#### Episode

###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Description

Jordan Ellenberg is a mathematician and author of Shape and How Not to Be Wrong. Please support this podcast by checking out our sponsors: - Secret Sauce: https://wondery.com/shows/secret-sauce/ - ExpressVPN: https://expressvpn.com/lexpod and use code LexPod to get 3 months free - Blinkist: https://blinkist.com/lex and use code LEX to get 25% off premium - Indeed: https://indeed.com/lex to get $75 credit EPISODE LINKS: Jordan's Website: http://www.jordanellenberg.com Jordan's Twitter: https://twitter.com/JSEllenberg PODCAST INFO: Podcast website: https://lexfridman.com/podcast Apple Podcasts: https://apple.co/2lwqZIr Spotify: https://spoti.fi/2nEwCF8 RSS: https://lexfridman.com/feed/podcast/ YouTube Full Episodes: https://youtube.com/lexfridman YouTube Clips: https://youtube.com/lexclips SUPPORT & CONNECT: - Check out the sponsors above, it's the best way to support this podcast - Support on Patreon: https://www.patreon.com/lexfridman - Twitter: https://twitter.com/lexfridman - Instagram: https://www.instagram.com/lexfridman - LinkedIn: https://www.linkedin.com/in/lexfridman - Facebook: https://www.facebook.com/lexfridman - Medium: https://medium.com/@lexfridman OUTLINE: Here's the timestamps for the episode. On some podcast players you should be able to click the timestamp to jump to that time. (00:00) - Introduction (06:44) - Mathematical thinking (10:21) - Geometry (14:58) - Symmetry (25:29) - Math and science in the Soviet Union (33:09) - Topology (47:57) - Do we live in many more than 4 dimensions? (52:28) - How many holes does a straw have (1:01:53) - 3Blue1Brown (1:07:40) - Will AI ever win a Fields Medal? (1:16:05) - Fermat's last theorem (1:33:23) - Reality cannot be explained simply (1:39:08) - Prime numbers (2:00:37) - John Conway's Game of Life (2:12:29) - Group theory (2:15:45) - Gauge theory (2:23:47) - Grigori Perelman and the Poincare Conjecture (2:33:59) - How to learn math (2:41:08) - Advice for young people (2:43:13) - Meaning of life

###### Chapters

###### In this podcast episode, the guest explains the importance of transparency and control in data usage and suggests using Blinkist to decide if a book is worth reading before summarizing it.

###### 00:00 - 06:24 (06:24)

###### Summary

In this podcast episode, the guest explains the importance of transparency and control in data usage and suggests using Blinkist to decide if a book is worth reading before summarizing it. The conversation also features Jordan Ellenberg, a mathematician and author of "How Not to Be Wrong" and "Shape".

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The speaker shares thoughts on whether or not visual thinking, in the form of diagrams and pictures, counts as a language, using a six by eight array of holes as an example.

###### 06:31 - 13:15 (06:44)

###### Summary

The speaker shares thoughts on whether or not visual thinking, in the form of diagrams and pictures, counts as a language, using a six by eight array of holes as an example.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### Learn about the different types of symmetry and how they can be used to transform images, spaces, or objects in mathematics.

###### 13:15 - 17:23 (04:07)

###### Summary

Learn about the different types of symmetry and how they can be used to transform images, spaces, or objects in mathematics.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The host dives deep into mathematics with two mathematicians, discussing the brain's ability to construct novel ideas, contemporary mathematical ideas, and the understanding of numbers.

###### 17:23 - 22:17 (04:53)

###### Summary

The host dives deep into mathematics with two mathematicians, discussing the brain's ability to construct novel ideas, contemporary mathematical ideas, and the understanding of numbers.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The study of modern geometry includes insights on studying complex problems like the three body problem and the importance of being methodical, like the Germans.

###### 22:17 - 25:47 (03:30)

###### Summary

The study of modern geometry includes insights on studying complex problems like the three body problem and the importance of being methodical, like the Germans.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### This episode delves into the brief yet significant life of French mathematician Évariste Galois, who developed the theory of groups and symmetries before dying in a duel at the age of 20.

###### 25:47 - 32:02 (06:15)

###### Summary

This episode delves into the brief yet significant life of French mathematician Évariste Galois, who developed the theory of groups and symmetries before dying in a duel at the age of 20.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The need for a flexible geometry that could talk about multi-dimensional spaces led to the development of 'differential geometry,' which describes mathematically complex curved shapes that are impossible to represent on a flat surface.

###### 32:02 - 39:15 (07:12)

###### Summary

The need for a flexible geometry that could talk about multi-dimensional spaces led to the development of 'differential geometry,' which describes mathematically complex curved shapes that are impossible to represent on a flat surface.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The speaker poses the question of whether it is possible to determine whether or not you are living on a Mobius strip and discusses the features of a strip that lead to the inability to distinguish between living inside of a cylinder, on the surface of a cylinder, or on the surface of a Mobius strip.

###### 39:15 - 45:52 (06:37)

###### Summary

The speaker poses the question of whether it is possible to determine whether or not you are living on a Mobius strip and discusses the features of a strip that lead to the inability to distinguish between living inside of a cylinder, on the surface of a cylinder, or on the surface of a Mobius strip.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The hosts discuss what a four-dimensional version of themselves would be like by following a sterile yet philosophical argument between a square and a sphere.

###### 45:52 - 52:36 (06:43)

###### Summary

The hosts discuss what a four-dimensional version of themselves would be like by following a sterile yet philosophical argument between a square and a sphere. They question what it would be like to be pulled out of the earthly world to see the truth from a higher perspective.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The debate about whether a straw has one or two holes is similar to the debate about high dimensional space, where individuals struggle to define where one hole ends and the other begins.

###### 52:36 - 56:34 (03:58)

###### Summary

The debate about whether a straw has one or two holes is similar to the debate about high dimensional space, where individuals struggle to define where one hole ends and the other begins.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### In this episode, Rabih Rizk discusses the equation of pants, where the amount of milkshake in each leg of the pants is equal to the amount coming out of the waist, and how the waist is just the total of the two leg holes stuck together.

###### 56:34 - 59:44 (03:09)

###### Summary

In this episode, Rabih Rizk discusses the equation of pants, where the amount of milkshake in each leg of the pants is equal to the amount coming out of the waist, and how the waist is just the total of the two leg holes stuck together.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The conversation delves into specialized skills like writing and creating eye-catching videos.

###### 59:44 - 1:05:07 (05:23)

###### Summary

The conversation delves into specialized skills like writing and creating eye-catching videos. It also suggests that it is possible to teach math through heavy metal music.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The mathematics community is recognizing the importance of visualization and its potential to create new knowledge in the field.

###### 1:05:07 - 1:09:40 (04:33)

###### Summary

The mathematics community is recognizing the importance of visualization and its potential to create new knowledge in the field. Institutes like the Institute for Computational Experimental Mathematics at Brown are embracing this concept as part of their work.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### An AI developed by researchers at the University of Liverpool has discovered a solution to the Conway knot problem, which had remained vexed to human mathematicians.

###### 1:09:40 - 1:12:51 (03:10)

###### Summary

An AI developed by researchers at the University of Liverpool has discovered a solution to the Conway knot problem, which had remained vexed to human mathematicians. The achievement highlights the potential for machine learning to aid and enhance mathematical and scientific research.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The beauty of mathematics lies in the human struggle of perseverance and the drama of not knowing if the proof is correct.

###### 1:12:51 - 1:18:31 (05:40)

###### Summary

The beauty of mathematics lies in the human struggle of perseverance and the drama of not knowing if the proof is correct. The story of Andrew Wiles' proof of the Fermat's Last Theorem is a perfect example of this.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The Fermat problem was a major breakthrough in number theory, resulting in the development of incredible richness in the field.

###### 1:18:32 - 1:22:59 (04:26)

###### Summary

The Fermat problem was a major breakthrough in number theory, resulting in the development of incredible richness in the field. Clever methods have been used to solve equations like this, including Andrew Wiles and Richard Taylor's proof of the theorem.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The two-adic number theory considers two numbers to be close if their difference is a multiple of a large power of two.

###### 1:23:00 - 1:27:32 (04:31)

###### Summary

The two-adic number theory considers two numbers to be close if their difference is a multiple of a large power of two. This idea is different from the conventional understanding of numbers being close.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The potential insights about the source of violence, entrepreneurial success, invention or economic success, communism or capitalism can be gained by thinking outside the box about totally new distance functions that could unlock profound space-related discoveries.

###### 1:27:32 - 1:33:18 (05:46)

###### Summary

The potential insights about the source of violence, entrepreneurial success, invention or economic success, communism or capitalism can be gained by thinking outside the box about totally new distance functions that could unlock profound space-related discoveries.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The validity of the quote attributed to physicist Richard Feynman regarding the connection between mathematics and classic rock is disputed.

###### 1:33:19 - 1:39:03 (05:44)

###### Summary

The validity of the quote attributed to physicist Richard Feynman regarding the connection between mathematics and classic rock is disputed. However, the idea of personal affinity and social networks influencing the perception of distance is relevant to the discussion.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The host discusses prime numbers and the significance of their discovery of the beauty of bounding gaps for the future of math.

###### 1:39:03 - 1:42:01 (02:58)

###### Summary

The host discusses prime numbers and the significance of their discovery of the beauty of bounding gaps for the future of math.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The Fermat primality test is a reliable way to determine whether a number is composite.

###### 1:42:02 - 1:45:25 (03:23)

###### Summary

The Fermat primality test is a reliable way to determine whether a number is composite. Although it's not perfect, it can help identify potential prime offenders such as 57 and 51.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### This podcast talks about the concept of twin primes and the question of whether or not they are infinite, exploring the history of prime numbers and their distribution.

###### 1:45:25 - 1:48:52 (03:27)

###### Summary

This podcast talks about the concept of twin primes and the question of whether or not they are infinite, exploring the history of prime numbers and their distribution.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The Twin Prime Conjecture is the idea that there are infinitely many prime numbers that differ by two; some mathematicians believe there may be a deeper structure involved in this pattern.

###### 1:48:52 - 1:55:45 (06:52)

###### Summary

The Twin Prime Conjecture is the idea that there are infinitely many prime numbers that differ by two; some mathematicians believe there may be a deeper structure involved in this pattern. However, as more primes are discovered, it is not necessarily a new discovery or experience.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The modern theory of infinity studies the cardinality of infinite sets and involves dividing lines infinitely to yield smaller segments.

###### 1:55:45 - 2:00:42 (04:57)

###### Summary

The modern theory of infinity studies the cardinality of infinite sets and involves dividing lines infinitely to yield smaller segments. Mathematics can treat infinity as a real or ambiguous concept while studying the continuum.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The speaker talks about the iPad app Golly which has different variants of Conway's rule and can be used to learn cellular automata and reduce textures from images, similar to analyzing organisms on a higher level of abstraction like a biologist would.

###### 2:00:43 - 2:09:03 (08:19)

###### Summary

The speaker talks about the iPad app Golly which has different variants of Conway's rule and can be used to learn cellular automata and reduce textures from images, similar to analyzing organisms on a higher level of abstraction like a biologist would.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### This podcast episode explores the beginnings of group theory and the general theory of symmetry in mathematics, as well as the work and legacy of mathematician John Horton Conway, who is famous for his contributions to automata theory and the invention of the Game of Life.

###### 2:09:03 - 2:13:06 (04:03)

###### Summary

This podcast episode explores the beginnings of group theory and the general theory of symmetry in mathematics, as well as the work and legacy of mathematician John Horton Conway, who is famous for his contributions to automata theory and the invention of the Game of Life.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### This episode explores the idea of the fundamental group of symmetries before Einstein and after, making connections with computer science and physics.

###### 2:13:06 - 2:18:30 (05:23)

###### Summary

This episode explores the idea of the fundamental group of symmetries before Einstein and after, making connections with computer science and physics.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The speaker emphasizes the importance of having multiple approaches to understanding intricate mathematical concepts to cater to different people's thought processes and to discover new ideas.

###### 2:18:30 - 2:23:56 (05:26)

###### Summary

The speaker emphasizes the importance of having multiple approaches to understanding intricate mathematical concepts to cater to different people's thought processes and to discover new ideas.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The Ricci flow allows mathematicians to understand the geometry of the space of all three-dimensional geometries and determine if an unknown space is similar to the standard three-dimensional space.

###### 2:23:57 - 2:27:50 (03:53)

###### Summary

The Ricci flow allows mathematicians to understand the geometry of the space of all three-dimensional geometries and determine if an unknown space is similar to the standard three-dimensional space.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### While awards may not be important in the grand scheme of things, there will be moments in life where society will pressure you to go against your principles, and it's important to stand your ground.

###### 2:27:50 - 2:33:50 (05:59)

###### Summary

While awards may not be important in the grand scheme of things, there will be moments in life where society will pressure you to go against your principles, and it's important to stand your ground. This is something that many people may experience at some point in their lives.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### Tai-Danae Bradley, a mathematician and data scientist, discusses the persistence required to learn complex subjects like algebraic geometry, and the challenge of starting even when you don't feel like it.

###### 2:33:51 - 2:43:00 (09:08)

###### Summary

Tai-Danae Bradley, a mathematician and data scientist, discusses the persistence required to learn complex subjects like algebraic geometry, and the challenge of starting even when you don't feel like it.

###### Episode###### #190 – Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

###### Podcast###### Lex Fridman Podcast

###### The podcast discusses the relationship between spirituality and mathematics, specifically referring to mathematician Blaise Pascal's mystical experience and his writings on the connection between the two.

###### 2:43:00 - 2:47:17 (04:17)

###### Summary

The podcast discusses the relationship between spirituality and mathematics, specifically referring to mathematician Blaise Pascal's mystical experience and his writings on the connection between the two.