Freya Holmér
9 HN recommenders · recommended in 7 months, 2023–2025
51:16 · 525K views · Oct 26, 2023
Freya Holmer's Dutch Game Day 2023 talk asking why vectors can't be multiplied, building from complex numbers to a single axiom and arriving at geometric algebra, quaternions and multivectors.
HN treats it as the best entry point to geometric algebra and an intuitive route to understanding quaternions.
11 comments from Hacker News
Freya Holmer, of The Continuity of Splines [0] fame, recently gave a great conference talk about how quaternions naturally arise from the desire to define the multiplication of two vectors if you just agree to one new axiom : v^2 = ||v||^2, or "the square of a vector is equal to the square of its norm". It was fascinating and entertaining and I highly recommend anyone who finds this stuff interesting but confusing (like I do) to check it out: this video [0] [LINK]
One of my favorite math/graphics YouTube creators Freya Holmér did an excellent intro to Geometric Algebra not that long ago [1]. If you have any interest in 3d graphics (especially but no limited to splines/Bezier curves) then be sure to check out all of their videos. I personally have always struggled with linear algebra and I tend to find these Clifford Algebra approaches much more intuitive. 1. this video
I know people have jumped here to try to explain it to you as a comment, but I highly recommend a recent talk by Freya Holmer. It's a fun little exploration of a seemingly simply question about vectors that motivates the answer you asked. It also culminates with an introduction to Clifford Algebras, which you may or may not want to learn about after the talk. this video
In another comment, they pointed out this video in which Freya did an excellent job reducing the number of axioms to 1: If you define multiplying a vector by itself to be equal to the length of the vector, squared, then everything else falls out of simple polynomial multiplication. It's quite lovely.
The absolute first video to watch should be the (ironically titled) Why can't you multiply vectors? this video