
The Exterior Algebra and Central Notions in Mathematics (2015) [pdf] - jpelecanos
https://www.ams.org/notices/201504/rnoti-p364.pdf
======
tw1010
This stuff really isn't that hard once it is described well (which doesn't
need to involve any simplification). I can't wait to see what happens to the
programming community once these notions are embraced with the love and
respect that it deserves. If anything is going to make a big dent in the
field, on a 5-10 year horizon, it'll be this. What we need to do is not to
think of it (modern concepts in pure mathematics, especially from algebra) as
something separate from programming, but to allow it to mix and skew the
fundamental language that we use to reason about coding.

~~~
Retra
I don't think this will make a big dent on account of geometry being
fundamentally simpler and easier to reason about than language in general, and
thus mathematicians dealing in geometry have a much more workable framework
for employing an algebraic approach than do computer scientists.

I think if these ideas were going to be revolutionary, they'd already have
been. They're not new, unfamiliar, or esoteric, they are in standard
curricula.

~~~
trashtoss
Yes but also no. One of the most fascinating applications I've seen is here:
[http://versor.mat.ucsb.edu](http://versor.mat.ucsb.edu) (and in subsequent
work).

There's already a decent tl;dr of the approach on that page, but the tl;dr of
that tl;dr is that conformal geometric algebra seems to provide not only a
rich language of geometric operations but--importantly!--these operations seem
to compose intuitively and interpolate well ("well" in the same way
interpolation between quaternions afford the most-natural interpolation
between 3D transforms).

The work there uses it to develop tools for parametric design...I'd be very
curious to see it extended (e.g. to include time dimensions).

What is notable though is that whereas a lot of the "hype" around geometric
algebra revolves around the ostensible intuitiveness--"look, we only need
Maxwell's equation, singular, in GA"\--actually using it effectively seems to
require acquisition of a lot of vocabulary and concepts (e.g. to make use of
versor you need to know about blades, rotors, etc.)...the learning curve to
use the material is actually steep. That wouldn't prevent it being used as the
backend of some tool, but the idea that GA is intuitive and "easier" than
alternatives doesn't seem to hold up in practice (IMHO)...at least for uses
like these.

Then there's the efficiency issues in that, in general, each term in a
k-dimensional geometric algebra will have 2^k coefficients (and thus adding
two terms is ~ 2^k operations and multiplying is ~ 2^(k+1))...a good
implementation will have a lot of difficult tradeoffs to consider.

~~~
jacobolus
It is intuitive and easier in the sense that when you don’t have that
vocabulary and bag of concepts – trying to work in some other formalism – all
of your arguments end up being much more complicated and cluttered.

As for the number of scalar parameters involved: if anyone wanted to put
significant time and money in, as has been done with matrix computation
libraries, most of the 0×0 multiplications can be skipped, and the rest can be
efficiently SIMDized, etc.

If the concern is bandwidth over the wire, there are often ways to compress
things. E.g. we can take the stereographic projection of a rotor and then
reduce the precision to save a lot of I/O without losing accuracy.

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adamnemecek
I’ve been looking into this stuff as of late (I bought like 400 bucks worth of
books on it). I always kinda liked math but always felt like some parts don’t
make any fucking sense (cross product only in 3d? Nice algebra bro). Also wtf
are complex numbers really about? Geometric algebra resolves all this and it
has insane applications esp for any space related reasoning. Space time
algebra is very much related and it’s also balls to the wall insanity.

Maxwells four equations reduce to a single equation, a simple fraction. It’s
beyond me we aren’t going hard on this.

~~~
tw1010
Don't start with geometric algebra. Start with abstract algebra. It is the
assembly language, if you will, of geometric algebra (and of a lot of other
stuff). It'll also give you a new (from first principles) understanding of
complex numbers. And don't read explanations by programmers of abstract
algebra, those tend to be very shallow and only touch on a few definitions.
Also, don't just read these books. The real gold comes after you've struggled
with proofs and a mountain of exersizes.

~~~
jacobolus
I respectfully disagree with most of this (except for the struggling with
problems part).

A ring/field-theoretic construction of complex numbers as a quotient R[X]/(X²
+ 1), an extension of the field R by the roots of the polynomial, is
interesting/neat, and can be generalized in various interesting ways for
number theory, algebraic geometry, etc., but is not an effective/intuitive
early understanding for someone who wants to use them in geometry, physics,
general data processing, etc., because most of its content is purely
formal/abstract.

Thinking of the complex numbers as quotients of vectors in the Euclidean plane
(under the geometric product) is a much, much more fruitful and enlightening
interpretation, and makes the main theorems of complex analysis make clear
geometric sense.

Most importantly, the basics of geometric algebra are very accessible to e.g.
high school students. It could be used to completely replace typical courses
in trigonometry, partially replace & supplement a Euclidean geometry course,
partially replace & supplement instruction in solving systems of equations and
basic matrix algebra, etc. A curriculum which was wholesale infused with GA
tools and reasoning would give students more powerful tools and more
vocabulary earlier, would help them solve harder problems in most of their
math/physics courses, and would ultimately unify and simplify a lot of the
mathematical modeling tools used throughout any undergraduate technical
education.

To learn group theory, ring theory, etc., you want to start with a nice stable
of explicit concrete examples, explored deeply (I would recommend Nathan
Carter’s book _Visual Group Theory_ as a start for someone with a high school
math or undergraduate engineering background). Otherwise the theorems and
proofs are just symbol twiddling and pattern-matching to some assignment
handed down from above. Ideally students would start in on some examples in
primary/secondary school (tessellations, transformation geometry, modular
arithmetic, some basic number theory, polynomials treated as vectors, ...),
and be pretty well prepared for an abstract treatment by the time they are
undergraduates. Unfortunately there’s not much time for this in a typical
undergraduate mathematics sequence, so most classes in those subjects are done
entirely abstractly from axioms, with sparse examples only examined
superficially. Many if not most students leave these courses very confused
about the meaning of what they have been doing. GA provides some great
examples for an abstract algebra course.

If you care about powerful abstract tools, (both finite and Lie-type) groups
can be embedded in a geometric algebra and studied using GA tools in a similar
way they can be embedded in linear algebra and studied as matrix groups
(“representation theory”). In many cases the geometric algebra representation
is both easier to understand and gives more tools to use. Both projective and
affine geometry can be done with geometric algebra, and differential geometry
can be recast as “geometric calculus” which reveals some structure usually
hidden in the typical differential form version. See the papers at
[http://geocalc.clas.asu.edu/html/GeoAlg.html](http://geocalc.clas.asu.edu/html/GeoAlg.html)
[http://geocalc.clas.asu.edu/html/GeoCalc.html](http://geocalc.clas.asu.edu/html/GeoCalc.html)

~~~
adamnemecek
> Thinking of the complex numbers as quotients of vectors in the Euclidean
> plane (under the geometric product) is a much, much more fruitful and
> enlightening interpretation, and makes the main theorems of complex analysis
> make clear geometric sense.

On point! Like I feel my understanding is grounded in something I can reason
about myself as opposed to trying to see the relationships that someone else
came up with.

I also feel like most commenters misunderstood what I meant when I said “what
complex numbers are about”.

What other math books can you recommend?

~~~
jacobolus
What $400 of books did you start with, and what else have you read? And what
is your other mathematical background?

------
unao
_The neglect of the exterior algebra is the mathematical tragedy of our
century._

Why has it been neglected?

~~~
auggierose
If you look at the paper you will see that it uses a relatively advanced
mathematical language. Did you understand the significance and meaning of
exterior and geometric algebra from that? If not, then there you have a
partial explanation for it. It is much easier to just understand vectors and
matrices and maybe tensors for practical and applied work.

~~~
yiyus
I do not agree with this. The simplest mathematical concepts can look very
complicated when presented in a formal way.

Learning the geometric product in high school wouldn't be more difficult than
learning the dot and cross products, and would make obvious difficult to grasp
concepts as complex numbers and even quaternions.

There are historical reasons for which we do not learn this from another point
of view, and in my opinion it is, indeed, a great tragedy. A tragedy that I
hope will be remedied some day.

Disclaimer: I deal everyday with 3D rotations. Euler angles have been
traditionally used in my field, but they present many problems. Everybody
knows we could do better with quaternions, but very few people understand
them. I have shown many people how to interpret what quaternions are from
geometric algebra concepts and I have not yet found anybody who doesn't think
it is much more approachable that way.

~~~
auggierose
I do not disagree with you! For example the book
[http://faculty.luther.edu/~macdonal/laga/](http://faculty.luther.edu/~macdonal/laga/)
gives an elementary introduction to the subject.

I have learnt about geometric algebra just this year, and applied it to
compute the graphics in an app I wrote for a customer. It was a real eye
opener, concepts that I struggled with before were really simplified by using
geometric algebra.

BUT: I would never have guessed the usefulness of it for me from this paper.

~~~
zardo
Second recommendation for laga.

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daniel-levin
For a gentle appetiser, see Spivak's Calculus on Manifolds. It introduces
exterior algebra as a means to and end in doing computations on (embedded)
manifolds. At first, the definitions seem opaque and the formalism clunky. At
the end, Spivak recovers Stokes' theorem as a computation. This was a breath
of fresh air for me after I took a warped version of Calc III.

------
tobbe2064
Doesn't the example show that

    
    
        (ii): e1□e2 = -e2□e1
    

and

    
    
        (iii): (e1□e2)□e3 = e1□(e2□e3)
    

contradict each other? In the end we end up with

    
    
        (e1□e3)□e5 = e5□(e1□e3) 
    

which seems to go against (ii)

~~~
aquamongoose
No: (ii) only holds when e1, e2 come from the original vector space (not the
product).

