cognitive and creative ability we have. p. 110
He’s approaching the problem just the way a mathematician would – starting from the end of the game. That’s no surprise; we are all mathematicians in the deep strategic parts of our brain, whether it says that on our business cards or not. p. 115
If we played more games in math class, would students learn more math? Yes. Also, no. I’ve been teaching math for more than 20 years now. When I started, I was driven by questions like this: what’s the right way to teach a mathematical concept? Examples first, then explanation? Explanation followed by examples? Letting students discover principles by examining the examples I present, or stating principles on the blackboard and letting students discover examples? Wait, are blackboards even good? I’ve come to feel there’s no one right way. (Though there are certainly some wrong ways.) Different students are different and there is no one true teaching method that will ring everyone’s saliva bell. …..Math teachers, I think, ought to adopt every teaching strategy they can and shuttle through them in quick succession. That’s the way to maximize the chance each student at least sometimes feels that their teacher is finally, after so much boring hop-hah, talking about things in a way that makes sense. p. 122
THAT is stuff that ALL math teachers from preschool through grad school need to know. After 27 years of teaching K, 1st, 4th, and mostly 5th and 6th grade math (where you get to some good stuff like why did your parents say to invert and multiply???) I whole heartedly agree. Please stick with me for a few more:
p. 126 Here’s something that happens a lot in math. You sit down to solve one problem, and when you finish, the next day or month or year, you realize you’ve sold a lot more problems at the same time. When a nail requires you to invent a truly new kind of hammer, everything looks like a nail worth hitting with that hammer, and lots of things actually are.
p. 139 to Tinsley‘s way of thinking, even though he and Chinook were carrying out the same task, they were fundamentally different kinds of beings. “ I had a better programmer than Chinook,” he told the newspaper before the two met in the 1992 tournament. “His was Jonathan, mine was the Lord.”
p. 142-3 Kasparov says,” I was amazed by the beauty of this geometry.” The tree geometry tells you how to win; it doesn’t tell you what makes a game beautiful. That’s a subtler geometry, and for now it’s not one a machine can compute step-by-step with a short list of rules. Perfection isn’t beauty. We have absolute proof that perfect players will never win and never lose. whatever interest we can have in the game is there only because human beings are imperfect. And maybe that’s not bad. Perfect play isn’t play at all, not in the plain English sense of that word. To the extent we are personally present in our game playing, it’s by the virtue of art in perfectness. We feel some thing when our own in perfectness scrape up against the imperfections of another.
p. 145 This is one of the questions I hear most as a math teacher: how do I even start this? I’m always happy to hear it, no matter how stricken the student looks as they ask, because the question is an opportunity to teach a lesson. The lesson is that it matters much less how you start than THAT you start. Try something. It might not work. If it doesn’t, try something else. Students often grow up in a world where you solve a math problem by executing a fixed algorithm. You’re asked to multiply two 3 digit numbers and the first thing you do is multiply the first number by the last digit of the second number and you write that down and you’re off. Real math (like real life) is nothing like this. There’s a lot of trial and error. That method gets looked down on a lot, probably because it has the word “error” in it. In math we are not afraid of errors. Errors are great! An error is just an opportunity to run another trial.
p. 200-1 Our students are afraid to ask questions in class because they’re afraid of “looking stupid.” If we were honest about how difficult and deep mathematics is, even the mathematics that appears in a high school geometry classroom, this would surely be less of a problem; we could move toward a classroom we’re asking a question meant not “looking stupid” but “looking like someone who came here to learn something.” And this doesn’t just apply to students who find themselves struggling. Yes, some have no trouble picking up the basic rules of algebraic manipulation or geometric constructions. Those students should still be asking questions, of their teachers and of themselves. For example: I have done with the teacher asked, but what if I tried to do this other thing that the teacher didn’t ask of me, and, for that matter, why did the teacher ask for one thing and not the other? There’s no intellectual vantage from which you can easily sight a zone of ignorance, and that’s where your eyes should be pointed, if you want to learn. If math class is easy, you’re doing it wrong.
p. 205 An autonomous vehicle may be able to make the right choice 95% of the time, but that doesn’t mean it’s 95% of the way to making the right choice all the time; that last 5%, those outlier cases, might well be a problem our sloppy brains are better equipped to solve than any current or near future machine.
A REASON why AI will always have a tad of struggle. One more!! You can do it!!
p. 149 And the tang of contradiction fills are nostrils once again.
THESE are things people need to know, math isn’t right/wrong, math is thinking, discussing, contradicting, working and working again, questioning, and a continuous path of these for all learners regardless if you’re 6 or 60 years old. Had my 7th grade math teacher had a shred of this approach, I’d have enjoyed and learned way more math than I did before teacher math conferences got a hold of me in my 30s.
I’d say 3.5 stars in truth. I’m not a person whose brain grasps and wrestles around mathematics easily, but because I’ve learned to make sense of enough mathematics to enjoy it I did enjoy this book. I’m a firm believer that math is more about experience and less about ability ( also conceding that some have a more natural tendency towards understanding than others in the same way that people all have strengths in various areas of life).
Confession: I glossed over some parts when my energy at the time wasn’t into full grasping of the topic in that section.
I was unable to stop collecting quotes from various sections of this book. I will share them because many people will not read a book on geometry but may read a review and I think these thoughts are worth sharing with more people. Stay with me here:
Mathematics is a fundamentally imaginative enterprise, which draws on every cognitive and creative ability we have. p. 110
He’s approaching the problem just the way a mathematician would – starting from the end of the game. That’s no surprise; we are all mathematicians in the deep strategic parts of our brain, whether it says that on our business cards or not. p. 115
If we played more games in math class, would students learn more math? Yes. Also, no. I’ve been teaching math for more than 20 years now. When I started, I was driven by questions like this: what’s the right way to teach a mathematical concept? Examples first, then explanation? Explanation followed by examples? Letting students discover principles by examining the examples I present, or stating principles on the blackboard and letting students discover examples? Wait, are blackboards even good? I’ve come to feel there’s no one right way. (Though there are certainly some wrong ways.) Different students are different and there is no one true teaching method that will ring everyone’s saliva bell. …..Math teachers, I think, ought to adopt every teaching strategy they can and shuttle through them in quick succession. That’s the way to maximize the chance each student at least sometimes feels that their teacher is finally, after so much boring hop-hah, talking about things in a way that makes sense. p. 122
THAT is stuff that ALL math teachers from preschool through grad school need to know. After 27 years of teaching K, 1st, 4th, and mostly 5th and 6th grade math (where you get to some good stuff like why did your parents say to invert and multiply???) I whole heartedly agree. Please stick with me for a few more:
p. 126 Here’s something that happens a lot in math. You sit down to solve one problem, and when you finish, the next day or month or year, you realize you’ve sold a lot more problems at the same time. When a nail requires you to invent a truly new kind of hammer, everything looks like a nail worth hitting with that hammer, and lots of things actually are.
p. 139 to Tinsley‘s way of thinking, even though he and Chinook were carrying out the same task, they were fundamentally different kinds of beings. “ I had a better programmer than Chinook,” he told the newspaper before the two met in the 1992 tournament. “His was Jonathan, mine was the Lord.”
p. 142-3 Kasparov says,” I was amazed by the beauty of this geometry.” The tree geometry tells you how to win; it doesn’t tell you what makes a game beautiful. That’s a subtler geometry, and for now it’s not one a machine can compute step-by-step with a short list of rules. Perfection isn’t beauty. We have absolute proof that perfect players will never win and never lose. whatever interest we can have in the game is there only because human beings are imperfect. And maybe that’s not bad. Perfect play isn’t play at all, not in the plain English sense of that word. To the extent we are personally present in our game playing, it’s by the virtue of art in perfectness. We feel some thing when our own in perfectness scrape up against the imperfections of another.
p. 145 This is one of the questions I hear most as a math teacher: how do I even start this? I’m always happy to hear it, no matter how stricken the student looks as they ask, because the question is an opportunity to teach a lesson. The lesson is that it matters much less how you start than THAT you start. Try something. It might not work. If it doesn’t, try something else. Students often grow up in a world where you solve a math problem by executing a fixed algorithm. You’re asked to multiply two 3 digit numbers and the first thing you do is multiply the first number by the last digit of the second number and you write that down and you’re off. Real math (like real life) is nothing like this. There’s a lot of trial and error. That method gets looked down on a lot, probably because it has the word “error” in it. In math we are not afraid of errors. Errors are great! An error is just an opportunity to run another trial.
p. 200-1 Our students are afraid to ask questions in class because they’re afraid of “looking stupid.” If we were honest about how difficult and deep mathematics is, even the mathematics that appears in a high school geometry classroom, this would surely be less of a problem; we could move toward a classroom we’re asking a question meant not “looking stupid” but “looking like someone who came here to learn something.” And this doesn’t just apply to students who find themselves struggling. Yes, some have no trouble picking up the basic rules of algebraic manipulation or geometric constructions. Those students should still be asking questions, of their teachers and of themselves. For example: I have done with the teacher asked, but what if I tried to do this other thing that the teacher didn’t ask of me, and, for that matter, why did the teacher ask for one thing and not the other? There’s no intellectual vantage from which you can easily sight a zone of ignorance, and that’s where your eyes should be pointed, if you want to learn. If math class is easy, you’re doing it wrong.
p. 205 An autonomous vehicle may be able to make the right choice 95% of the time, but that doesn’t mean it’s 95% of the way to making the right choice all the time; that last 5%, those outlier cases, might well be a problem our sloppy brains are better equipped to solve than any current or near future machine.
A REASON why AI will always have a tad of struggle. One more!! You can do it!!
p. 149 And the tang of contradiction fills are nostrils once again.
THESE are things people need to know, math isn’t right/wrong, math is thinking, discussing, contradicting, working and working again, questioning, and a continuous path of these for all learners regardless if you’re 6 or 60 years old. Had my 7th grade math teacher had a shred of this approach, I’d have enjoyed and learned way more math than I did before teacher math conferences got a hold of me in my 30s.
July 4, 2021
Enjoyed! Strangely wanted more geometry (he defines geometry loosely) but got a bunch of teaching ideas
Read
September 20, 2021I didn’t get far at all. There’s a reason I was an English major.
August 29, 2021
Good meandering explanation of geometry in our everyday lives. (Geometry being a much broader slice of math than I realized. ) I found the first few chapters a little too all-over-the-place but after a while I started enjoying the style and by the end of the book I was really appreciating it. The penultimate chapter about congressional districting and gerrymandering was great - he shows some of the math aspects but also really explores the political and philosophical aspects as well - defining districts is a lot more interesting of a problem then I had realized.
There were a few pages here and there that I just skimmed. It wasn’t that he was doing complex math, he was just going over something relatively simple in a step-by-step way and it felt tedious so I skipped some of those. But that was just maybe a dozen or two pages out of 400.
There were a few pages here and there that I just skimmed. It wasn’t that he was doing complex math, he was just going over something relatively simple in a step-by-step way and it felt tedious so I skipped some of those. But that was just maybe a dozen or two pages out of 400.
October 15, 2023
I’d rate it a solid 4.5. The author hits the peak in the chapter on gerrymandering, weaving together all the meticulously crafted concepts from earlier sections to unveil a mathematical perspective on the insidious nature of this political maneuver. The book is peppered with biting sarcasm and wit, at times channeling the vibe of a self-narrated Larry David episode.
(Thanks chatgpt for the extra “frills”)
(Thanks chatgpt for the extra “frills”)
August 19, 2024
Genuinely funny math dad jokes, but I think a clearer overarching structure would’ve helped me file away the information better. There wasn’t a narrative thrust—the chapters felt like the random walk some of them described! Also think the COVID-focused chapters already feel dated in a way the rest of the material doesn’t.
August 7, 2021
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.
September 27, 2023
Dense read but charming and funny throughout.
June 3, 2021
This hit my sweet spot for popular science books. High school geometry was the last math course I understood and I don't remember much from that.
Ellenberg starts from basic Euclid stuff and traces it to artificial intelligence, Gerrymandering, Covid-19 plus games, magic tricks and logic problems. I was able to follow him down most of the rabbit holes. A few times I couldn't follow him all the way. That is perfect. I want this kind of book to push me right to the edge of what I can understand.
Ellenberg likes to start by taking simple models of simple problems, what move to make in a tic-tac-toe game? or, how many holes are there in a pair of pants?. Next he explains a way to solve the problem. He then shows how that explanation can be used to solve big problems like how to be the world's best chess computer or how to solve the hardest topological problems.
He has a relaxed writing style. He mocks his own nerdiness and likes to drop in wisecracks but, at the same time, this is a serious and successful attempt to explain some very sophisticated applications of geometry to neural networks, cryptography, epidemiology, planetary physics and a bunch more.
Ellenberg throws in big chunks of history, mini-biographies and puzzles as he goes along.
I would have been helped by a discussion of what geometry is. At several points I was interested in what he was explaining, but I didn't really understand why it was part of geometry.
Ellenberg starts from basic Euclid stuff and traces it to artificial intelligence, Gerrymandering, Covid-19 plus games, magic tricks and logic problems. I was able to follow him down most of the rabbit holes. A few times I couldn't follow him all the way. That is perfect. I want this kind of book to push me right to the edge of what I can understand.
Ellenberg likes to start by taking simple models of simple problems, what move to make in a tic-tac-toe game? or, how many holes are there in a pair of pants?. Next he explains a way to solve the problem. He then shows how that explanation can be used to solve big problems like how to be the world's best chess computer or how to solve the hardest topological problems.
He has a relaxed writing style. He mocks his own nerdiness and likes to drop in wisecracks but, at the same time, this is a serious and successful attempt to explain some very sophisticated applications of geometry to neural networks, cryptography, epidemiology, planetary physics and a bunch more.
Ellenberg throws in big chunks of history, mini-biographies and puzzles as he goes along.
I would have been helped by a discussion of what geometry is. At several points I was interested in what he was explaining, but I didn't really understand why it was part of geometry.
April 5, 2022
This is a great book, with Ellenberg's characteristic engaging writing and ability to make technical mathematical ideas accessible. The chapter on gerrymandering is clearly the best part of this book (and at over 70 pages, a LARGE part of it!) Ellenberg is clearly passionate about the problem of gerrymandering as a resident of Wisconsin, where the state legislature has been so gerrymandered that no plausible majority of Democratic votes could possibly result in a majority of Democratic legislators. (A Republican wouldn't be as concerned about this as Ellenberg clearly is, though anyone can appreciate that the situation offends his sense of fairness.) The demonstration of the magnitude of the problem, and Ellenberg's evident disgust at the federal Supreme Court's refusal to act on this are clearly emotionally charged for him, and as a result the chapter is quite engaging.
There are a few chapters dealing with the COVID pandemic (which was unfolding as he wrote it), and epidemiology more generally, as well. All very interesting to read from inside the pandemic, though I wonder if their topic will retain its interest as the pandemic recedes. (Maybe I should say _if_ it ever recedes!)
The book has occasional oblique references Ellenberg's first popular book on mathematics, _How Not To Be Wrong_. While reading it first is not essential, I'd advise it. Partly this is because the ideas in Shape are generally a little more difficult to grapple with, though the range of difficulties of the two books overlap quite a bit. Maybe it is just nostalgia, but I recall HNTBW hanging together just a little better than this book. The transitions between chapters and the structure of multi-chapter parts of the book felt better thought-out and designed. That is, Ellenberg established an amazingly high bar with his first popular offering and this one doesn't quite meet it. Still people wanting to see how math appears in their lives or discover how contemporary mathematicians (or at least some of them) view their discipline, this is an enjoyable read.
There are a few chapters dealing with the COVID pandemic (which was unfolding as he wrote it), and epidemiology more generally, as well. All very interesting to read from inside the pandemic, though I wonder if their topic will retain its interest as the pandemic recedes. (Maybe I should say _if_ it ever recedes!)
The book has occasional oblique references Ellenberg's first popular book on mathematics, _How Not To Be Wrong_. While reading it first is not essential, I'd advise it. Partly this is because the ideas in Shape are generally a little more difficult to grapple with, though the range of difficulties of the two books overlap quite a bit. Maybe it is just nostalgia, but I recall HNTBW hanging together just a little better than this book. The transitions between chapters and the structure of multi-chapter parts of the book felt better thought-out and designed. That is, Ellenberg established an amazingly high bar with his first popular offering and this one doesn't quite meet it. Still people wanting to see how math appears in their lives or discover how contemporary mathematicians (or at least some of them) view their discipline, this is an enjoyable read.
March 27, 2021
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.
February 2, 2022
This is a difficult book to rate, but is in the 3-4 range for me. The issue is that of the difficulty of the material itself. I don't believe the book makes claims of being accessible to non-mathematicians, and it certainly reads well in a general sense, but the material covers so many aspects of geometry that it is very difficult to stay with it. Ellenberg's other book that I have read, How Not To Be Wrong, is much more accessible overall.
The tail end of the book addresses some of the risks that we face as a democracy, and some of the issues surrounding some of the purported solutions. Ellenberg does it gently, but he effectively skewers most of the Supreme Court justices for being unable or unwilling to tackle the problem of rampant corruption in the way that our legislatures are selecting their voters in naked attempts to garner and keep power. It appears that they (the justices) preferred to kick the can to someone else by deliberately misunderstanding or mis-stating the nature of the problem before them.
I liked a lot of the material covered in the book, and the book itself may deserve a re-read sometime in the future.
The tail end of the book addresses some of the risks that we face as a democracy, and some of the issues surrounding some of the purported solutions. Ellenberg does it gently, but he effectively skewers most of the Supreme Court justices for being unable or unwilling to tackle the problem of rampant corruption in the way that our legislatures are selecting their voters in naked attempts to garner and keep power. It appears that they (the justices) preferred to kick the can to someone else by deliberately misunderstanding or mis-stating the nature of the problem before them.
I liked a lot of the material covered in the book, and the book itself may deserve a re-read sometime in the future.
December 7, 2021
I wanted to love this book. Ellenberg has a knack for writing, and there are many humorous turns of phrase and edifying sentences in this book.
But there are many, MANY sentences in this book. Mixed in with the geometry are discussions of history and art and pretty much every other science. It's a style designed to make math palatable to the disinclined, and to show how geometric ideas are woven into the fabric of society. Unfortunately, it's also a style that allows for a lot of rambling, and a more aggressive editorial pen (perhaps an editorial scythe?) was needed to shape the long-winded chapters about subjects ranging from game theory to random walks to political science.
I enjoyed this book. I kept picking it up, again and again, until I got to the end of it. But I would have enjoyed it a lot more if there had been less of it, and that's not something I usually say about leisure reading.
But there are many, MANY sentences in this book. Mixed in with the geometry are discussions of history and art and pretty much every other science. It's a style designed to make math palatable to the disinclined, and to show how geometric ideas are woven into the fabric of society. Unfortunately, it's also a style that allows for a lot of rambling, and a more aggressive editorial pen (perhaps an editorial scythe?) was needed to shape the long-winded chapters about subjects ranging from game theory to random walks to political science.
I enjoyed this book. I kept picking it up, again and again, until I got to the end of it. But I would have enjoyed it a lot more if there had been less of it, and that's not something I usually say about leisure reading.
March 23, 2022
I strongly dislike the structure of this book. The author considers himself a "fun professor" and meanders from topic to topic. This panders to those with short attention spans. And bothers me.
A meaningful potential thesis of the book is the mathematics of gerrymandering. It seems to me the author was afraid this is too boring a topic. Which it's not. There are onion layers to the problem of determining if a district map has been gerrymandered. It's really interesting!
I recommend you skip to chapter 14 titled "How Math Broke Democracy (and Might Still Save It)".
It's the only chapter worth reading.
A meaningful potential thesis of the book is the mathematics of gerrymandering. It seems to me the author was afraid this is too boring a topic. Which it's not. There are onion layers to the problem of determining if a district map has been gerrymandered. It's really interesting!
I recommend you skip to chapter 14 titled "How Math Broke Democracy (and Might Still Save It)".
It's the only chapter worth reading.
June 22, 2021
This is an American political rant disguised as a maths book. It starts out fine, but the latter half of the book is the author's explanation (albeit mathematical) of what's wrong with the American political system.
April 11, 2023
This was the most fun math book I have read! My new favorite! I love math so much!
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