
Clifford Algebra: A visual introduction (2014) - mcbits
https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/
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imh
I went a little off the deep end a bunch of years ago learning geometric
algebra. For a similar, but slightly different and cool take, check this out
[0]. It's a take on multivectors, dyadics, and differential forms, which
really shine in E&M. It's a shame this isn't taught to undergrads with linear
algebra and vector calc.

[0]
[http://onlinelibrary.wiley.com/book/10.1002/0471723096](http://onlinelibrary.wiley.com/book/10.1002/0471723096)

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ajkjk
I like the idea of Clifford Algebras a lot, though this article is written
like the whole thing is crackpottery and has a lot of weird irrelevant rants.

It feels extremely natural to incorporate bi- and tri-, etc, vectors into
physics -- far more natural than the awkward Pauli matrices and pseudovectors
and pseudoscalars that it's normally explained in terms of. I'm totally
convinced that it's correct to, for instance, interpret the magnetic field as
a bivector-valued field.

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westoncb
I'm assuming you're referring to:

> _This entire analysis of Clifford Algebra was based on my own foundational
> assumption that mathematics is not a human invention, but more of a
> discovery of the essential principles of computation in the brain._

I don't see what the issue with that is. It's an interesting hypothesis, and
he's not alone in thinking about it. It's already a common stance that
effective/valuable/interesting mathematical structures are direct
consequences/reflections of the universe's own structure, and are seen more as
discoveries than inventions. The author's proposal is just refining the
question: have we come to this conclusion because it's an intrinsic property
of the universe, or does it always seem present because we can only
contemplate the universe through the human brain, which does its own
structuring?

As an example of respected thinkers thinking along these lines, Sir Arthur
Eddington's _The Philosophy of Physical Science_ works pretty well (Though, to
be fair, it's not as if he's never been called 'out there').

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j2kun
The nonstandard definitions and notation, the lack of proper typesetting,
random clip-arty images, the precociousness of "Grand Unified" whatever, and
the general pretense that this isn't covered in most intro algebra graduate
courses, all of that doesn't help his case. :(

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westoncb
I guess if it were purporting to be a text book rather than a blog post, these
would be issues. As it stands, he's just someone on the internet sharing his
thoughts.

I'm not sure what you mean by 'precociousness' there, since I imagine you're
not attempting to compliment the author—but he's hardly alone in considering
Geometric Algebra to be a kind of foundation/'unified theory' for large swaths
of mathematics.

The other critiques seem mostly along the lines of, "he doesn't dress like
us", which, while maybe a little off-putting to the in-crowd, shouldn't earn
him an automatic dismissal or derogation.

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j2kun
Of course people can write and wear whatever they like. I am simply explaining
why it smells like crackpottery, since it seems you thought it was only due to
that one sentence.

And, for the record, anything aspiring to be a grand unified theory of
mathematics has to go far beyond algebra.

~~~
jacobolus
Here, start with these: (1)
[http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf](http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf)
(2)
[http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf](http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf)
(3) [http://geometry.mrao.cam.ac.uk/wp-
content/uploads/2015/02/Im...](http://geometry.mrao.cam.ac.uk/wp-
content/uploads/2015/02/ImagNumbersArentReal.pdf) (4)
[https://arxiv.org/pdf/1205.5935v1.pdf](https://arxiv.org/pdf/1205.5935v1.pdf)
(5)
[http://www2.montgomerycollege.edu/departments/planet/planet/...](http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/bookGA.pdf)

This book is also great if you are interested whatsoever in computer modeling
of geometry, e.g. for robotics, computer vision, computer graphics, VR, etc.
[http://www.geometricalgebra.net](http://www.geometricalgebra.net) The
“conformal geometric algebra” model developed there is a really amazing recent
idea. If you don’t want to buy a book see the papers at
[http://geocalc.clas.asu.edu/html/UAFCG.html](http://geocalc.clas.asu.edu/html/UAFCG.html)
or this Ph.D. thesis
[http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pd...](http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pdf)

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sullyj3
"The truth, or validity of Clifford Algebra is confirmed by Occam’s Razor, it
provides a simpler model of mathematical objects than does vector algebra
[...]"

What? No. No. That's not how Math works.

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mathgenius
I think this stuff is all well known, and goes under the modern heading of
"multilinear algebra".

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macmac
Besides its application to graphics, I wonder if this could be applied to type
algebra.

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analognoise
Sounded interesting until the crank shit at the end.

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GFK_of_xmaspast
The author writes: "Clifford Algebra reveals, for example, that the apparent
chirality in electromagnetism, i.e. the right-hand rule for electric
generators, and the left-hand rule for electric motors, turns out to be
actually an artifact of the math used to describe the world, not a property of
the world itself. It turns out that electromagnetism has no chirality, as
revealed by Clifford Algebra"

but I'm a little bit skeptical of this given that properties of things like
"electric motors" can be observed in the real world.

