Exploring Physics with Geometric Algebra, Book I [pdf] (2016)
peeterjoot.com
peeterjoot.com
Basically it unifies the geometric concepts of a line and a circle (in GA, a line is just a circle with points on the circle at infinite distances). As a result, you can do very useful computations very easily. E.g. a point times a point gives you a line connecting the points. It kind of makes sense, because generally, you care more about objects around you as opposed to all objects in one direction to infinity.
It's also a generalization of the complex plane beyond 2D. I finally kinda understand complex numbers.
So much shit is easier in this formalism, esp. anything related to reasoning about space. This includes but isn't limited to computer vision, crystallography, graphics, physics (all the quantum shit makes sense in GA, Maxwell's equations too, in GA, Maxwell's equations are reduced from 4 to 1!!!, https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-v...).
For example there's something called CSVM (Clifford SVM) which learns any given manifold, unlike an SVM which learns a hyperplane.
It also replaces quite a bit of differential geometry[0], as you you can answer questions from differential geometry without resorting to calculus.
I feel like some sort of burden has been lifted from me, because I feel like so much higher mathematics is a lot more accessible in this formalism. But your mileage might vary.
[0] pls note that some time ago I didn't give a fuck about differential geometry but somehow I do now.
Check out this project https://github.com/reloZid/algeosharp it’s c# but cross platform.
I think the day is coming for an explosion of GA in everything, since it has been simmering behind the scenes for many years thanks to many, Dave Hestenes[2] in particular. I love his book New Foundations for Classical Mechanics.
I program in APL and J[3], and hope that the array programming will segue into a winning GA library/module and way of approaching coding GA to make it more of a player in my work rather than my hobby.
EDIT: Versor.js is also available for a spin! [4]
[1] http://versor.mat.ucsb.edu/
[2] http://geocalc.clas.asu.edu/
[3] jsoftware.com
[4] https://github.com/weshoke/versor.js/* 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.
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.
4*pi is of course, the most reasonable circle constant.
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.
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.
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.
Geometric algebra is pretty cool, but it is a potential replacement for vector algebra, not linear algebra. It is defined on a vector space, which means that the theory of linear algebra (concepts like linear transformation, bases, etc.) applies.
As a sidenote, reducing Maxwell's equations in the 4-equation Heaviside form to a single equation is not necessarily a boon. Each of the four has a clear physical meaning in the vector calculus formulation, which may be lost when moving to a single equation.
Geometric Algebra is a beautiful subject (best explained in Hestenes' Oersted lecture). On the other hand, in practice, as the proposed "geometry of physics" it has been long replaced by the machinery of differential forms, with a vast amount of literature devoted to it; a short introduction to this you can find, for example, in an article by Ted Frankel: http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.p....
(The Geometric Algebra being discussed here is not to be confused with the subject of the marvelous book by Artin that bears the same title.)
> These notes are more journal than book. You’ll find lots of duplication, since I reworked some topics from scratch a number of times. In many places I was attempting to learn both the basic physics concepts as well as playing with how to express many of those concepts using GA formalisms. The page count proves that I did a very poor job of weeding out all the duplication. These notes are (dis)organized into the following chapters
2.) I don't know how things are now but in the past G.A. had the feeling of a cult theory. Unlike most cults, this one had substance. The problem learning the material first with G.A. is that you won't be able to understand other papers, books or solved problems of others. You'll be dependent on using G.A. and unable to make sense of Goldstein or Spivak.
3.) I'm all for alternative ways of doing things. If you have time to burn there's a "Bird Tracks" method of doing calculations in Lie algebras which is very different than the traditional Cartan matrix, Dynkin diagrams. I advise you to avoid learning these topics the first time around in a weird mathematical language. After you know you're way around these topics go for it.
2.) Diversify. I picked it up because I felt like I can mold this better for my problems than LA. And I'm more about publishing code than publishing papers. So I care more about making it "my own" than explaining things to people.
3.) Coincidentally Lie algebras are also easier in GA. Look at this https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8... and don't tell me that the symmetries seem more circular than linear.
Because you seem to suffer from survivor bias, that's why. By comparison, I am a physics drop-out. Allow me to give you the view from below.
First, I know this pdf is about GA in relation to physics, but you're throwing all kinds of names and terms around that only make sense to people who already have studied physics; conscious or not, that's basically gatekeeping via jargon.
Claiming that we'll be unable to make sense of Goldstein and Spivak is also directly contradicting your claim that it's easier once you know the "more conventional stuff" - so learning one thing after the other is easier, but not the other way around? Why would it not be easier to learn the more conventional stuff after you have a solid GA fundation?
Both my own experience and that of many commentators here seem to indicate GA is inherently more intuitive to grasp than the disparate mathematics it connects, precisely because it all seems more logically structured and connected. And perhaps the people who didn't manage to get through have a better idea of what is more intuitive to learn that the people who did here. Similarly, given that GA connects so many things together, I have a hard time believing that it is a "specialised tool", but since I never had to deal with most of the stuff you mention I guess I can't be a good judge of that (did I mention the gatekeeping?).
Anyway, why should people first have to slog through all the other stuff, sometimes learning it more by rote than insight? We don't force kids to learn assembly or punch-cards before we let them play with a higher level programming language either.
I'm not suggesting we throw all the old physics books out the window, but we also managed to stop doing science in Latin. That removed a huge barrier of exclusion, has not lead to a loss of knowledge, and making the field more accessible to everyone has only benefited humankind.
One of my favorite undergrad classes was linear algebra. I wonder if there are any linear algebra books out there that follow this approach of "build the world, one axiom at a time". My sweet spot would be something aimed at someone who is not a mathematician but has some math background (like an engineering unndergrad) and can follow along with a bit of effort and study. Any recommendations?
Patience. It relies on mathematics that are probably foreign to you, so it makes sense.
Basically any math book—so most textbooks don't qualify as math books—will do things this way. SUMS (http://www.springer.com/series/3423) and most MAA publications (https://www.maa.org/press/books/book-series) are good places to look. I didn't see one in a glance at the MAA series, but SUMS has Further Linear Algebra (http://www.springer.com/us/book/9781852334253). I haven't used it personally, but it looks like it has a chance of being the sort of thing you want.
(Edit: thought it looks like it's not using small-caps variants for numerals, which is a shame)