Geometric Algebra

Currently Playing: The quaternions | Geometric algebra episode 10

#geometricalgebra #quaternions #complexnumbers Earlier, we discovered the complex numbers as the even-graded elements in 2D geometric algebra. In a very similar way, the quaternions turn out to be the even-graded elements in 3D. We explore some of their properties, and we ask ourselves why the quaternions use a sandwich product. The answer to this intriguing question will be given in upcoming videos. You can support us on Patreon, where you can already watch all the Geometric Algebra videos and get access to exclusive content. Unfortunately, we won't be publishing any new videos after Geometric Algebra, at least not in the near future. Your support is still more than welcome of course: https://www.patreon.com/user?u=86649007 I have collected many interesting resources for you, where you can learn much more about quaternions in general, and specifically for how they emerge out of geometric algebra: [3B1B 1] https://www.youtube.com/watch?v=d4EgbgTm0Bg This one is not about the algebra of the quaternions, but about their geometry. In particular, it tries to give you a sense for the 4-dimensional space that they live in. [3B1B 2] https://www.youtube.com/watch?v=zjMuIxRvygQ A short follow-up about quaternions, and in particular how they help us describe rotations in 3D. [PENG 1] https://www.youtube.com/watch?v=jTgdKoQv738 Good introduction to the sandwich product and why it performs rotations. [MOM 1] https://www.youtube.com/watch?v=jlskQDR8-bY Quick introduction to the algebra of the quaternions, without any geometric insight. [MOM 2] https://www.youtube.com/watch?v=nJnWM7R6Y5A Very good overview of how the even-graded sub-algebra of 3D GA gives you the quaternions. The only unfortunate thing, is that the video uses e_31 instead of e_13 as a basis bivector, so that it requires an extra minus sign on one of the quaternion components. [WIKI 1] https://en.wikipedia.org/wiki/Quaternion#Quaternions_as_the_even_part_of_Cl3,0(R) The wikipedia page lists a large number of advantages of geometric algebra over quaternions. 0:00 Recap of complex numbers in geometric algebra 1:47 Every plane has a copy of the complex numbers 4:32 Discovering the quaternions 7:36 A few properties of quaternions 10:25 Advantages of geometric algebra over quaternions This video is published under a CC Attribution license ( https://creativecommons.org/licenses/by/4.0/ )