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BenoitP

This post was brought to you by the Geometric Algebra gang.

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Somehow, in your links, there isn't a single explanation anywhere (that's not in the form of a video that I don't have time or inclination to watch) about what "geometric algebra" is supposed to mean. I'm trying to gather what this is supposed to be from the catchphrases on the front page, and honestly, this just looks like linear algebra.

Can you explain what "geometric algebra" is supposed to be? Or just link to written explanations? I'm a mathematician (algebraic topology), so don't be afraid to get technical. How is this different from linear algebra? The only things I can see are some tensor products and exterior products, and a few couple of division algebra structures. Can you enlighten me? Is there any actually new math behind all this?

Geometric algebras = Clifford algebras over R.

And they made dozens of hour long videos about that? That's insane. And the arrogance of calling it "geometric algebra" like they invented a new field of math?!

One could argue it is more about physics than math. GA has simpler expressions for a lot of physical topics.

Perhaps, but it's not new math (Clifford algebras are almost 200 years old). Shrouding it in a veil of buzzwords and pizzaz makes you look more like con artists than scientists. A sentence like this one is just absurd:

> Clifford's Geometric Algebra enables a unified, intuitive and fresh perspective on vector spaces, giving elements of arbitrary dimensionality a natural home.

My take is a bit different. May I quote but edit myself :

Once you accept that imaginary number is not about a straight real number but a number about rotation. And you have two number that is not the same, living in different dimensions. One about straight line and one about rotation I.e.

X0 ops X1(another number system ) where X0, X1 is real

You then have complex number or a+ bi where i^2 = -1 And you can have split complex number a+bj where j^2 = 1 etc

The next move is not only the usual quaternions which one can, but to Clifford algebra

x0 ops x1n1 ops x2n2 ops …

xi is all real but ni is a different number system represent things not on the real number but a different and higher dimension object.

For example, the Clifford Geometry define the 2nd not a i or j but a plane, so you have

x0 + x1<plane> … and so on

with more high dimensional object (especially if one may say x0 is x0<line>).

Like complex number it simplify a lot of things. Spinsor would be easier to define this way say.

After seeing the video I was a bit surprise by her two comments. This is after the most understandable introduction to this king of geometry. Shocking last 5 minutes.

The first one that it is not that useful just different way of grouping. IaAware of this taking by both physics and computer people. But it is a total different way of looking and should have been taught in high school or at least undergrad. A bad cycle here. It has a lot of use to understand physics for normal people whilst the tensor …

Also her struggle with thinking about shift to another game engine. Unreal … too complex and Godot … to engineering … her remark basically said she may have to go back to game programming (to get a living).

Both honest comments and reflect hard reality.

I never did quite 'get' geometric algebra. I mean the exterior algebra gives a couple of clear generalisations, and for the ones that require a metric you can typically use the Hodge star to find a generalisation.

Geometric algebra then blends all of this together, and I'm still not entirely convinced that this improves things. Is it actually ever handy to have to deal with mixed degree multivectors?

Even without using mixed-degree multivectors, the fact that the existence and the properties of all the different kinds of multivectors results from a small set of simple axioms is very satisfying in itself.

Before all these concepts were unified by geometric algebra, the system of physical quantities as taught in most places was a huge mess of many different kinds of quantities, scalars, polar vectors, axial vectors, pseudoscalars, tensors, pseudotensors, complex numbers, quaternions, spinors and so on.

It was not at all obvious why there are so many kinds of quantities, which are the relationships between them, are there any other kinds of quantities besides those already studied, etc.

Geometric algebra has brought order in this chaos and it has enabled a much deeper and more complete understanding of physics, by reducing a long list of seemingly arbitrary rules to a much smaller set of axioms, and by deriving all the many kinds of physical quantities from the vectors in the strict sense, i.e. from the translations of the space, which are themselves derived from the points of the affine space (as equivalence classes of point pairs).

(Both logically and historically, in physics the vectors are more fundamental than the scalars. The scalars are obtained by dividing collinear vectors, i.e. they are equivalence classes of pairs of collinear vectors. This division operation is a.k.a. measurement and what are now named as "real numbers" were named as "measures" in the past, for more than two millennia. For any Archimedean group it is possible to define a division operation using the Axiom of Archimedes, generating a set of scalars over which the original group is a vector space).

Yeah but like all of that is true about the exterior algebra as well. Except exterior algebra is a bit more explicit about where you use the metric, which is really convenient when the metric starts to become important.

I'm not versed at all into the Hodge star operator but it feels like Maxwell's equations expressed in both systems[1] might locate the answer. With Hodge star:

    dF = 0
    d * F = J

Expressed in GA:

    ∇F = J

The eyeball-difference of which (dF = 0) would be your "how GA then blends all of this together", if I understand correctly. I'd guesstimate that dF = 0 to be akin to Gauss' law; and that maybe GA somewhat incorporates that the curl of a gradient is the zero field.

[1] https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...

These are not quite equivalent, since using the hodge dual explicitly incorporates the geometry of space in a way left implicit by your geometric algebra formulation. The appropriate exterior algebra analogue here is simply `F = dA`. F is the electromagnetic field tensor, d is the exterior derivative, A is the 4-potential.

Clifford (i.e. "geometric") algebras are of some mathematical interest, but for physics the only real value I've seen from them is in offering a fairly nice presentation of spin groups, which is ultimately where the Dirac matrices come from. The geometric algebra advocacy, so far as I can tell, comes entirely from engineers who were never given a proper account of tensors in the first place: it's certainly an improvement over whatever godawful basis-dependent stuff gets taught there.

I'd be interested to know your opinion on space-time algebras. They seem like they would provide a nice way of unifying spatial rotations and Lorentz boosts, but that may be blind optimism as your last sentence seems to have been written to describe me specifically...

Spacetime algebra is just the Clifford algebra Cl_1_3(R), so all of the above applies.

> They seem like they would provide a nice way of unifying spatial rotations and Lorentz boosts

Yes and no: the right way to unify rotations and boosts is to consider them as the orientation-preserving elements of the Lorentz group (sometimes called the proper Lorentz group). You can construct this from the corresponding Clifford algebra, but it's somewhat technical and not physically well-motivated until you start dealing with spinors. It's also the group of symmetries of spacetime that leave the origin unchanged, which is far, far more natural.

I found it a much nicer way to think about physical concepts.

Torque and magnetism, for example, make much more sense as bivectors.

And never having to do the right-hand rule is a nice plus.

Yes. In physics. E.g. Maxwell's equation reduces to a single equation in geometric Algebra: https://peeterjoot.com/archives/geometric-algebra/maxwells_g...

Thank you so much for the links! I just happen to be endeavoring to seriously learn this stuff at the moment, driven by interests in physics, differential geometry, and gamedev. I'm seriously jazzed to check these out, thank you.

Based.

Based and Grassmann-pilled?

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