#bivector — Public Fediverse posts

Geometric Algebra — https://bivector.net/
#HackerNews #Geometric #Algebra #Mathematics #AlgebraicStructures #MathEducation #Bivector

How to even conceptualize them?

In {1,e1, e2, e12} the bivector e12 acts exactly like the imaginary # i

Does that mean that I could plot the scalar and bivector components of a multivector on the complex plane?

The "amount of #bivector" is itself a dimension? "The planar area is shown in this linear dim" seems insane

And I'd still have two components left. Do I plot [1, e12] and [e1, e2] as two vectors? Or a 4D quantity?

I guess I'll have to see what's useful.

#GeometricAlgebra #math

I wrote Vector, BiVector and TriVector #python classes separately. Now I see how they can be combined into a generic #MultiVector class with a "simple" computation of the outer product.

I'm trying to figure out how to #visualization them, tho.

#GeometricAlgebra #bivector

#python #pyqtgraph #3d #visualization

'normal' vectors in x,y,z

u = [2 0 1]
v = [0 2 -1]

via wedge product forming #GeometricAlgebra #bivector

u ∧ v = [ 4 -2 -2] (blue)

made of unit bivectors, i.e 1x1 plane segments in xy, xz, yz

Original "edge vectors" don't matter, all bivectors with same area and orientation are equal. Therefore a given bivector could be drawn an infinite number of ways. Here I've chosen

u2 = [1 1 0] (green)
v2 = [ 0 4 -2]

bivector.net

Consulting #pyqtgraph to determine where it puts x, y, z axes from #python #numpy vectors and comparing that with conventional #GeometricAlgebra basis vectors e1, e2, e3 I think I have my coordinate system straightened out

There's an #algebra trick to turn component bivectors into a single #bivector. Trick has two singularities (no e13 component and then either has or hasn't e23)

I either covered both those cases or hacked it until it didn't crash, I'm not 100% sure...

Now to prettify.

I think I now understand #vector and #bivector in their most basic forms. #python #code can make a BiVector given two vectors but represents it internally as bivector components. Can then ask for two vectors that it would be the outer product of for the purposes of drawing it in #3d in #pyqtgraph

Need to fix up some coordinate system agreement issues and some singularities plus add color and transparency.

Then possibly time to move to #GeometricAlgebra #multivectors and then some operators

unexpected #geometricalgebra

https://youtu.be/jvPPXbo87ds?t=1893

#bivector #math

Scratch "assemble mixer pcb" off the list--it works as designed! Which isn't much, but I'm a n00b!

Also scratch "two #muvco voices" off the list--they work as intended! Which isn't amazing (only square really makes an impact...?), but I'm a n00b!

Also also scratch #GeometricAlgebra off the list--I did a bunch of problems! I got semi-stuck, but I'm still a n00b!

Video of some of this later.

#bivector (what tags are my geomalg buds hanging out...at?) #math #synthdiy #esp32 #micropython #python

Still learning more about #geometricalgebra. I can definitely see how it works for geometrical theory ideas (including computer graphics).

How well does it work for geometrical *numerical* ideas? I mean like "fitting a best plane to measured values" and things of that nature? Is there development of a least squares concept for instance?

#bivector

So...#geometricalgebra. It seems pretty neat. But all the info on it is either very mathematical or very philosophical. Not much of it is practical.

"We can make a game engine!" A page of math follows and then an animation. OK, but exactly how did those mathematical objects become pixels? "A motor in Rn" is not helpful.

How can I turn my problems into geometric algebra and then back into solutions? Are there worked examples for dummies like myself?

#math #bivector