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A few nits: * GA is not a replacement for linear algebra. It's a vector space equipped with a certain product. All of linear algebra still applies to it. There
by madhadron 9y ago
A few nits:
* GA is not a replacement for linear algebra. It's a vector space equipped with a certain product. All of linear algebra still applies to it. There's just more structure. What it does replace is Gibbs-Heaviside vector analysis.
* Unifying a line and a circle is a classical part of projective geometry. It can be written in Clifford algebra straightforwardly, but the identification is centuries old at this point.
* Quaternions and octonions are the generalization of complex numbers beyond the plane. A lot of stuff just happens to reduce to the same thing in 2D.
* Maxwell's equations are reduced to one in any relativistic formulation. The hard part is rewiring your brain to work in spacetime instead of space plus time.
* It doesn't replace differential geometry. It's a convenient notation for parts of it, the way that bra-ket notation in quantum mechanics doesn't replace the underlying structure, it's just a notation adapted to it.
But, yes, it's cool stuff.
- zardo 9y ago>* Quaternions and octonions are the generalization of complex numbers beyond the plane. A lot of stuff just happens to reduce to the same thing in 2D. Would an octonion by any other name, not smell as sweet? The even graded sub-algebras in GA are isomorphic to complex numbers, quaternions, etc... For me at least, coming at the same structure from a completely different direction is the best bang for the buck in increasing understanding.
- vanderZwan 9y agoDo you happen to be a tauist? ;)
- zardo 9y agoI was reading Shakespeare yesterday. I guess it leaked. 4*pi is of course, the most reasonable circle constant.
- vanderZwan 9y agoAh yes, the "when in doubt, annoy everyone equally" approach. I approve.
- vanderZwan 9y agoOf course, you haven't lived until you did as the astronomers do, and did exams stating that you may consider Pi = 4 and Pi^2 = 10
- adamnemecek 9y ago> All of linear algebra still applies to it. Why would I use it tho? Sure, if I upgrade from a tricycle to a warp drive space ship, I can still use the tricycle by why would I when I need to haul shit across space. > Unifying a line and a circle is a classical part of projective geometry. Sure, but in Euclidean geometry, you don't really do this. Like it's essential go GA. > Quaternions and octonions are the generalization of complex numbers beyond the plane. Yes, I meant that it generalizes the concept of complex plane, not the plane itself. > It doesn't replace differential geometry. That is correct, however I still believe that you can rephrase certain DG questions as GA questions so I said that it replaces some of it.
- vanderZwan 9y ago> * GA is not a replacement for linear algebra. It's a vector space equipped with a certain product. All of linear algebra still applies to it. There's just more structure. What it does replace is Gibbs-Heaviside vector analysis. Perhaps GP meant to say "GA subsumes linear algebra". Yes, LA is still valid, but a lot more intuitive within the context of GA if you ask me.
- ajkjk 9y agoIf I recall correctly Maxwell's equations are usually reduced to 2 in relativistic formuliations: dF = 0 and d⭑F = J. GA purports to be useful in combining F and ⭑F into a single object (or rather having a derivative operator D = (d + d⭑)) so that DF = J captures everything. But I don't find that that insightful.