The most beautiful formula not enough people understand
A mathematician presents what he considers one of the most beautiful yet underappreciated formulas in mathematics - the formula for the volume of high-dimensional spheres. He derives this formula using an extension of Archimedes' method and explores the surprising behavior of spheres in higher dimensions.
Summary
The lecture begins with two probability puzzles to motivate the study of high-dimensional geometry. The first involves choosing random points in a square and determining if they fall within a circle, demonstrating how geometric intuition helps solve analytical problems. This extends to higher dimensions, showing the practical relevance of high-dimensional geometry in machine learning and language models like ChatGPT.
The second puzzle reveals how counterintuitive high-dimensional geometry can be. By placing unit spheres at the corners of a cube and finding the largest sphere that fits in the center, the speaker shows that in 10 dimensions, this inner sphere actually extends outside the bounding cube - a seemingly impossible result that highlights how our 3D intuitions fail in higher dimensions.
The core of the lecture focuses on deriving the formula for volumes of high-dimensional spheres using an extension of Archimedes' cylinder method. Starting with familiar 2D and 3D formulas, the speaker builds a systematic approach using what he calls "knight's moves" - jumping two dimensions at a time in a chart relating boundary areas to interior volumes. This method reveals that each step up in dimension involves multiplying by 2π and dividing by the dimension number.
The resulting general formula involves π raised to half the dimension divided by the factorial of half the dimension, requiring an extension of factorial to non-integers through the gamma function. Remarkably, when examining actual numerical values, sphere volumes initially increase with dimension but peak at 5 dimensions, then decrease rapidly. By 100 dimensions, the volume becomes infinitesimally small (around 10^-40), meaning that choosing 100 random numbers and having their squares sum to less than 1 is extraordinarily unlikely.
The lecture concludes by exploring why this happens and what it means geometrically - essentially all volume in high-dimensional spheres concentrates near the boundary, and spheres become negligible compared to cubes. This has practical implications in machine learning, cryptography, and quantum mechanics.
About this episode
On the volumes of higher-dimensional spheres Explore the 3b1b virtual career fair: See https://3b1b.co/talent Become a supporter for early views of new videos: https://3b1b.co/support An equally valuable form of support is to simply share the videos. Home page: https://www.3blue1brown.com Thanks to UC Santa Cruz for letting me film there, and special thanks to Pedro Morales-Almazan for arranging everything. My video on Numberphile with a fun application of this problem: https://youtu.be/6_yU9eJ0NxA Timestamps: 0:00 - Introduction 1:01 - Random puzzle 6:16 - Outside the box 14:35 - Setting up the volume grid 21:14 - Why 4πr^2 25:21 - Archimedes in higher dimensions 36:17 - The general formula 40:40 - 1/2 factorial 44:58 - Why 5D spheres are the biggest 50:16 - Concentration at the surface 54:27 - A unit-free interpretation 57:50 - 3b1b Talent 59:13 - Explaining the intro animation ------------------ These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim Music by Vincent Rubinetti. https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly. Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Bluesky: https://bsky.app/profile/3blue1brown.com Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com
Key Insights
- The speaker argues that high-dimensional geometry is real and useful, not just abstract nonsense, because machine learning represents data as points in high-dimensional space
- Large language models work by converting text chunks into long lists of numbers that researchers interpret as points in high-dimensional space to understand model behavior
- The speaker claims that spheres are not the problem in high-dimensional geometry - cubes are, because spheres are perfectly round by definition while cubes become counterintuitively spiky
- In high-dimensional cubes, corners are much farther from the origin than the edges, creating a situation where unit spheres at corners are 'way, way, way farther away than the edge'
- Archimedes' cylinder method works because two competing effects cancel out perfectly - stretching in one direction and shadow-casting compression in another direction
- The speaker demonstrates that the relationship between sphere volume and surface area follows calculus principles, where the derivative of volume gives surface area
- The volume formula for n-dimensional spheres follows a recursive pattern where each dimension multiplies by 2π and divides by the dimension number, creating 'knight's moves' in a dimensional chart
- Sphere volumes peak at 5 dimensions because that's where the numerator factor 2π and denominator factors are closest, with 2π being slightly larger than 5
- A 100-dimensional unit sphere has volume around 2.37 × 10^-40, making it 'absolutely nothing' despite intuitions about containing lower-dimensional spheres
- In high dimensions, essentially all volume sits right at the boundary - scaling a 10,000-dimensional sphere by 99% reduces its volume by factor 0.99^10,000, which is effectively zero
- The speaker reveals that factorial can be extended to non-integers through the gamma function, where one-half factorial equals square root of π divided by 2
- Mathematical beauty often requires seeing familiar concepts like factorials or circle areas in more general contexts, where hidden connections like π appearing in factorial calculations become visible
Topics
Transcript
[0:00] [Submit subtitle corrections at criblate.com] Thank you very much. It is good to be here. I don't know if you people realize what a beautiful campus you have and how you basically just study in heaven. Today, I want to talk with you about what I think is one of the most underappreciated formulas, not because those who know it don't appreciate it, but because not enough people know about this. And more importantly, not enough people understand where it comes from. And this should be one of the gems. It's this should be the e to the pi i of the mathematical community. [0:33] But I really want you to come away knowing not just what it is,…
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