Talk:Plane-based geometric algebra

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✠ SunDawn ✠ (contact) 04:44, 14 September 2023 (UTC)[reply]

Thank you very much! 😁 Hamishtodd1 (talk) 12:48, 20 October 2023 (UTC)[reply]

Notes on "Projective" versus plane-based

"Projective Geometric Algebra" is, it must be admitted, a more widely-used term/system than plane-based geometric algebra. But typing that into wikipedia redirects to a section of this page.

Please note that Plane-based GA and PGA are different (but one is contained in the other). As this article states, PGA is a generalization of plane-based GA allowing for distance measurement and "joins". One could have made a separate article for PGA, but that seems like a bad idea. Here are the reasons why:

Avoiding complexity/controversy. Different people have somewhat different views of what the phrase "Projective Geometric Algebra" refers to. But at least we can say nobody is in doubt that PGA is a generalization of the plane-based geometric algebra, and that PGA involves duality. Name aside, as of 2023, the book is not closed on what the most productive view of PGA duality is. Even the list of three definitions present in the article today is not really comprehensive - there is also the shuffle product view, the inertial duality view, and the elliptic geometry view. It is not productive for people to be confronted with this complexity early on.
The geometric product is very important and newcomers need to internalize it. If you think about planes first, the geometric product appears as compositions of planar reflections. Planar reflections are familiar and simple, so this is a helpful thing to see early on, even if your eventual goal is to understand PGA. It must be admitted that for people accustomed to building objects from points, it can be hard to internalize - but that does not change the fact that the geometric product is the most fundamental operation, so makes sense to draw attention to plane-based thinking ASAP.
Relatedly, whether we are talking about PGA or plane-based GA, our main concern is transformations that are Euclidean. PGA contains Euclidean transformations, but it can be seen as containing some non-Euclidean ones, which Gunn and Lengyel have written about. If these are to be taken as part of PGA, PGA must be seen as a rather more complicated system. "Point-based PGA" is, mathematically, a coherent thing, but its transformations are non-euclidean and unfamiliar. The fundamental reason for this can be seen geometrically: a point reflection is the composition of three planar reflections, but a planar reflection is not the composition of three point reflections. That is unless you are in a strange non-euclidean space.
"PGA" is a bit of a misnomer, since it does not contain the group of projective transformations (whereas Conformal GA does contain the group of Conformal transformations). It was suggested at various points to be called Homogeneous geometric algebra or Euclidean, either of which might have been better. So its name may even change. Whether or not this happens, "Plane-based GA" will remain a coherent subset of it.
Aside from Clifford/Hamilton/Grassman, the first people to publish on the ideas that have evolved into PGA were Ian Porteous and James Brooke; neither of them used duality, so they were in some sense confined to the plane-based subalgebra of GA. Therefore, in some sense plane-based GA precedes PGA.

Hamishtodd1 (talk) 12:48, 20 October 2023 (UTC)[reply]