Nice intro! I agree that the cross-product should be deprecated in favor of the wedge product in almost every physical application.
I like Geometric Algebra, but I find that its proponents tend to oversell it (you aren’t doing that here, I just mean in general). Which is unfortunate, since it increases the entropy (on all sides) of pretty much all discussions of it. On the other hand, it does seem to add more energy toward people learning it.
Anyway, here are some of my observations on potential blindspots.
I think the way it mixes types (e.g. the geometric product of two vectors is the sum of a bivector and a scalar) sometimes adds more confusion and complexity than it removes. As an example of this from Hestenes himself, from page 18 of this, he describes trying to find the right way to use GA to model kinematics (i.e. translations, rotations, and their combination: screws). At first, this seems like the perfect excuse to add a vector to a bivector, and get a coherent geometric meaning out of it! However, he found that it was actually better to add some null basis elements, so that translations and rotations both end up being bivectors. Another case where I think type conflation is happening is in the identification of the dual space with the primary space; these have different physical units (but to be fair, standard math is terrible about conflating these too)!
None of this is to say that there aren’t a bunch of great insights from thinking about things from the GA viewpoint! In particular, I find thinking of spinors as exponentiated bivectors is especially enlightening! Just a note of caution about some blindspots of the community that I’ve noticed since first being interested in it.
Thank you for your insightful comment. The concept of a screw is new to me, so I’ll have good look at the article you shared and I will try to think carefully about how physical units relate to types, as well as what constitutes true geometric meaning.
Nice intro! I agree that the cross-product should be deprecated in favor of the wedge product in almost every physical application.
I like Geometric Algebra, but I find that its proponents tend to oversell it (you aren’t doing that here, I just mean in general). Which is unfortunate, since it increases the entropy (on all sides) of pretty much all discussions of it. On the other hand, it does seem to add more energy toward people learning it.
Anyway, here are some of my observations on potential blindspots. I think the way it mixes types (e.g. the geometric product of two vectors is the sum of a bivector and a scalar) sometimes adds more confusion and complexity than it removes. As an example of this from Hestenes himself, from page 18 of this, he describes trying to find the right way to use GA to model kinematics (i.e. translations, rotations, and their combination: screws). At first, this seems like the perfect excuse to add a vector to a bivector, and get a coherent geometric meaning out of it! However, he found that it was actually better to add some null basis elements, so that translations and rotations both end up being bivectors. Another case where I think type conflation is happening is in the identification of the dual space with the primary space; these have different physical units (but to be fair, standard math is terrible about conflating these too)!
None of this is to say that there aren’t a bunch of great insights from thinking about things from the GA viewpoint! In particular, I find thinking of spinors as exponentiated bivectors is especially enlightening! Just a note of caution about some blindspots of the community that I’ve noticed since first being interested in it.
Thank you for your insightful comment. The concept of a screw is new to me, so I’ll have good look at the article you shared and I will try to think carefully about how physical units relate to types, as well as what constitutes true geometric meaning.