What will I learn about: how studying geometry can improve your rhetorical and deductive reasoning skills, a method famously used by Abraham Lincoln to construct meticulous arguments?

Discover how studying geometry can improve your rhetorical and deductive reasoning skills, a method famously used by Abraham Lincoln to construct meticulous arguments.

What will I learn about: that geometry exists anywhere there is a notion of distance, allowing you to measure the relational space between everything from alphabetical lists to Hollywood actors?

Understand that geometry exists anywhere there is a notion of distance, allowing you to measure the relational space between everything from alphabetical lists to Hollywood actors.

What will I learn about: Learn to solve paradoxical questions by modifying your initial intuition with mathematical logic, revealing why topology dictates that a straw has exactly one hole?

Learn to solve paradoxical questions by modifying your initial intuition with mathematical logic, revealing why topology dictates that a straw has exactly one hole.

What will I learn about: Explore the geometric principle of the 'random walk' to understand spatial paths, which explains why a randomly moving mosquito is statistically most likely to die exactly where it was born?

Explore the geometric principle of the 'random walk' to understand spatial paths, which explains why a randomly moving mosquito is statistically most likely to die exactly where it was born.

What will I learn about: how the same geometric theories used to track insect flight paths can be applied to finance, helping traders predict bond prices and stock market behavior?

Find out how the same geometric theories used to track insect flight paths can be applied to finance, helping traders predict bond prices and stock market behavior.

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Shape

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

Jordan Ellenberg

★ 3.9 / 5(77 ratings)

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Science Politics Philosophy Technology & the Future

If You're Curious About These Questions...

You should listen to this audiobook

💡Did you know that the secret to a fair democracy might actually be hidden in the geometry of voting districts—and that its misuse can manipulate an entire election?
💡Have you ever wondered why some viruses spread like wildfire while others fizzle out, and how the hidden 'shape' of our social networks holds the answer?
💡What if you could master the complex logic behind artificial intelligence and modern technology by simply changing the way you look at basic geometric shapes?

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Key Takeaways from Shape

✓Discover how studying geometry can improve your rhetorical and deductive reasoning skills, a method famously used by Abraham Lincoln to construct meticulous arguments.
✓Understand that geometry exists anywhere there is a notion of distance, allowing you to measure the relational space between everything from alphabetical lists to Hollywood actors.
✓Learn to solve paradoxical questions by modifying your initial intuition with mathematical logic, revealing why topology dictates that a straw has exactly one hole.
✓Explore the geometric principle of the 'random walk' to understand spatial paths, which explains why a randomly moving mosquito is statistically most likely to die exactly where it was born.
✓Find out how the same geometric theories used to track insect flight paths can be applied to finance, helping traders predict bond prices and stock market behavior.

Learning Tools

Reinforce what you learned from Shape

Mind Map

Shape

Redefining Geometry+

Intuition and Logic+

Probability and Space+

Modeling Spread+

The Geometry of Games+

Machine Learning+

Political Geometry+

Quiz — Test Your Understanding

Question 1 of 8

According to the book, what is the fundamental requirement for a space or set to possess a geometry?

A. It must involve physical shapes like triangles, circles, or squares.
B. There must be a defined metric to signify the distance between any two points.
C. It must be measurable using a straight-line 'crow-fly' metric.
D. The points must represent geographic locations on a physical map.

Question 2 of 8

How does the geometric field of topology answer the internet debate about how many holes a straw has?

A. A straw has two holes because there is an opening at each end.
B. A straw has zero holes because it is essentially a rolled-up rectangle.
C. A straw has one hole because shortening it creates an annulus, a shape with a single hole.
D. A straw has negative one holes because poking it creates a zero-hole surface.

Question 3 of 8

What did French mathematician Louis Bachelier conclude by applying the 'random walk' principle to the stock market?

A. A stock option's price will inevitably grow exponentially over time.
B. Traders are guaranteed to lose money if they hold bonds for too long.
C. The market price of a bond is most likely to end up exactly where it started.
D. Stock prices follow a predictable Markov chain based on previous days' performance.

Question 4 of 8

How does a Markov chain differ from a simple random walk?

A. A Markov chain relies on decisions that are completely independent of past events.
B. In a Markov chain, the next step or decision is dependent on the current state.
C. Markov chains can only be applied to abstract spaces like language, not physical spaces.
D. A Markov chain always results in a geometric progression.

Question 5 of 8

In the context of geometric progressions and pandemics, what does an R0 (R-naught) value of less than one indicate?

A. The disease is spreading exponentially.
B. The pandemic is undergoing exponential decay and dying away.
C. The disease has mutated into a new variant.
D. The number of new infections is exactly equal to the number of recoveries.

Question 6 of 8

Which of the following is a required parameter for a game to be geometrically considered a 'tree'?

A. The game must involve an element of random chance, like dice rolls.
B. The game must allow for an infinite number of possible moves.
C. The game must be played by more than two players.
D. The game's outcome must not be determined by random chance.

Question 7 of 8

How does the concept of 'gradient descent' apply to machine learning?

A. It helps a computer continuously tweak its strategy to minimize its 'wrongness score.'
B. It allows computers to sort images alphabetically based on pixel density.
C. It prevents a computer from making any errors during its initial learning phase.
D. It uses Markov chains to predict the exact pixels of an upcoming image.

Question 8 of 8

How can mathematicians use computer-generated maps (ensembles) to prove that a voting district map has been gerrymandered?

A. By creating a single, perfectly fair map and forcing politicians to adopt it legally.
B. By calculating the exact number of 'wasted votes' and ensuring it is zero for both parties.
C. By generating thousands of random, legal maps and showing that the actual map's political outcome is an extreme statistical outlier.
D. By drawing maps that rely exclusively on straight lines and perfect geometric shapes like squares.

Shape — Full Chapter Overview

1Recommendation
2Wherever there’s a sense of distance, geometry is present.
3Geometry invites us to begin with intuition and refine it using logic.
4The idea of the random walk lets us make sense of journeys through space.
5Markov chains appear in many different kinds of spaces.
6Pandemics grow through geometric progressions, with complicating factors.
7Some games have geometric structures that let us foresee their results.
8Machines rely on gradient descent to improve by being less wrong.
9Politicians have used mathematics to secure political advantage.
10Computer-created maps can reveal when gerrymandering has taken place.

Shape Summary & Overview

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.

Who Should Listen to 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

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.