Geometric Algebra, Geometric Calculus

[hermann]

Here is a fine overview on Clifford Algebra/Geometric Algebra
http://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/

GEOMETRIC ALGEBRA of one and many multivector variables
http://www.helsinki.fi/~jmpesone/index_files/GA.htm

Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics - David Hestenes
http://geocalc.clas.asu.edu/pdf-preAdobe8/VeSp%26ComUpdated.pdf

Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf - David Hestenes

Spacetime Physics with Geometric Algebra - David Hestenes
http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf

Gauge Theory Gravity with Geometric Calculus - David Hestenes
http://geocalc.clas.asu.edu/pdf/GTG.w.GC.FP.pdf

[kram1032]

I feel like that blog post puts a bit too much magic behind the simple, straight forward idea of closure. And it really shouldn't be a mystery that i is a 90° rotation. That's already clear from how it works in complex numbers. Nor is it particularly strange that there is no difference between reflecting a single vector in the origin and 180° rotation of that vector around the origin. (If you consider more points, there actually is a difference. The same difference as being mirrored vs. being upside down. That seemed to be a little unclear from the exposition in that text)

All of those things, of course, are great to have in Clifford/Geometric Algebra. And some aspects are much easier to deal with and clearer therein. For instance, reflections that are NOT in the origin and rotations defined as double-reflections. (That's still not special in its own right but the corresponding notation in CA/GA is much more compact than the corresponding Vector Algebra analog)

Other than that, though, it's a nice read. Thanks for sharing

EDIT: Oh, you edited in a bunch of links. I am referring to the first.
As for the other links, I might not have linked them directly but I'm pretty sure I linked the corresponding pages that collected them.

[Roquen]

And it really shouldn't be a mystery that i is a 90° rotation.

I'd state this different.  The 'i' of complex numbers (and generalizations) is the rule (and not a number) that makes the two numbers independent and how they interact over multiplication.  The complex rule of i² = -1 geometrically makes them orthogonal in a Euclidean sense and indirectly contains all of the rules of plane trigonometry.  Change the rule for 'i' and you complete different notions of independence. 

[kram1032]

I'm not sure what you mean by that.
I was just referring to how slehar wrote up what he had to say about geometric algebra.
He makes some good points but much of it is neither specific to geometric algebra nor special.

That i = 90° rotation thing is pretty darn straight forward and should be taught in any semi-decent course on/ introduction to complex numbers.

What's really remarkable, and he did mention that too, is the seamless generalization of that idea to arbitrary dimensions. And a lot of objects to get a clearer, more straight forward description using these ideas. But that particular part of how the fact that i corresponds to a 90° rotation is rather mysterious seemed a bit far-fetched to me.
If you can't learn that from a course on complex numbers, a course on geometric algebra will probably not suddenly give you the epiphany to understand it.

[Roquen]

I'll say the same thing in a different way.  'i' makes the two values linearly independent (same for the 'n' components of any system formed from a set of basis with no special rules for addition: clifford,quaternions,hypercomplex,etc)

In complex numbers the multiplication rule of 'i' causes multiplication to be the trig identity sum-of-angles (plus composition of scale)  So multiplying by 'i' is a 90 degree rotation.  Conjugation negates the angle and combined with multiply difference-of-angles (plus composition of scale). etc.  Trig identities naturally fall out.

[kram1032]

Another paper
Geometric Algebra for Electrical and Electronic Engineers
As the title may suggest, this one focuses on electromagnetism. It's rather physics-heavy, but it looks nice.

[David Makin]

While this thread's getting pretty heavy mathematically - does anyone know about proofs relating to mathematical fields in higher dimensions other than that it's proved that there's no R3 form that constitutes a field that isn't simply rewritable as an R2 form ?
e.g. is there a similar proof that says there are no mathematical fields in R4 or R5+ ?
All I know for sure is I can't find any true Rn fields documented for n>2.

[kram1032]

what do you mean by field? There are fields of any number of dimensions.
http://en.wikipedia.org/wiki/Field_(mathematics)

[David Makin]

what do you mean by field? There are fields of any number of dimensions.
http://en.wikipedia.org/wiki/Field_(mathematics)

I mean infinite like reals and complex - not restrictive like p-adics etc.

Or perhaps I mean standard Euclidean ? Not sure, I'm not great on formal math

[hermann]

Hallo David,

the basic trick is, that you have algorithems to work with unit vectors. For example expand your mathematical universe with the following equations (relations between unit vectors):







With these definitions you can bring structure into complex numbers:
Define the complex value i in the following way:

          

now do squaring this equation (very detailed steps, inserting the above equations):



It is possible to redefine is this way also quaterions, octerions and other hyper complex numbers.

For the geometric interpretation (what are this unit vectors ) and as starter on this issue I propose the paper by Alan Miller:
http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=7943&context=etd_theses
Look carefully on the geometric interpretation of the elements of this algebra. It opens a new way to do geometry

The paper of Alan Miller was for me an opener to other papers.

Hermann

[Roquen]

Do you mean division ring?

[hermann]

Do you mean division ring?

I don't know what a division ring is. I talk about geometric algebra. May be it is a division ring.
In geometric algebra is is possible doing devision by vectors!

[Roquen]

That was to David.  It sounded to me like he's asking about Frobenius theorem.

[kram1032]

@David

p-adics are infinite just like the reals are. In fact, there is only one kind of real but there are countably infinitely many p-adics (one per prime number)

And standard-Euclidean...
I think you'll have to clear up your own vocabulary there, because that makes little sense in the context

@hermann

a division ring is a ring that also has a multiplicative inverse.
A ring is some kind of number structure which has a form of addition and a form of multiplication which includes the classical rules:
addition is commutative and associative, multiplication is associative, and together, they are distributive; addition has the neutral element 0 and multiplication has the neutral element 1. Addition also has its inverse, while multiplication doesn't necessarily have to have that.

Examples:

the "classical" algebra over R is a commutative division ring.
R³-vector algebra with the cross product is an anti-commutative ring.
Geometric Algebras are division-rings.
etc.

[hermann]

Thanks kram1032,

I expected that Geometric Algebra is a division ring.

p-adic numbers.
http://en.wikipedia.org/wiki/P-adic_number

On a computer I prefer to have a number writen in the form (Rationa Number) instead of 0.333333333333333333333333333333333333333333333333333333333333333333333333...
Three dots ... mean go on to infinity. The first presentation has also the advatage to require less storage space on a computer.
A decimal presentation on a computer means that one has to cut digits and loses precision.

Prime Factorization
Integers can be then be split into prime factors:
1482 = 2 * 3 * 13 * 19
1488 = 2 * 2 * 2 * 2 * 3 * 31
http://www.wackerart.de/mathematik/primfaktoren_zerlegung.html

Devision can then be made much easier when one has stored integers in the form of prime products and as rational numbers only.
The following form can be implemented very effectively on a computer.




Donald E. Knuth has worked this out in detail in one of his books on "The Art of Computer Programming".

I tried to implement a package for rational number operations but then discovered that intergers implemented on a computer are limited.
So I need to implement a package of unbounded integers to make it work well.
(I know that such implementaions already exist but I prefer to program it by my self. It is for me more fun to program instead of understanding code of other programmers)

Hermann
P.S Thanks for the inspiration.