Euler's Formula Beyond Complex Numbers

Go to https://brilliant.org/Morphocular to get started learning STEM for free. The first 200 people get 20% off an annual premium subscription. The famous Euler's Formula for complex numbers provides an elegant way to describe 2D rotation, but is there a way to make it work for 3D or higher dimensions? Previous video about complex numbers: https://youtu.be/4KlvI_uK9zs 3Blue1Brown's video on the matrix exponential: https://youtu.be/O85OWBJ2ayo =Chapters= 0:00 - Intro 3:03 - 3D vs 2D 5:07 - A Brief Overview of Matrices 8:47 - How To Exponentiate a Matrix 12:06 - 3D Rotations via Matrix Exponentials 17:54 - How to Build Rotation Generators 24:54 - 3D Euler 27:10 - A Look Ahead 28:25 - Brilliant ad =============================== CREDITS The music tracks used in this video are (in order of first appearance): "Checkmate", "Dream Escape", "Ascending", "Rubix Cube", "Orient", "Falling Snow" The track "Rubix Cube" comes courtesy of Audionautix.com =============================== This video was supported in part by these patrons on Patreon: Marshall Harrison, Michael OConnor. To support future videos, become a patron at https://www.patreon.com/morphocular Thank you for your support! =============================== The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here: https://github.com/morpho-matters/morpholib