# 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…

# 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…

# How pi was almost 6.283185…

From 3Blue1Brown. Happy pi day! Did you know that in some of his notes, Euler used the symbol pi to represent 6.28…, before the more familiar 3.14… took off as a standard? Plushie creatures now available: http://3b1b.co/store The idea for this video, as well as the live shots, came from Ben Hambrecht, with the writing…

# Why is pi here? And why is it squared? A geometric answer to the Basel problem

From 3Blue1Brown. A most beautiful proof of the Basel problem, using light. Plushie pi creatures now available: http://3b1b.co/store More problem-driven learning: https://brilliant.org/3b1b Brilliant’s principles list that I referenced: https://brilliant.org/principles/ Special thanks to patrons: https://www.patreon.com/3blue1brown http://3b1b.co/basel-thanks The content here was based on a paper by Johan Wästlund http://www.math.chalmers.se/~wastlund/Cosmic.pdf Check out Mathologer’s video on the many cousins…

# The more general uncertainty principle, beyond quantum

From 3Blue1Brown. The Heisenberg uncertainty principle is just one specific example of a much more general, relatable, non-quantum phenomenon. Apply to work at one of my favorite math education companies: http://aops.com/3b1b Special thanks to the following Patrons: http://3b1b.co/uncertainty-thanks You are the ones making this possible: http://3b1b.co/support For more on quantum mechanical wave functions, I highly…

# But what is the Fourier Transform? A visual introduction.

From 3Blue1Brown. An animated introduction to the Fourier Transform, winding graphs around circles. Supported by viewers: https://www.patreon.com/3blue1brown Special thanks to these Patrons: http://3b1b.co/fourier-thanks Follow-on video about the uncertainty principle: https://youtu.be/MBnnXbOM5S4 Puzzler at the end by Jane Street: https://janestreet.com/3b1b Music by Vincent Rubinetti: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown —————— 3blue1brown is a channel about animating math, in all senses of…

# Science YouTubers attempting a graph theory puzzle

From 3Blue1Brown. A mug with some unexpectedly interesting math. Vihart response: https://youtu.be/CruQylWSfoU More problem-solving at Brilliant: https://brilliant.org/3b1b https://brilliant.org/3b1bflipping Special thanks to these patrons: http://3b1b.co/mug-thanks Thanks to all the following channels for participating. Standup Maths https://www.youtube.com/user/standupmaths Wendover Productions https://www.youtube.com/user/Wendoverproductions Welch Labs: https://www.youtube.com/user/Taylorns34 MinutePhysics: https://www.youtube.com/user/minutephysics Ben Eater: https://www.youtube.com/user/eaterbc Mathologer: https://www.youtube.com/channel/UC1_uAIS3r8Vu6JjXWvastJg Singing Banana: https://www.youtube.com/user/singingbanana Numberphile: https://www.youtube.com/user/numberphile Looking Glass…

# Q&A #2 + Net Neutrality Nuance

From 3Blue1Brown. Questions: http://3b1b.co/questions Answers: https://youtu.be/1nxF19qDBQ0 Full net neutrality video: https://youtu.be/hKD-lBrZ_Gg Ben Eater’s channel: https://www.youtube.com/user/eaterbc First Q&A Answers: https://youtu.be/8r5WKpK9-m8 CGPGrey’s (excellent) video on Net Neutrality: https://youtu.be/wtt2aSV8wdw —————— 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…

# The hardest problem on the hardest test

From 3Blue1Brown. A difficult geometry puzzle with an elegant solution. Practice more problem-solving at https://brilliant.org/3b1b Solution to the puzzle mentioned at the end: https://brilliant.org/3b1bindicator/ Special thanks to the following patrons: http://3b1b.co/putnam-thanks These videos exist thanks to Patreon: https://www.patreon.com/3blue1brown A different write-up of this solution: http://lsusmath.rickmabry.org/psisson/putnam/putnam-web.htm 1992 Putnam with this problem: http://kskedlaya.org/putnam-archive/1992.pdf A problem with a…