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#geometricalgebra#quaternions#vectors

Why can't you multiply vectors?

Freya Holmér

291K subscribers • 6 videos on Hallway

YouTube
  • Recommended by 9 people: 9 on Hacker News.
  • Presented by Freya Holmér

Intermediate. Assumes working knowledge.

524K views3 years ago9 recommendations

Graphics and audioComputer scienceMath for programmersGame programmingRendering

Freya Holmér explores vector multiplication, showing how geometric algebra and quaternions naturally arise from a single geometric axiom.

..or can you? A deceptively simple question with a complex answer – come join a mathematical journey into madness and wonder, in search of answers that might just give you a new perspective on the mathematical constructs we use in our games Recorded at Dutch Game Day 2023, October 4th Timestamps: Intro Talk Start Anatomy of a Vector Products Numbers of different kinds Complex Numbers Code for complex numbers Complex multiplication visualized Multiplying Vectors (an attempt) Frustration A Leap of Faith A divine axiomatic truth Squaring Basis Vectors The Diagonal of Bases Vector multiplication unraveling Dot/Cross Byproducts Complex Byproduct Quaternionic Byproduct The Answer Beyond the Algebra Basis Bivectors Conceptualizing Vectors & Bivectors Wedge Product Curvature Bivector Mental Gymnastics Geometric Algebra & Multivectors What does the multiplication represent? Outro Q: When is this useful? Q: Does your math lib support geometric algebra? Q: Do you plan on making tools for other engines? AV team kicking us out 💖 Patreon ❱ patreon.com/acegikmo 🐦 Twitter ❱ twitter.com/FreyaHolmer 📺 Twitch ❱ twitch.tv/acegikmo 💬 Discord ❱ discord.gg/v5VWuga 📧 Mailing list ❱ acegikmo.substack.com/ 🌸 Instagram ❱ instagram.com/freya_holmer

Why it's here

9 people recommend this

What people say. Highly recommended as an intuitive and fascinating introduction to geometric algebra and quaternions for game developers.

mkaicon Hacker News • Recommends

On “Exploring Quaternions (2018)”

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: link to this video 0 link
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Syntonicleson Hacker News • Recommends

On “Linear Algebra Done Right – 4th Edition”

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. link to this video
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zoogenyon Hacker News • Recommends

On “Look, ma, no matrices”

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. link to this video
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simpaticoderon Hacker News • Recommends

On “Look, ma, no matrices”

In another comment, they pointed out link to 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.
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gowldon Hacker News • Recommends

On “Poor Foundations in Geometric Algebra”

The absolute first video to watch should be the (ironically titled) Why can't you multiply vectors? link to this video
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conradludgateon Hacker News • Recommends

On “Exploring Quaternions (2018)”

Coincidentally, this conference talk1 released yesterday regarding vector multiplication. In it, it describes how quaternions can be seen as an extension of 3d vector spaces, like how complex numbers can be seen as an extension to 2d vector spaces. More specifically, quaternions are a 4d vector with the basis of {1,yz,xz,xy} where those "bivectors" result from 3d vector multiplication, e.g. (ax+by+cz)(dx+ey+fz) 1: link to this video
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infogulchon Hacker News • Recommends

On “Rapier is a set of 2D and 3D physics engines written in Rust”

I got sucked down a Geometric Algebra rabbit hole a few months ago, which seems like a remarkably concise and intuitive way to work in a very wide variety of geometries, including 2D, 3D, 4D+, non-euclidean etc. I've wondered if GA would make a good foundation for a physics engine... There are a few small rust libraries that look interesting 12, but none with a lot of traction. Has anyone looked into this? 1: link 2: link If you dare, a good place to jump in might be Freya Holmér's Why can't you multiply vectors? - link to this video - Then link
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GuB-42on Hacker News • Recommends

On “Quaternion Knowledge Graph Embeddings (2019)”

If you are still interested, try this: link The idea is that quaternions are the wrong way to look at the problem and that a better approach would be to use geometric algebra, bivectors and rotors. The formulas are essentially the same as with quaternions, but at least for me, this approach make more intuitive sense. It also work in dimensions other than 3, which matter to the author as he is the author of "4D toys" and hopefully, Miegakure. Here is another talk on the subject, by a different author: link to this video It is not specifically about quaternions, it is about multiplying vectors and what you get from that, and it includes quaternions.
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BenoitPon Hacker News • Recommends

On “Quaternions in Signal and Image Processing”

You should jump straight to geometric algebra. It's actually simple there. Quaternions and Complex Numbers are about just some not so special multivectors. Here are 2 45mn videos you'll not regret: link to this video link
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kurthron Hacker News • Mentions

On “Why is Maxwell's theory so hard to understand? (2007) [pdf]”

…really liked Kathy Joseph's historical reviews of vector physics and the people who developed it, which explain some of the reason's for how it's taught. Most texts don't even develop electrodynamics from relativistic electrostatics as a demonstration. link I think I fell down the rabbit hole from Freya Holmer's "why you can't multiply vectors". link to this video The key being that all of the Hamiltonian fields can be found in a single quaternion equation, which is just what happens when you start multiplying vectors together.…
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elbearon Hacker News • Mentions

On “Functions Are Vectors (2023)”

Thanks for mentioning this. I think I encountered Geometric Algebra in a video related to making video games. It's this one (Why you can't multiply vectors): link to this video
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