[HN Gopher] Reforming the Mathematical Language of Physics (2002...
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Reforming the Mathematical Language of Physics (2002) [pdf]
Author : alokrai
Score : 57 points
Date : 2021-02-14 21:39 UTC (2 days ago)
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(TXT) w3m dump (worrydream.com)
| ktpsns wrote:
| I think the author proposes geometric algebra for widespread use
| in teaching fundamental physics. That's a good idea, it unifies a
| lot of things and brings new insights! Hestens seems to be quite
| popular with his opinion, see for instance
| https://link.springer.com/chapter/10.1007/978-94-011-5036-1_...
| and https://en.wikipedia.org/wiki/Geometric_algebra
|
| Very interesting! But it requires really a skilled lecturer to
| make students understand the concepts. I've heard advanced
| mathematical physics by a, well, bad lecturer, who taught
| everything on some abstract level which I never heard from
| previously (and after). I had to learn differential geometry and
| all that later on my own in order to graduate in some topic in
| general relativity.
| abelaer wrote:
| (Disclaimer: haven't read the whole thing) This just seems like
| one of these situations: https://xkcd.com/927/
|
| The fact that the same physical system can be described by very
| different kinds of mathematics is often very illuminating and
| interesting, rather than confusing. It's one of the reasons that
| physics can inspire mathematics, rather than maths just being a
| tool that physicists use. Different types of mathematics often
| allow you to emphasise different properties of the same system,
| so being comfortable in more than one language brings you freedom
| and new insights.
| zentiggr wrote:
| I've never had a chance to dive deeply into most 'advanced'
| math topics, but I've readd more than one of Hestenes's papers
| (and cohorts) and honestly GA looks to be so much clearer,
| concise, and more easily worked than having to blend together
| vectors, matrixes, and other constructs in order to do the same
| work in more complicated ways, with extra notational layers.
|
| Again, I don't use any of this in real life, so YMMV, but it
| looks very clean, fast, and intuitive compared to (waves at
| everything else).
| Ceezy wrote:
| There is some very good propositions in this paper. Physics
| should definitively have less redundant mathematics formulations
| of problems. Hamiltonian physics is quite close to Newtonian in
| comparison quantum physics.
| jojobas wrote:
| First I thought it would be about dumbing down the existing maths
| so more people can understand.
|
| Some 10 pages in I realized it's explaining the concepts I've
| internalized years ago in a way I can't comprehend at all.
| mncharity wrote:
| Hestenes; The Genesis of Geometric Algebra: A Personal
| Retrospective (2017)[1]
|
| [1]
| https://link.springer.com/content/pdf/10.1007/s00006-016-066...
| carapace wrote:
| Hestenes on Victor's radar, that seems promising! :)
|
| I like to mention on such threads:
|
| "Geometric Algebra for Electrical and Electronic Engineers"
|
| > In this paper, we explicate the suggested benefits of
| Clifford's geometric algebra (GA) when applied to the field of
| electrical engineering. Engineers are always interested in
| keeping formulas as simple or compact as possible, and we
| illustrate that geometric algebra does provide such a simplified
| representation in many cases. We also demonstrate an additional
| structural check provided by GA for formulas in addition to the
| usual checking of physical dimensions. Naturally, there is an
| initial learning curve when applying a new method, but it appears
| to be worth the effort, as we show significantly simplified
| formulas, greater intuition, and improved problem solving in many
| cases.
|
| https://ieeexplore.ieee.org/document/6876131
| yesenadam wrote:
| I recently read John Vince's _Geometric Algebra for Computer
| Graphics_ , which seemed excellent - written, the author
| explains, because he realized you had to virtually be a
| mathematician to understand the books in the field, and he wanted
| to write something much more accessible. It sounded great until
| Chapter 11, _Conformal Geometry_.. Various
| arguments have been proposed for adopting this model such as,
| conformal space: * is homogeneous and consequently removes
| the origin as being a special point * supports points and
| lines at infinity * provides a single geometric mechanism
| for representing lines, circles, planes and spheres, which
| introduces a rare quality of elegance to problem solving *
| enjoys all the normal features of Euclidean space. A model
| of space that possesses such a range of positive features sounds
| too good to be true! But remember, in mathematics there is no
| such thing as a free lunch. So what price must we pay for this
| model?
|
| Then Vince explains that the conformal model, "an alternative
| space to Euclidean space for solving 3D geometric problems", is
| covered by a patent..of Hestenes, _US Patent 6,853,964 System for
| encoding and manipulating models of objects_. Vince says
| "Licences to employ this system for commercial purposes are
| available through the patent holders, although there are no
| restrictions for academic research and educational use. All of
| the above may be a small price to pay for the benefits associated
| with the conformal model, and I will do my best to explain the
| model in this chapter."
|
| Uh what?! So it actually costs money to use... That rather put me
| off Hestenes' noble mission. It seems...bizarre that what seems a
| whole subfield of mathematics is patented, rather different to
| patenting a particular algorithm or technique. Very strange,
| after reading so much of Hestenes' wondering why the superior GA
| isn't used everywhere. (Is it to make the world a better
| place..or to make Hestenes rich?..) I stopped reading at that
| point.
|
| As far as using Geometric Algebra in a Euclidean setting goes,
| most of my programming is in 2D, but when I do 3D next I will
| consider using it, it seems very cool. Most of the
| features/differences/advantages over the usual vector/matrix
| geometry only appear in 3 or more dimensions. Vince's book is
| great at explaining it, highly recommended.
| gg80 wrote:
| It seems the patent has been discontinued:
| https://patents.google.com/patent/US6853964B1/en
| yesenadam wrote:
| Oh thank you. Hmm "Status Expired - Fee Related .. 2022-07-13
| Adjusted expiration" .. is that like when your power gets cut
| off from not paying the bill? p.s. Vince's book was published
| in 2008.
| benrbray wrote:
| Is this... legal? How on earth was a patent like that approved,
| and surely it wouldn't hold up to scrutiny, right?
| dmitrybrant wrote:
| Most people familiar with the US patent office will agree
| that it's totally broken. The reviewers are not scientists,
| and aren't required to understand what's being proposed in a
| patent. They simply perform a search for prior art.
|
| As a result, all kinds of pseudoscience has gotten
| "patented", for example this patent for detecting
| "tachyonized" materials (you know, materials that have been
| infused with tachyons, the hypothetical particles that travel
| faster than light):
|
| https://patents.google.com/patent/US7760357B2/en
| jcranmer wrote:
| It looks like the patent was approved in between the CAFC
| decision of State St. (subject matter is patentable if it
| produces a "useful, concrete, and tangible result") and
| SCOTUS's Bilski decision (State St. is on crack and not good
| case law). Bilski and later Alice have made it rather clear
| that these kinds of abstract patents aren't going to fly, not
| that it stops patent attorneys from pretending otherwise.
|
| So this patent very likely wouldn't hold up in court. It was
| approved in a time when the USPTO more or less gave up on
| adjudicating patentability (thanks to the State St.
| decision).
| WildParser wrote:
| Good that people sometimes remember that a more modern base for
| physics would be nice.
|
| Doing electrodynamics without using alternating products and
| differential forms feels so wrong...
| wolfram74 wrote:
| This line on page 6 confuses me a bit --For example, for
| applications to rotations, quaternions are demonstrably more
| efficient than the vectorial and matrix methods taught in
| standard physics courses
|
| And then claims that aerospace engineers mostly use quaternions.
| It's from 2002, so it's sense of what's in vogue might be dated,
| but I'd understood at least in the graphics manipulation world,
| the vector/matrix methods were dominant. Those are separate
| industries, so both statements could be true, anybody have
| opinions?
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