
A Unified Mathematical Language for Physics and Engineering (1996) [pdf] - signa11
http://www.mrao.cam.ac.uk/~clifford/publications/ps/dll_millen.pdf
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jacobolus
For those interested in the history, I highly recommend Crowe’s book _A
History of Vector Analysis_. It’s written a bit from the perspective of the
dominant “modern” system of vectors from Gibbs/Heaviside, so it probably
somewhat undersells the significance/usefulness of both Hamilton’s quaternions
and Grassmann’s vectors (or what Hestenes calls “geometric algebra”), but the
historical and biographical details are fascinating.

Also recommended is all of Hestenes’s work (e.g. _A New Foundation for
Classical Mechanics_ , his Oersted Medal lecture, various papers), as well as
the book _Geometric Algebra for Computer Science_ by Dorst, Fontijne, & Mann.
The latter is a bit light on the mathematical formalities but gives a good
introduction that could get someone started using geometric algebra in
practice in e.g. robotics, computer graphics, etc.

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jwmerrill
Dorst, Fontijne, & Mann is the best pedagogical introduction to geometric
algebra that I have seen. I spent a couple years very excited about the ideas
in Hestenes' papers, and Doran and Lasenby's book, but unable to actually
calculate anything. Dorst, Fontijne, & Mann have very good exercises, and
build up the general framework slowly and clearly.

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jperras
When I took a graduate class in General Relativity under Robert Brandenberger,
one of the more unconventional (and brilliant) approaches he took with his
students was to introduce the formalism of geometric algebra quite early on,
along with tetrads and spinors. While confusing at first, it proved to be
extremely enlightening when Maxwell's equations were brought into play.

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madengr
Were Maxwell's equations still condensed to 4 equations (Heaviside's equations
in Gibb's vector notation), or something less concise, as I believe they were
with quanternions?

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phkahler
I believe they become Maxwell's Equation, and it's super simple.

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throwaway283719
Expressing Maxwell's equations using the six-component electromagnetic tensor
F, they become

    
    
      dF  = μJ
      d*F = 0
    

where μ is the magnetic permeability of the vacuum and J is the
electromagnetic 4-current. The operator 'd' is the differential operator from
exterior algebra, and the '*' is the Hodge dual.

Using the bivector field F = E + iB from geometric algebra, they become

    
    
      DF = μJ
    

where μ and J are as before, and D is the covector derivative.

For comparison, using the traditional Gibbs/Heaviside notation Maxwell's
equations are

    
    
      ∇ . E = ρ/ε
      ∇ . B = 0
      ∇ x E = -∂B/∂t
      ∇ x B = μ(J + ε ∂E/∂t)

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MisterMashable
I studied several of Hestenes' papers around ten years ago. Most of them are
computational in nature and do not proceed to do geometry in an axiomatic way.
This isn't to say he didn't write such a paper but I never found one and I
read nearly all his extent papers. The thing I liked most was his derivation
of the Kerr metric for a rotating black hole. I couldn't find anything wrong
with it and I could actual understand every step, unlike Kerr's original
paper. Other than that GA has done absolutely nothing to help gain a better
understanding of QFT, GR, string theory, Lie groups etc. (for that an in depth
understanding of differential forms is best. Physicists, even mathematicians
who truly understand the magic of differential forms are even more rare than
programmers with a deep understanding javascript!!!) IMO Geometric algebra is
a useful collection of interesting computation tools. It reminds me of Pedrag
Cvitanovic's "bird track" diagrams to calculate representations of Lie
algebras. Nothing unique or fundamental is gained but it is remarkable that a
completely orthogonal viewpoint to some very traditional topics exist. Other
examples (from mathematics) of surprisingly novel viewpoints of traditional
topics include Kuratowski who has a very unique approach to general topology
and F. Riesz came up with the notion of "nearness" which simplifies difficult
theorems in functional analysis but it could be used to recreate all of basic
mathematical analysis.

~~~
westoncb
It seems like the point isn't to gain anything unique or fundamental in
physics or mathematics directly, but rather that, since it has the potential
to unify the language used across a number of fields in mathematics and
physics, that the adoption of a common language could eventually lead to
significant progress. So (if it is in fact an effective unifier), it is
something unique and fundamental in the realm of tools—though not within the
fields the tools would be applied to. Or do you not think it would be an
effective unifier (in the sense of communication) after all?

~~~
MisterMashable
From experience I would have to say 'differential forms' seem to fit the bill
for a unifying mathematical language as applied to geometry and physics. GA
seems to me as more of a computation tool. Differential forms are pretty
standard in many maths and physics texts. Many paper on the preprint Arxiv use
differential forms. The only thing about DFs is that they are very efficient
for communicating ideas and doing proofs. For practical calculations they
don't really simplify anything (actually get in the way) but it's easy to
transform them into the usual vector tensor notation. GA seems less flexible
in this regard, you take the product it uses and you either like or lump it.

~~~
jwmerrill
There's a bit of discussion of how differential forms fit into the geometric
calculus framework in [1], which is probably the most concise and readable
introduction to the geometric calculus approach to differential geometry. All
the machinery of forms is available as part of geometric calculus.

Geometric algebra/calculus has a more direct way to deal with metrical
information using the dot product. Forms only use the wedge product: in
problems where the dot product would be useful, forms simulate it by applying
the hodge dual twice, which is a less intuitive and less direct way to get the
job done.

[1] The Shape of Differential Geometry in Geometric Calculus, see section 19.4
for the bit about forms
[http://geocalc.clas.asu.edu/pdf/Shape%20in%20GC-2012.pdf](http://geocalc.clas.asu.edu/pdf/Shape%20in%20GC-2012.pdf)

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irickt
This 1996 paper is an introduction to Geometric Algebra, which is in effect a
coordinate system where some manipulations are greatly simplified.

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jl221
My resident core-stability expert (who might now have to learn about GA)
alerted me to this thread. Can't say how happy I am that people are looking at
it -- am happy even with the useful criticisms (have just emailed the link to
the other Lasenbys, Doran, Dorst, Hestenes). Another similar semi-popular IEEE
article (without equations) will appear soon.

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lapsedengineer
I really enjoyed reading it but I got stuck on the equation at the bottom of
page 2. Why is the condition satisfied equal to -1? Why not 1 or 0?

Thanks :-)

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Osmium
The equation is a definition; the letters i, j and k do not represent
variables :)

Edit: If you want a more detailed explanation, the Wikipedia entry on
Quaternions might be helpful (specifically the multiplication table at the
top).

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tel
Grassmann/Exterior Algebra is really neat. It also gives a natural deduction
of a lot of basic multidimensional calculus concepts and has a nice lead-in to
differential geometry.

~~~
MisterMashable
Gauss-Bonet all the way! :)

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_delirium
The survey of historical background here is nicely done. Accessible and quite
concise given the amount of material it surveys, but with substance.

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yeureka
I remember being exposed to Geometric Algebra by Geomerics who were bragging
of their new approach to global illumination when they were still a relatively
new company.

Their demos were quite impressive and they sparked a lot of interest in GA in
the graphics programming community.

EDIT: actually, I just realized that the founder of Geomerics ( Chris Doran )
is quoted as one of authors in this paper.

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abdulhaq
For those who want to learn geometric algebra I strongly recommend Linear And
Geometric Algebra by Macdonald. I taught myself this subject entirely from
this book (making sure I did all the exercises) and even enjoyed the process!
Slightly weirdly an application of the algebra became useful at work shortly
after finishing the book.

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ccorcos
At the bottom of page 15 they have a reference from 1999 -- after the
publication of this paper!

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ISL
The wonderful thing about standards is that there are so many from which to
choose!

Edit: With regard for dang's call for substantive comments (are koans not
substantive?):

In physics, we emerged from a forest of units to a standardization on CGS
units, then we changed gears to SI. CGS remains no less elegant than the day
it was invented; electrodynamics is beautiful when viewed through the lens of
CGS.

Languages are tools, we shouldn't expect one of them to suit all situations.

~~~
todd8
Changing conventions for units and moving toward standardizing across all
scientific fields to the SI units (meters, etc.) is useful, but this article
isn't about changing units. This idea of a new set of abstractions for
representation and exploration of physics is a much much more powerful idea.

When I first learned special relativity, I remember struggling with my
intuition and having to rely on unfamiliar formula to find the answer to even
simple problems. The farther one goes in physics the more one has to trust the
equations (electo-magnetics, special and general relativity, quantum physics,
etc.) and the math itself starts to obscure whats happening. Having simpler
representations and more powerful abstracts is an exciting possibility. I saw
this again when trying to solve some simply stated problems (e.g. a particle
falling off of a frictionless sphere), mathematics like Lagrangians which I
didn't know when I first attempted this problem make the solution so much
easier.

As an analogy for those that aren't really interested in the mathematics,
these ideas are a bit like the jump from algebra and infinite series math to
integral calculus. Although, in theory, one could solve many problems of
physics without calculus (see for example [1]), the use of calculus
immediately opens up a better understanding and the ability to describe and
solve more realistic problems (like cars that don't travel at a constant
speed).

[1] "Feynman's Lost Lecture: The Motion of Planets Around the Sun" by David
Goodstein

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cwhy
Please.. Do the same for machine learning and statistics...

~~~
tel
What do you feel is the division between them? I feel like probability is
pretty close to the "unified language of ml and stat"

