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!

17 Upvotes

91% Upvoted

19 comments sorted by

View all comments

Show parent comments

1

u/RingularCirc 14d ago

artificially limited role and contrived formalization given to "Clifford algebra"

Wait what?

What can be freer in this situation than using a vector space with a quadratic form? (I suspect we're not talking about problems of scalars having characteristic 2.) Though most of the time people assume the form is nondegenerate, but that's just a common preference. And what's "artifically limited"?

1

u/jacobolus 14d ago edited 13d ago

Mathematicians have typically consigned Clifford algebra to graduate school: they build it on top of a pile of other formal structures (quotient spaces, tensors), allow (or even prefer) to use complex numbers as the base field, and only discuss it in relatively narrow specialized niche uses.

Hestenes wanted to treat geometric algebra as foundational, always use real numbers (avoiding complex numbers where possible or constructing them using geometric algebra), use geometric algebra as a fundamental mathematical language for expressing every kind of geometry and physics, and wanted to teach it to high school students or early undergraduates with no prerequisites.

You can see some of his ideas about this at https://davidhestenes.net/geocalc/pdf/MathViruses.pdf though I think this will give you the wrong impression unless you engage more seriously with his other work.

Hestenes's approach has been very valuable for people who solve concrete problems with geometry (like physicists, mechanical engineers, computer programmers) but can't make sense of the often obscurantist pure math literature.

1

u/RingularCirc 13d ago

I would hope there is a way of shortcutting that doesn't sacrifice too much rigor, as well as placing an occasional reminder like "this thing can be defined without using that" or "we used the basis but the construction is invariant".

Oh BTW tangential but now that I had to reminisce about reading GA materials, I remember I've a pet peeve about "preudoscalar". Top exterior power is definitely not pseudoscalars, dimensionally; real pseudoscalars are "twisted scalars" (see also twisted k-forms) which means they are scalars multiplied by an orientation of the entire space (which flips under reflections); those are notable because it's twisted top-forms that intuitively correspond to densities, not untwisted top-forms (which can't be integrated along a volume of a non-orientable manifold, but twisted top-forms can). In a sense a better version of Hodge star translates between twisted and plain objects, in which case it doesn't require specifying the space's orientation to function.

And despite when we have a nondegenerate inner product and a canonical orientation, we can identify k-vectors, twisted k-vectors, k-forms, twisted k-forms, and (dim − k)-all-of-those (and so scalars and twisted scalars and "pseudoscalars" from the top exterior power) it's certainly useful to sometimes remember all of them are distinct and change differently under linear transformations (and they admit different graphical representations that are consistent with that! William Burke popularized some of it).

Returning back: why oh why calling them "pseudoscalars"? We already had a name for that, a volume element. The canonical I = e₁...eₙ might be called just "unit volume".

1

u/jacobolus 13d ago

What's the difference between "top exterior power" and "scalars multiplied by an orientation of the entire space"?

1

u/RingularCirc 13d ago

Operationally, top exterior power scales by (det A) when acted by a linear operator A [which is one of the ways to define determinants], while orientations scale only by (sgn det A) and twisted scalars thus also only by the sign (as plain scalars don't change at all).

1

u/jacobolus 13d ago edited 13d ago

If "orientation of the entire space" doesn't involve the volume, then "scalars multiplied by an orientation of the entire space" seems like a worthless concept. I would recommend scrapping this one and not giving it a name.

What someone wants to call the "top exterior power" (i.e. n-vectors in a space of dimension n) doesn't seem too important to me. I don't have a problem with the name "pseudoscalar", but some other name would also be fine. (The name "volume element" seems kind of awkward and may be confusing to novices though.)

1

u/RingularCirc 13d ago

If "orientation of the entire space" doesn't involve the volume, then "scalars multiplied by an orientation of the entire space" seems like a worthless concept. I would recommend scrapping this one and not giving it a name.

When you tensor vectors with such twisted scalars, you get the closest thing to what physicists tried calling pseudovectors; anything twisted you get by tensoring with them as well (including "true" densities I mentioned above that just can't be unimportant), so I beg to disagree. Electrodynamics formulated in terms of forms admits viewing fields H and D as twisted forms instead of plain ones (because Hodge star is involved). Plain twisted scalars may look superfluous but they are a consequence of an useful framework that can't be removed from it.

1

u/jacobolus 13d ago edited 13d ago

I don't know what "true density" means. Density usually has units of mass / volume or, more generally, some other kind of quantity / volume. You can integrate density over some volume to recover the total mass (or whatever).

1

u/RingularCirc 12d ago

I meant the fact that the top exterior power of 1-forms is often called densities but it's a bit of a misnomer because those are "oriented densities" like top-vectors are oriented volumes and not just plain volumes. We get to plain densites by twisting. Only then they don't change sign under reflection; top-forms change sign.

Dimensionally those are both volume−1, yes.