
“Reforming the Mathematical Language of Physics” (pdf) - jacobolus
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
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sriram_malhar
Tangentially related is this lovely piece by Tom Apostol at Caltech: [A visual
approach to calculus](<http://www.mamikon.com/VisualCalc.pdf>).

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phaet0n
If you're interested in applications to computers science, and computer
graphics, I highly reccommend <http://geometricalgebra.org/>

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yobbobandana
As a former student of physics, this does appear quite interesting. The
article feels concise and well-structured, and the math seems very elegant.

For physics and math students I recommend starting at page 9. The 8 pages of
rationale seem a little unnecessary, with the math mostly speaking for itself.

I'm looking forward to giving this a more detailed read later.

~~~
maxbrunsfeld
As another former physics student, I'm excited about this too. I think my
understanding of physics would benefit greatly if the various mathematical
formalisms used to describe its different aspects were unified systematically.

I'm starting to read this article carefully right now.

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jacobolus
I ran across Geometric Algebra a couple months ago, and am convinced it’s one
of the most important things I’ve learned about in years.

Some quick summary / commentary to motivate reading the link:

\- By the 19th century, the complex numbers had become a generally acceptable
concept, for example with lots of attempts to prove the fundamental theorem of
algebra. The idea that the complex numbers could be thought of as 1–1 mapped
to a two-dimensional plane was widely known. Complex numbers used as an
algebraic proxy for points in the plane were useful for solving geometry
problems that Cartesian analytic geometry couldn’t manage.

\- Many thinkers were trying to extend these ideas to three-dimensional space.
Hermann Grassmann, who invented GA in the 1840s, was one of many, but in many
ways he got things “right”. Another thinker who worked extensively on this
problem was Hamilton, who devoted decades to the quaternions (a sort of
generalization of the complex numbers with three imaginary units i, j, k
instead of one) which can be quite useful for characterizing rotations and are
still used in niche physics applications even though they’ve been otherwise
marginalized by matrix approaches.

\- It turns out that there is a quite natural relationship between imaginary
numbers, the vector part of a quaternion, and geometric algebra. As Hestenes
describes in the linked essay, the imaginary unit, a “bivector”, can be
thought of either as an operator which performs a 90° rotation on vectors in
the plane, or as a “directed area” with unit magnitude. A quaternion’s vector
part is just a combination of three unit bivector components of geometric
algebra in 3-dimensional space. This gives complex numbers and quaternions a
nice intuitive geometrical interpretation, and makes them a lot less foreign
seeming to students.

\- The traditional/mainstream conception of vector algebra (which ended up
winning the 19th century fight among tools for the algebra of space) uses a
dot product and a cross product, which are defined to be, respectively, a
scalar and a vector. Cross products constantly come up in the physics of
rotations, and in trying to compute areas and volumes. Under GA, the “outer
product”, mathematically dual to the cross product, is seen to be a bivector.
This is a much better description because the behavior of vectors produced by
cross products is quite different than other vectors, which means that
considering them to be the same type of object results in a great deal of
confusion.

\- Geometric algebra has a natural relationship with projective geometry
(homogeneous coordinates, Plücker coordinates). Such mathematics is used
constantly in applications like computer graphics.

\- Physicist David Hestenes has spent most of his career since the 1960s
elaborating and advocating GA. In exploring various areas of physics, he found
that in almost every case GA was at least as convenient as the alternative
mathematical formulations, and often quite insightful. For instance, some
quantum phenomena which were previously considered “counterintuitive” or
strange become quite obviously necessary and natural conclusions of their
mathematical context in GA.

\- GA is still far from mainstream. As far as I can tell, it is taught to
almost no undergraduates even though it is pedagogically and operationally
superior to the alternatives. It mainly comes up in very technical advanced
physics courses and extremely abstract mathematical proofs. Reading some of
these, one would never realize how broadly applicable and powerfully
connective its ideas are.

\- Geometric Algebra is (I firmly believe) the future of geometric education.
It should be taught to all students of mathematics, physics, engineering,
computer graphics, computer vision, and similar fields, perhaps starting in
high school. It simplifies both understanding and computation; for instance,
many problems can be done coordinate free, making their solutions easier to
tackle and more intuitively relevant.

~~~
Dn_Ab
I am currently working my way through some geometric algebra (as well as
algebraic geometry and topology) texts and building a [programming] library
incorporating what I read to reinforce my mathematical intuition. I completely
agree about how awesome Geometric Algebra is. In fact its emphasis of tying
algebra with geometry makes it much more useful than for just teaching
geometry. It makes a wonderful foundation for mathematical intuition. For self
study I particularly like such subjects because they take a whole bunch of
stuff and put them under one roof.

Grassmann's story is semi-tragic. He was a school teacher who created a
powerful algebra of space where you could compute with not just points and
vectors but with lines, planes, surfaces and other vector subspaces without
being hampered by a particular coordinate system. Geometric Algebra can
subsume complex numbers, quaternions and plane geometry - projections,
rotations etc and easily extend them to higher surfaces. Matrices play less a
role here. Hestens also puts forward that Geometric Calculus can contain
matrix calculus, linear algebra, lie algebra and groups and Differential
forms, see [1]. But because Grassmann was so ahead of his time and did himself
no favours by writing a dense, super long text with lots of philosophy
interwoven not many bothered to read or could understand it. Later, Clifford
came and made extensions, cleaned it up and the subject gets named after him
as Clifford Algebras, though not by his choice. But Clifford died early so the
more geometrically unwieldy Vector Analysis won out. A setback for
physics?...Angular momentum makes so much more sense as a bivector. From [2]:

 _Fearnley-Sander writes in [27]:-

All mathematicians stand, as Newton said he did, on the shoulders of giants,
but few have come closer than Hermann Grassmann to creating, single-handedly,
a new subject._

To read more about Grassmann I recommend: A History of Vector Anaylsis by
Michael Crowe, well written, I couldn't put it down.

[1] <http://geocalc.clas.asu.edu/pdf/DIF_FORM.pdf>

[2] [http://www.gap-
system.org/~history/Biographies/Grassmann.htm...](http://www.gap-
system.org/~history/Biographies/Grassmann.html)

~~~
copper
Perhaps the most interesting part, to me, was the goal of making geometric
algebra operations computationally efficient. While I'm still reading (and
enjoying) the original paper, I must confess an inability to see how it's more
efficient than a standard vector representation. Would you have any pointers
on that?

~~~
Dn_Ab
At the moment, the core of the gains are in the conceptually clean framework
at the cost of speed. Kind of like going from C to Python. It will help your
linear algebra by strengthening your geometric intuitions, stuff like how to
figure out if a plane intersects a sphere or if a line intersects a plane or
SLERP are much are easier for the novice. You solve it algebraically but the
objects you are reasoning with are geometric entities so you can't help but
build your intuition on both.

Computationally though, they take more operations and elements (2^dim). And
there is an order of magnitude difference in speed - 100x if implemented
naively. But there are optimizations that can get it to 2x - 7x [1]. But
apparently for full generality, 3D geometry is best done in a 5D conformal
geometry which requires 32 element representations but I haven't gotten that
far yet. You can make it more efficient by taking advantage of the sparseness
of most structures and being able to scale up to more information as required
by filling more indices. Something that cannot be handled as cleanly in
standard vector based methods. Strangely this method of conformal geometry is
covered by a Patent so commercial usage in the US requires licensing
arrangements [2].

[1]
[http://staff.science.uva.nl/~fontijne/phd/fontijne_phd_compr...](http://staff.science.uva.nl/~fontijne/phd/fontijne_phd_compressed.pdf)

[2][http://www.google.co.uk/patents?id=0CkWAAAAEBAJ&pg=PA7&#...</a>

~~~
copper
Thanks, that's interesting, and seems more in line with what I figured: a
_lot_ easier to reason about, and a much cleverer coder than me to do it
properly :)

(As an aside, one of the inventors listed on that patent happens to be the
person who wrote this article.)

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troymc
This paper / talk was given in 2002, so of course it doesn't tell you what
happened with the adoption of, or interest-level in Geometric Algebra (GA)
since then. A quick search on Amazon.com turns up over a dozen recent books
using GA in physics, computer science and engineering.

