Shape Book Summary - Shape Book explained in key points
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Shape summary

Jordan Ellenberg

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

3.9 (76 ratings)
26 mins

Brief summary

Shape by Jordan Ellenberg is a book that challenges the way we see mathematics. Through examples in sports, politics, and everyday life, Shape shows how math shapes the world around us and how we can better understand it.

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    Summary of 9 key ideas

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    Key idea 1 of 9

    Anywhere there’s a notion of distance, there’s geometry.

    In Greek, the word “geometry” means “measuring the Earth.” When we examine the geometry of something, that’s precisely what we’re doing –⁠ ⁠although not always literally.

    A patch of land, a set of people, a team of horses –⁠ we can assign a geometry to each of them. All we have to do is define our metric –⁠ that is, the number we’re using to signify the distance between any two points. Whatever metric we choose results in different geometry.

    For instance, we might look at two points on a map through the “crow-fly” metric, in which distance is the length of a straight line connecting those points. Or your metric could be the distance between two places on an alphabetical list of all US cities. By this metric, Los Angeles is closer to New York than it is to San Francisco.

    The key message here is: Anywhere there’s a notion of distance, there’s geometry.

    It’s not just places that can have a geometry. People can, too –⁠ say, actors. To study the geometry of actors, we just need a metric to measure how “far apart” they are.

    In fact, there’s already a metric for this called the costar distance. Two actors form a link when they appear in a movie together. The distance between any two actors is the smallest number of links that joins them. 

    So, for instance, the distance between George Reeves and Keanu Reeves is 2. How? Well, George appeared in From Here to Eternity with Jack Warden, who was in The Replacements with Keanu. 

    What about the geometry of a deck of cards? It’s just like the geometry of movie stars, except on a much bigger scale. The “points” are all the different ways a standard fifty-two card deck can be arranged –⁠ which is a huge, sixty-seven-digit number written as 52!, or 52 factorial.

    Now we’ve defined the points in our deck of cards. Next, we need a way to think about the distance between them. That’s where shuffling comes in, specifically a riffle shuffle, where you cut the deck into two stacks of any size and then recombine them. If you can riffle shuffle the deck once to get to another, specific ordering, those two orderings are linked. The distance between them is the number of riffle shuffles it takes to get from one ordering to the other. 

    Cities, actors, decks of cards –⁠ what else do you think has a geometry?

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    What is Shape about?

    Shape (2021) is a love letter to geometry, addressed to those who have –⁠ or thought they had –⁠ sworn off math forever. Accessible and fascinating, it shows how geometry underpins not just objects in the physical world, but also things like games, pandemics, artificial intelligence, and even American democracy. When we understand geometry, we understand a bit more about almost everything.

    Shape Review

    Shape (2021) by Jordan Ellenberg elegantly explores the math behind everyday life, revealing how understanding numbers can enrich our understanding of the world. Here's why this book is worth reading:

    • Delving into practical applications of mathematics, it provides insights into topics like politics, health, and sports, showing how math shapes our decisions.
    • Through the engaging exploration of real-life stories and examples, the book makes complex concepts accessible and relevant, captivating readers from start to finish.
    • With its fascinating blend of history, science, and sociology, this book presents a fresh perspective on numbers, proving that math can be both informative and entertaining.

    Who should read Shape?

    • Math haters convinced that geometry is boring
    • Math lovers interested in how geometry intersects with other subjects
    • The perennially curious

    About the Author

    Jordan Ellenberg was a child prodigy and is currently a professor of mathematics at the University of Wisconsin-Madison. He is also an author and blogger responsible for writing the “Do the Math” column for Slate; his own blog, Quomodocumque; and the best-selling book How Not to Be Wrong.

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    Shape FAQs 

    What is the main message of Shape?

    Shape explores how math can be the key to understanding the world around us.

    How long does it take to read Shape?

    The reading time for Shape varies, but the Blinkist summary can be read in just 15 minutes.

    Is Shape a good book? Is it worth reading?

    Shape is worth reading as it uncovers the fascinating role math plays in shaping our lives.

    Who is the author of Shape?

    The author of Shape is Jordan Ellenberg.

    What to read after Shape?

    If you're wondering what to read next after Shape, here are some recommendations we suggest:
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    • Inflation Matters by Pete Comley
    • The Signal and the Noise by Nate Silver
    • Sapiens by Yuval Noah Harari
    • A Brief History of Time by Stephen Hawking
    • Entangled Life by Merlin Sheldrake
    • The Magic of Reality by Richard Dawkins
    • Dopamine Detox by Thibaut Meurisse