
Exploring Physics with Geometric Algebra, Book I [pdf] (2016) - adamnemecek
http://peeterjoot.com/archives/math2015/gabookI.pdf
======
adamnemecek
I've been talking on HN basically non-stop about geometric algebra (aka
Clifford, Grassman, Exterior algebra) which I've been exploring as of late.
I'm not a math person but GA is possibly the most exciting math field to me.
It's a replacement for linear algebra but unlike LA it actually makes fucking
sense (e.g. cross product generalizes beyond 3 and 7 dimensions, but there's a
lot more).

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...](https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-
introduction/)).

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](https://github.com/reloZid/algeosharp)
it’s c# but cross platform.

~~~
madhadron
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
>* 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
Do you happen to be a tauist? ;)

~~~
zardo
I was reading Shakespeare yesterday. I guess it leaked.

4*pi is of course, the most reasonable circle constant.

~~~
vanderZwan
Ah yes, the "when in doubt, annoy everyone equally" approach. I approve.

------
Koshkin
To me, the text feels more like a set of notes that are meant to be read by
the author. Even the "introduction" must be completely incomprehensible to a
reader who is uninitiated in whatever topic the author is trying to present.

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...](http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.pdf).

(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.)

~~~
vanderZwan
The author is pretty much in agreement with you. From the preface:

> _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_

------
Mugwort
The main problem I have with geometric algebra is that it's an alternative
framework to express exactly the same things. Some things are expressed or
derived easier in G.A. e.g. the Kerr solution of General Relativity, electron
Zitterbewegung, etc. The problems with G.A. as I see it... 1) It's a
specialized tool that does not generalize well to simple everyday physics
problems. Don't get me wrong, I like G.A. and learned it from Hestenes' papers
almost 20 years ago. I just don't see the point in trotting out G.A. to solve
certain problems or illustrate some theory any more than I would abandon Gibbs
vector calculus and insist on doing all calculations in 4-vector notation
using field tensors and differential forms. I wouldn't even want to use it to
teach differential geometry. Once someone knows these subjects I'd say go for
it and learn G.A. It's actually very easy once you know the more conventional
stuff.

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.

~~~
vanderZwan
Sorry, but pretty much every argument you make looks like a different flavour
of "I already put a lot of effort into mastering other ways, and now that I
have they work fine for me, I don't see why we should switch to this one."

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.

------
elteto
I loved the axiomatic approach followed in the introduction! It builds
intuition from the very beginning. At some point I could not follow anymore as
the notation and the concepts started getting more complicated and I lack the
math background to keep up.

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?

~~~
JadeNB
> I wonder if there are any linear algebra books out there that follow this
> approach of "build the world, one axiom at a time".

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](http://www.springer.com/series/3423))
and most MAA publications ([https://www.maa.org/press/books/book-
series](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](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.

------
evo_9
Would be cool if it were interactive like this site:
[http://immersivemath.com/ila/index.html](http://immersivemath.com/ila/index.html)

------
agumonkey
what a beautiful pdf

~~~
Osmium
Agreed! I think I've seen it before as a theme, but I'm not sure what it's
called. Wish there was a CSS version available somewhere.

(Edit: thought it looks like it's not using small-caps variants for numerals,
which is a shame)

~~~
wrs
It's a variation on a LaTeX style called classicthesis
([http://www.miede.de/#classicthesis](http://www.miede.de/#classicthesis))
based on Robert Bringhurst's _Elements of Typographic Style_. Unfortunately
this particular version has messed up some things (notably line length), so
use the original!

