Jack RusherOct 11, 2023

My talk on geometric algebra from this year's Strange Loop/@[email protected] Conference is up! Here's a mini-site with the video, a transcript, and links to all the papers I mention.

https://jackrusher.com/pwl-2023/

Geometry to Algebra and Back Again

4000 Years of Papers

Jack Rusher
Show threadJack RusherOct 16, 2023

For those who would like a video introduction to GA that shows loads of lovely algebra, check out this one!

https://www.youtube.com/watch?v=60z_hpEAtD8

A Swift Introduction to Geometric Algebra

YouTube
Show threadJack RusherOct 20, 2023
Excellent video on efficient implementation of this approach in Julia:
https://www.youtube.com/watch?v=zq3wHwWf4WA
SimpleGA. A lightweight Geometric Algebra library | Chris Doran | JuliaCon 2023

YouTube
Show threadJack RusherOct 29, 2023

This talk from the always great
@acegikmo provides a very nice explanation of the geometric product:

https://www.youtube.com/watch?v=htYh-Tq7ZBI

Show threadBenjohn
@jack @acegikmo It is going to be hilarious but also brilliant if GA breaks in to the maknstream through games and graphics, then trickles down in to physics which seems to still resist it (?).
Oct 29, 2023 at 10:03amWeb
Show threadFreya HolmérOct 29, 2023
@benjohn oh huh, I thought physics was way ahead in this regard!
Show threadBenjohnOct 29, 2023
@acegikmo :-) Honestly, I’m not sure. To me it feels like it ought to be A level (pre degree) content as an ongoing mathematical language for anything in the domain of mechanics. Similar to how we use position value, not roman numerals to write values! I don’t think it’s had anything like that impact yet, but I could be very mistaken!
Show threadJack RusherOct 29, 2023
@acegikmo @benjohn It’s still outsider stuff, sadly. My hope is that the tide may finally be turning 🤞🏻
Show threadBenjohnOct 29, 2023
@jack @acegikmo are multi vector closed under exponential (of another MC)?
Show threadFreya HolmérOct 29, 2023
@benjohn @jack ah gosh that seems complicated, I have no idea!
Show threadBenjohnOct 30, 2023

@acegikmo @jack :-) Me neither, it just a curiosity itch!

… more practically, it seems like “PGA” is the algebra nearest to what graphics practitioners currently use? It does homogeneous coordinates, I think.

I have not found a clear account of it yet though, and it seems to use the simplest vectors to represent planes (not points), unless I’m very confused? Which I would like to have explained as if to a child to me!

Show threadJack RusherOct 30, 2023

@benjohn @acegikmo the set of basis blades is closed under the geometric product, if that's what you mean :)

PGA reverse the order, yeah, so it starts with a plane, then defines a line as the intersection of two planes, and a point as the intersection of three planes

Sudgy has an explainer video:

https://www.youtube.com/watch?v=0i3ocLhbxJ4

Show threadBenjohnOct 30, 2023

@jack @acegikmo cool, thank you, will watch!

The earlier question, I was meaning powers. Can I raise a multi-vector to the power of another multi-vector? Do I still get a multi-vector?

I know (I think I know!) Complex are closed under this?

I don’t have any use for it at all! I am just curious :-)

Show threadFreya HolmérOct 30, 2023
@benjohn @jack individually complex numbers and quaternions are closed under exponentiation, but I don't know what happens when you involve the vector, trivector and the real part in the exponent. it's possible it's closed!
Show threadBenjohnOct 30, 2023
@acegikmo @jack 😆 I always imagine there will be a mathematician oracle about who can say “YES” or of course “NO”. Or just maybe “yeah, we’re not sure yet - still working on that one, but tricky you see.”