<p>Who is the author of Shape?</p>

<p>The author of Shape is Jordan Ellenberg.</p>

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Shape Book Summary - Shape Book explained in key points

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Shape summary

Name: Shape
Rating: 3.9 (77 reviews)
Author: Jordan Ellenberg

Jordan Ellenberg

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

3.9 (77 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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Shape
Summary of 9 key ideas

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Key idea 1 of 9

What’s in it for me? A riveting exploration of how geometry underpins practically everything.

For many people, the word geometry conjures memories of a torturous snooze-fest in a high school math class.

But geometry is much more than just calculating the hypotenuse of a triangle or writing proofs line by line. For instance: Did you know that American president Abraham Lincoln attributed his famed rhetorical skills to geometry? By studying the subject, Lincoln learned how to construct an argument meticulously based on careful deductive reasoning.

That’s only the tip of the iceberg, because geometry can be found just about everywhere. There’s geometry hidden in games like checkers and chess. There’s geometry that determines the flight paths of mosquitoes. And there’s even geometry underpinning American democracy! These blinks will cover all of this⁠ and more.

In these blinks, you’ll learn

how many holes a straw has;
why a mosquito will probably die in the same place where it was born; and
how Russian literature can be boiled down to percentages.

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?

Geometry asks us to start with an intuition and then modify it with logic.

If you’ve spent any time on the internet, you may have stumbled onto an argument over the question How many holes does a straw have?

You’re probably already leaning toward an answer that feels intuitive. It’s obvious a straw has two holes –⁠ just look at it! But wait –⁠ where does one hole start and the other end? Maybe that means a straw has just one hole that goes all the way through. Or does it have zero holes, since it’s just a rolled-up rectangle and rectangles don’t have holes?

The principle of the random walk helps us understand paths through space.

Picture a diagram of five concentric circles. A mosquito is born on the outermost ring. What’s the probability of finding the mosquito’s dead body, at the end of its life, anywhere else on the diagram?

This is the question that Sir Ronald Ross aimed to answer at the beginning of the twentieth century. Ross’s study of a mosquito’s flight path gave birth to a geometric theory that eventually permeated all sorts of other fields, from physics to finance and even to poetry. That theory is called the random walk.

Markov chains exist in all kinds of different spaces.

Sir Ronald Ross wasn’t the only early twentieth-century mathematician spending time thinking about mosquitoes. Halfway across the world, a Russian named Andrei Markov was doing the same.

Markov wasn’t thinking about mosquitoes whose every decision was random. Instead, he considered a mosquito whose decision on where to fly was dependent on its decision the previous day.

Pandemics spread as geometric progressions, with some added wrinkles.

Let’s go back to Sir Ronald Ross. Ross was the person who aimed to describe the probability of a mosquito ending up a particular distance from wherever it was born.

We’ve left out part of the story, because in reality, Ross’s goals went much further. Ultimately, he wanted to quantify disease itself – and, furthermore, to quantify the spread of just about anything, from religions to military enlistments. In the course of that effort, he, along with a mathematician named Hilda Hudson, determined that pandemics spread in a sequence called a geometric progression, sometimes called exponential growth.

Games have geometry that, in some cases, allows us to predict their outcomes.

What do games like checkers, Connect Four, and chess have in common? Well, geometrically speaking, they’re all considered trees.

Games are considered trees when they observe a few parameters. Those are: they involve two players taking turns; the outcomes aren’t determined by random chance –⁠ that means no coin flips, spinners, or card draws;⁠ and every game ends after a finite number of steps.

Machines use gradient descent to learn how to be less wrong.

Picture yourself as a mountaineer trying to reach a high summit. You lost your map at some point along the way. Now how do you get to the top?

One way might be to look down at your feet and assess the slope. When you step northward, you see that the grade goes up a little. As you inch toward the south, the grade goes down. Then, you try turning northeast, and the grade suddenly goes up by a lot. Having identified the direction with the steepest upward slope, you walk a few steps northeast and then repeat the process over and over until you reach the top.

Politicians have exploited math for political gain.

In November 2018, Democrats in the US state of Wisconsin were celebrating. Their gubernatorial candidate had won out over the Republican incumbent, they’d taken over the attorney general and state treasurer positions, and a Democratic senator had won reelection.

But it wasn’t quite a victory across the board. They’d picked up just one seat in the Wisconsin State Assembly –⁠ the lower house of the state legislature –⁠ allowing the Republicans to keep their 63–⁠36 majority. Meanwhile, the Republicans actually gained a seat in the state senate.

Computer-generated maps can help expose gerrymandering.

What does a gerrymandered district map look like? Sometimes, it’s pretty obvious.

Take, for instance, Pennsylvania’s Seventh District, which looked like a mangled depiction of the cartoon character Goofy kicking Donald Duck. The district was so obviously drawn to capture specific voters from specific regions that the courts threw it out in 2018.

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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.

Categories with Shape

Science Politics Philosophy Technology & the Future

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