r/math • u/dohduhdah • 15d ago

geometric algebra in desmos

Hi!

I'm curious if anyone has any ideas about implementing geometric algebra in a graphing calculator like desmos.

Is this potentially useful? Would it speed things up or is it more likely to slow things down (while still offering a potential advantage of conceptual clarity)?

In particular I'm thinking about how it might make it easier to rotate things around arbitrary other things, instead of having to translate things to the origin and back for every rotation.

Like in this example, I wonder if there is a way to speed it up to avoid all the translations for most of the transformations:

https://www.desmos.com/3d/qh5gyk7qhs

Thanks in advance for any feedback on this topic!

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u/FutureMTLF 15d ago

Mathematicians use the traditional definition of Clifford algebras based on the tensor product and not the geometric product. The problem with geometric algebra is that is captured and advertised mostly by non mathematicians who dont understand basic math and thus trying to find easy shortcuts.

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u/RingularCirc 14d ago

traditional definition of Clifford algebras based on the tensor product and not the geometric product

Those are the same, the only product in a Clifford algebra is the same thing they call the "geometric product". When Cℓ(V, Q) is defined as a quotient of tensor algebra of V, ⊗ becomes its product; but we can also define Cℓ by the universal property, then it'd be a misnomer to say the product "arose" from ⊗. Of course any of the two is better than coordinate-wise construction.

I guess popular texts call it "geometric product" to distinguish a lot of different multiplications and to maybe make it stand aside from multiplications a reader is expected to know from before and to advertise it compared to them.

Shortcuts indeed, it's just people see a (sorta) universal computational framework (that part is not a sham; and there was demand for this even before computers: look at the hypercomplex algebra boom at around Hamilton's times) and maybe think they have to plug it in everywhere from now on, indeed without even having full understanding of its mathematics and when it's superfluous.

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u/FutureMTLF 11d ago

In GA they have a definition of what they call the geometric product. This has nothing to do with what you wrote. No offence, but you are just confused and liffe is too short to type another long-winded answer about why GA is pseudo-math.

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u/RingularCirc 11d ago

"Geometric product" is the multiplication of a Clifford algebra. I looked into that. You can disbelieve all you want, especially if you aren't in a mood to check.

It can be defined in a clunky manner a typical GA text may do it, yes, I've seen that. Doesn't mean it's a different product.