Stream #3: Different approaches/notation for Projective Geometric Algebra from Eric Lengyel and Jon

Google "Projective Geometric Algebra" and you see that there are two different notations people are using. This video tries to clarify things around that!
Predictably, I made a few mistakes! I did a stream clarifying these but the connection let me down. So, here are those mistakes:
1. I say in this video that you can measure the angle between anything and anything, without dualization. That's not true! There is an interesting exception, which is that you can't measure the angle between points or lines at infinity.
2. I keep saying "polar space" here, but that's muddling the terminology. Ordinary space is "euclidean space", the other space (the tetris window full of lines) is actually called "dual euclidean", and then the term "polar" is meant to be used for the two of them added together. So "polar tetris" is the right name for the game, since it has both, but the window full of lines alone is "dual euclidean tetris", not "polar tetris"
3. I didn't use the word "antispace" here at all, which is an oversight. So I'll say it here: "anti space", which is Eric's term, is another term for the tetris window, whereas the tetris window full of lines is what he calls "space". Of the things that start with "anti-" in PGA (anti geometric product, anti reverse), all of them are specific to Eric's approach, with two exceptions: the antiwedge, which others call the join or regressive product, and the antinorm, which others call the ideal norm. -- Watch live at / hamish_todd

[@AndrewBrownK]

really great talk again, great history.
I'm still pretty new to this and biased because I learned from Eric's material first, however...
There's lots of stuff in history, "arbitrary choices" as you would put it, that ossify and set in stone as convention unless they are quickly overturned. And so even if there is a small but compelling reason to arbitrate differently, it's worth considering, especially in a field so (dare I say) young.
For some examples... qwerty keyboards. Maybe back in the day they couldn't have the foresight to predict dvorak or other formats would have some marginal advantages. You can very well argue "we'd all be better off if the better convention took root". But how much better off? only a marginal bit. And people today can still customize their keyboards as individuals if they prefer. And yes that customized keyboard won't be easier for others to immediately pick up. But it is what it is.
Then there's driving on the left vs right side of the road. Why does it have to be different? idk. But it's not the end of the world. Just have to know which territory you're in and which rules apply. Would it be nicer if everywhere followed the same convention? sure.
Anyway all that to say.... Personally, I like Eric's arbitration. I think the grade of the object matching the shape itself makes a lot of sense. If taking the wedge increases grade, and performing a join adds dimensionality of the objects, it's only reasonable that joins should be the wedge (and meets should be anti-wedge). You say it was completely arbitrary. And to some extent, you're right. It's all opinion and preference at the end of the day. But it's exactly in that way that I say my opinion and preference is Eric's approach.
So which convention would be better for everyone to adopt? or rather what should we teach to new people? For every point of conflict Eric's approach has with other dialects of PGA, I think it gains just as many points for staying closer to Vanilla GA. Increasing grade = Increasing dimensionality. In the bottom left corner of the video I see the e1 e2 e3 as planes without orientation arrows and e31 e23 e12 as lines with spinny arrows. It feels wrong. Aesthetics is one of the appeals of GA. Everything changed when I realized rotations happen in planes (bivectors), not around axes (vectors (univectors, sorry lol)). Not staying true to that feels like a step backward, or as Eric might put it, a hack.
Anyway. Again, there's no reason we can't all get along and translate between conventions with a little extra work. But if there is a conversation or hypothetical wishful thinking about "if only we had one convention instead", then my vote is in Eric's formulation.
Apologies if I made any mistakes or miscomprehensions. No grief intended. These things just take time to digest.

[@hamish_todd]

I'm afraid you might have made a mistake here! I think you are imagining that all spaces are euclidean spaces. This is why I showed polar tetris.
Planes-grade-1 vs points-grade-1 is only an arbitrary choice **if you use both spaces**. The only people who use both spaces are Eric and Charles. Using both spaces means you have euclidean space and you also have "dual" euclidean space, as I show with Polar tetris (polar = euclidean+dual euclidean). Dual euclidean space is hard to learn, but those two are interested in it.
If you only care about *euclidean* geometry, which is all of engineering, **you have no choice but to make planes be grade 1**. The fundamental reason for this is that a point reflection is a composition of planar reflections, but a planar reflection is *not* a composition of point reflections.

[@hamish_todd]

Here's the game that takes place in elliptic space! store.steampowered.com/app/485680/sphereFACE/