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Less Weird Quaternions Using Geometric Algebra

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Re: Less Weird Quaternions Using Geometric Algebra

#31

Post by abainbridge »

defen wrote: post #21
abainbridge wrote: post #20

Earlier quoted context omitted.

That hasn't helped!

In a geometric setting - if you have two vectors, you can position them so that both have one end at the origin. This spans a plane (test it out yourself in 2D or 3D space with two pencils, put the eraser at the origin for each; it's a parallelogram). The area of the plane will depend on the length of the pencils. You can assign an orientation to the plane by imagining a rotor embedded in the plane that spins either…

Hmmm.

"The area of the plane will depend on the length of the pencils". Surely the area of the plane is infinite? The area of the _parallelogram_ will depend on the length of the pencils.

And I can't see how "you can assign an orientation to the plane" other than by changing the directions of the pencils. Again this description sounds like it refers to the parallelogram, not the plane.

And I don't know what a rotor is.

But other than that, I'm doing great.

Re: Less Weird Quaternions Using Geometric Algebra

#32

Post by auggierose »

klodolph wrote: post #26
auggierose wrote: post #11

Earlier quoted context omitted.

I'd say alternative is an unlucky choice of words. I'd rather say geometric algebra (GA) is an extension of linear algebra (LA). In order to really understand GA you need first to firmly understand LA. Then it becomes clear that all that GA does is to turn a Hilbert space into an algebra called a Clifford algebra, and to examine the geometric semantics of the various operations that pop up in the process. Here are th…

My personal recommendation for a book on geometric algebra is the one by Hestenes, New Foundations for Classical Mechanics ( https://www.amazon.com/dp/0792355148/ ). I was disappointed by Geometric Algebra for Computer Science and I recently got rid of my copy when I moved to a new apartment, but I have a mathematics background and tend to prefer denser books. I would say that "alternative" is a viable word here. Yes…

I guess we have to agree to disagree. GA is not an alternative to LA, as LA is the foundation of GA.

The main point of LA is not matrices, but linear operators, dimensionality, linear independence, bases, etc. Matrices flow naturally from that. If all you have been taught in LA is to manipulate matrices, then I can see why you feel about the relationship between LA and GA the way you do.

Re: Less Weird Quaternions Using Geometric Algebra

#33

Post by tnone »

This is great insight, but it seems a bit silly to act like you don't need a 4D / hypersphere representation when the 4th one is hiding in plain sight. For the not-quaternion to describe a rotation, it needs unit length in 4D, with the two components scaled as a sine/cosine pair.

Re: Less Weird Quaternions Using Geometric Algebra

#34

Post by klodolph »

abainbridge wrote: post #31
defen wrote: post #21

Earlier quoted context omitted.

In a geometric setting - if you have two vectors, you can position them so that both have one end at the origin. This spans a plane (test it out yourself in 2D or 3D space with two pencils, put the eraser at the origin for each; it's a parallelogram). The area of the plane will depend on the length of the pencils. You can assign an orientation to the plane by imagining a rotor embedded in the plane that spins either…

Hmmm. "The area of the plane will depend on the length of the pencils". Surely the area of the plane is infinite? The area of the _parallelogram_ will depend on the length of the pencils. And I can't see how "you can assign an orientation to the plane" other than by changing the directions of the pencils. Again this description sounds like it refers to the parallelogram, not the plane. And I don't know what a rotor i…

The length of a line is infinite too. If you think of a vector as a line + magnitude, it's a bit more natural to think of a bivector as a plane + magnitude.

Re: Less Weird Quaternions Using Geometric Algebra

#35

Post by klodolph »

auggierose wrote: post #32
klodolph wrote: post #26

Earlier quoted context omitted.

My personal recommendation for a book on geometric algebra is the one by Hestenes, New Foundations for Classical Mechanics ( https://www.amazon.com/dp/0792355148/ ). I was disappointed by Geometric Algebra for Computer Science and I recently got rid of my copy when I moved to a new apartment, but I have a mathematics background and tend to prefer denser books. I would say that "alternative" is a viable word here. Yes…

I guess we have to agree to disagree. GA is not an alternative to LA, as LA is the foundation of GA. The main point of LA is not matrices, but linear operators, dimensionality, linear independence, bases, etc. Matrices flow naturally from that. If all you have been taught in LA is to manipulate matrices, then I can see why you feel about the relationship between LA and GA the way you do.

You're saying things that I agree with 100% which makes me think that there's something missing from my explanation.

I'm not talking about linear algebra as a field of mathematics in some kind of ideal sense here. Yes, obviously, it's a foundation for geometric algebra. You don't need to convince me of that.

However, elementary linear algebra classes don't teach you about linear operators, they teach you about things like matrixes and cross products. In these basic classes, a "vector" is a "thing with X, Y, and Z coordinates". So when you get to physics, you use the cross product to write a formula for magnetic field. You have to remember that the magnetic field is transformed differently from other vectors according to some special rules. And engineers call this stuff "linear algebra". Mathematicians agree that it's linear algebra, but we know that there's a lot more to linear algebra that goes beyond that.

Alternatively, they could calculate the magnetic field using geometric algebra, and express it as a bivector, at which point all of those special rules vanish.

That's why Hestenes's book is called "New Foundations for Classical Mechanics". It's not that linear algebra is not the foundation for geometric algebra. It's that classes taught in colleges which are called "linear algebra" teach you the concepts used by Gibbs and Wilson in the book Vector Analysis, and these concepts don't generalize to different numbers of dimensions. GA does. Maybe the problem here is that we don't have a special name for that field of study which uses cross products, if had a different name for that stuff, say "vector analysis" after the book first appeared in, we wouldn't have a problems saying that "geometric algebra is an alternative to vector analysis".

GA is a nice alternative to the stuff they teach engineers scientists under the "linear algebra" banner.

Another example… look at Stokes' Theorem. The version with differential forms is a nice alternative to the version with just a cross product.

Re: Less Weird Quaternions Using Geometric Algebra

#36

Post by Govindae »

logfromblammo wrote: post #23
klodolph wrote: post #12

A couple things to add. For notation, we would often see the basis vectors named (e_1, e_2, e_3) instead of (x, y, z). The quaternions are the even-ordered subalgebra of the 3D exterior algebra. The exterior algebra has scalars (1), vectors (x, y, z), bivectors (xy, yz, zx), and pseudoscalars (xyz). The even-ordered subalgebra is scalars and bivectors (1, xy, yz, zx). Adding or multiplying two even-ordered multivecto…

The odd-ordered subalgebra (x, y, z, xyz) is symmetrical to the even-ordered subalgebra (1, xy, yz, zx), and can also represent quaternions.

That doesn't work, the odd space isn't closed.

xx = 1

Re: Less Weird Quaternions Using Geometric Algebra

#37

Post by Govindae »

abainbridge wrote: post #20
defen wrote: post #19

Earlier quoted context omitted.

A bivector is a plane spanned by two vectors, with an associated orientation.

That hasn't helped!

Do you know what a vector is? Vector:Line == Bivector:Plane

A vector is an oriented (+,-) magnitude(length) _in_ a line. A Bivector is an oriented (+,-) magnitude(area) _in_ a plane.

That area does not have any particular shape.

Re: Less Weird Quaternions Using Geometric Algebra

#38

Post by [deleted] »

madhadron wrote: post #24
klodolph wrote: post #7

Earlier quoted context omitted.

It's definitely an alternative in the sense that it gives you an alternative framework for concepts that are taught under the banner of linear algebra in school. For example, it gives an alternative construction for quaternions as a subalgebra, and it gives the exterior product as an alternative to the cross product.

No, it's not an alternative. You're conflating the grab bag of topics in an undergraduate linear algebra class with the subject of linear algebra. Linear algebra is the study of linear operators on vector spaces over fields (a special case of modules over rings). Some vector spaces are inner product spaces, but most are not. Exterior algebra is an example of multilinear algebra. Clifford (or geometric) algebras are c…

[deleted]

Re: Less Weird Quaternions Using Geometric Algebra

#39

Post by Govindae »

While we all learned in middle school geometric algebra that the even subalgebra of G3 is isomorphic to the quaternions, what is the relationship between the even subalgebra of G4 and the octonions?

If you write out a multiplication table, it seems that it's isomorphic. But... Octonions aren't associtive. Does the even subalgebra of G4 somehow lose associativity? Is it equivalent to Octonions with a cannonical multiplication order?

Re: Less Weird Quaternions Using Geometric Algebra

#40

Post by yequalsx »

klodolph wrote: post #27
yequalsx wrote: post #16

Earlier quoted context omitted.

I think there is a lot of unintentional irony in what you wrote. You start out saying, "There's a lot of hand waving in that phrase..." and then go on to write: "Start by using bivectors to represent reflections, then take the closure of your bivectors and you get the even-ordered subalgebra." It reminds me of the running joke we had in graduate school. Any book whose title starts off with "An Elementary Introduction…

I aiming that explanation at people who had read and understood the article. The article explains bivectors and how they can be used to represent reflections, and "closure" is a fairly common concept, so once you put those two together you should get a mathematical object which I've called "the even-ordered subalgebra". I haven't explained why it's even-ordered or what a subalgebra is, but I used those terms so you c…

I was a math Ph.D. student at Purdue University and studied commutative algebra. I understand what you were getting at. My comment was mostly tongue in cheek. For someone not versed in mathematics what you wrote could be ironic in a slightly humorous way. I.E. that the hand wavy way explanation is more understandable to a layman than subalgebras, and whatnot. That's the ironic difference between mathematicians and non-mathematicians. What is hand wavy to us is concrete to them and vice versa.
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