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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:    • Complex Numbers Have More Uses Than Y...  
3Blue1Brown's video on the matrix exponential:    • How (and why) to raise e to the power...  


=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


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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


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This video was supported in part by these patrons on Patreon:
Marshall Harrison, Michael OConnor.

To support future videos, become a patron at   / morphocular  
Thank you for your support!


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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/morphomatters/mor...