# But WHY is a sphere’s surface area four times its shadow?

From 3Blue1Brown. Two lovely ways of relating a sphere’s surface area to a circle. Fourier socks, pi plushies, and more: http://3b1b.co/store Happy holidays! The first proof goes back to Greek times, due to Archimedes, who was charmed by the fact that a sphere has 2/3 the volume of a cylinder encompassing it, and 2/3 the…

# Sneaky Topology

From 3Blue1Brown. Solving a discrete math puzzle using topology. Want more fair division math fun? Check out this Mathologer video https://youtu.be/7s-YM-kcKME (Seriously, Mathologer is great) These videos are supported by the community. https://www.patreon.com/3blue1brown The original 1986 by Alon and West with this proof https://m.tau.ac.il/~nogaa/PDFS/Publications/The%20Borsuk-Ulam%20Theorem%20and%20bisection%20of%20necklaces.pdf VSauce on fixed points https://youtu.be/csInNn6pfT4 EE Paper using ideas related to…

# Visualizing turbulence with a home demo

From 3Blue1Brown. Vortex rings with Physics Girl: https://youtu.be/N7d_RWyOv20 Here we look at some of the order amidst chaos in turbulence. Special thanks to these supporters: http://3b1b.co/turbulence-thanks https://patreon.com/3blue1brown Thanks to Dan Walsh for many great ideas, and thanks to Mike Hansen for many helpful conversations. Introduction to turbulence: http://www.astronomy.ohio-state.edu/~ryden/ast825/ch7.pdf More details on vortex stretching: https://www.math.nyu.edu/faculty/tabak/vorticity.pdf Video…

# What DO we know about turbulence?

From 3Blue1Brown. Here we look at some of the order amidst chaos in turbulence. Vortex rings with Physics Girl: https://youtu.be/N7d_RWyOv20 Special thanks to these supporters: http://3b1b.co/turbulence-thanks https://patreon.com/3blue1brown Thanks to Dan Walsh for many great ideas, and thanks to Mike Hansen for many helpful conversations. Error correction: I meant to describe Kolmogorov as a “20th-century mathematician”…

# Quaternions and 3d rotation, explained interactively

From 3Blue1Brown. Go experience the explorable videos: https://eater.net/quaternions Ben Eater’s channel: https://www.youtube.com/user/eaterbc Previous video on Quaternions: https://youtu.be/d4EgbgTm0Bg Special thanks to these supporters: http://3b1b.co/quaternion-explorable-thanks Nice explanation of Gimbal Lock: https://youtu.be/zc8b2Jo7mno Great videos comparing Euler angles and quaternions, from the perspective of an animator: https://youtu.be/syQnn_xuB8U https://youtu.be/4mXL751ko0w Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown —————— 3blue1brown is a channel about…

# What are quaternions, and how do you visualize them? A story of four dimensions.

From 3Blue1Brown. How to think about this 4d number system in our 3d space. Thanks to my modern Medicis: https://www.patreon.com/3blue1brown With extra-special thanks here: http://3b1b.co/quaternion-thanks Quanta article on quaternions: https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/ The math of Alice in Wonderland: https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved/ Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown —————— 3blue1brown is a channel about animating math, in all senses of the…

# Q&A with Grant Sanderson (3blue1brown)

From 3Blue1Brown. To commemorate crossing 2^20 subscribers, I answer some questions people asked on Reddit. AMC problem shown on screen: https://artofproblemsolving.com/wiki/index.php?title=2002_AMC_12A_Problems/Problem_22 Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown —————— 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on…

# What does genius look like in math? Where does it come from? (Dandelin spheres)

From 3Blue1Brown. A beautiful proof of why slicing a cone gives an ellipse. Thoughts on going sponsor-free (for those who missed it): https://www.patreon.com/posts/going-sponsor-19586800 Special thanks: http://3b1b.co/dandelin-thanks New shirts/mugs available: http://3b1b.co/store Video on Feynman’s lost lecture: https://youtu.be/xdIjYBtnvZU The 3d animations in the video were done using Grapher, while 2d animations were done using https://github.com/3b1b/manim Music by…

# 3b1b featured creators #1

From 3Blue1Brown. Think Twice: https://www.youtube.com/channel/UC9yt3wz-6j19RwD5m5f6HSg LeiosOS: https://www.youtube.com/user/LeiosOS Welch Labs: https://www.youtube.com/user/Taylorns34 Infinity plus one: https://infinityplusonemath.wordpress.com/ Check out the ones on relativity! https://infinityplusonemath.wordpress.com/2017/03/11/a-mathematical-intro-to-special-relativity/ Music by Enoch Kim https://soundcloud.com/themusemaker —————— 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on…

# Divergence and curl: The language of Maxwell’s equations, fluid flow, and more

From 3Blue1Brown. Intuitions for divergence and curl, and where they come up in physics. Thoughts on going sponsor-free: https://www.patreon.com/posts/19586800 Special thanks to these supporters: http://3b1b.co/divcurl-thanks My former work on divergence and curl at Khan Academy: https://www.khanacademy.org/math/multivariable-calculus/multivariable-derivatives For more fun fluid-flow illustrations, which heavily influenced how I animated this video, I think you’ll really enjoy this…

# What they won’t teach you in calculus

From 3Blue1Brown. A visual for derivatives which generalizes more nicely to topics beyond calculus. Supported by viewers: https://www.patreon.com/3blue1brown Problem-driven learning by Brilliant: https://brilliant.org/3b1b Essence of calculus series: http://3b1b.co/calculus Special thanks to the following patrons: http://3b1b.co/alt-calc-thanks Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown —————— 3blue1brown is a channel about animating math, in all senses of the word animate.…

# Why does this product equal π/2? A new proof of the Wallis formula for π.

From 3Blue1Brown. A new and more circularly proof of a famous infinite product for pi. Try out Udacity: https://udacity.com/3b1b Here’s supplemental blog post, expanding some of the rigor of this proofs, along with other interesting tidbits about surrounding topics that we didn’t fit into the video: https://www.3blue1brown.com/sridhars-corner/2018/4/17/wallis-product-supplement-dominated-convergence Special thanks to the following Patreon supporters: http://3b1b.co/wallis-thanks…

# Why does this product equal pi/2?

From 3Blue1Brown. A new and more circularly proof of a famous Wallis product for pi. Here’s supplemental blog post, expanding some of the rigor of this proofs, along with other interesting tidbits about surrounding topics that we didn’t fit into the video: https://www.3blue1brown.com/sridhars-corner/2018/4/17/wallis-product-supplement-dominated-convergence Try out Udacity: https://udacity.com/3b1b Special thanks to the following Patreon supporters: http://3b1b.co/wallis-thanks…

# The Wallis product for pi, proved geometrically

From 3Blue1Brown. A new and more circularly proof of a famous Wallis product for pi. Here’s supplemental blog post, expanding some of the rigor of this proofs, along with other interesting tidbits about surrounding topics that we didn’t fit into the video: https://www.3blue1brown.com/sridhars-corner/2018/4/17/wallis-product-supplement-dominated-convergence Try out Udacity: https://udacity.com/3b1b Special thanks to the following Patreon supporters: http://3b1b.co/wallis-thanks…

# Solving 2D equations using color, a story of winding numbers and composition

From 3Blue1Brown. An algorithm for numerically solving certain 2d equations. Brought to you by…you! https://patreon.com/3blue1brown Special thanks to these supporters: http://3b1b.co/winding-thanks Even though we described how winding numbers can be used to solve 2d equations at a high level, it’s worth pointing out that there are a few details missing for if you wanted to…