#geometricalgebra — Public Fediverse posts

"Ah, yes, another article where the author bravely takes on the Herculean task of explaining why math is hard and algebra is scary. 🤯🤦‍♂️ Maybe the real problem isn't with Geometric Algebra, but with the author's ability to enable #JavaScript and render equations properly. 🖥️🔧"
https://alexkritchevsky.com/2024/02/28/geometric-algebra.html #mathproblems #GeometricAlgebra #education #techissues #learningchallenges #HackerNews #ngated

"Ah, yes, another article where the author bravely takes on the Herculean task of explaining why math is hard and algebra is scary. 🤯🤦‍♂️ Maybe the real problem isn't with Geometric Algebra, but with the author's ability to enable #JavaScript and render equations properly. 🖥️🔧"
https://alexkritchevsky.com/2024/02/28/geometric-algebra.html #mathproblems #GeometricAlgebra #education #techissues #learningchallenges #HackerNews #ngated

The case against geometric algebra

https://alexkritchevsky.com/2024/02/28/geometric-algebra.html

#HackerNews #geometricalgebra #mathematics #techdebate #criticalthinking #algebra

The case against geometric algebra

https://alexkritchevsky.com/2024/02/28/geometric-algebra.html

#HackerNews #geometricalgebra #mathematics #techdebate #criticalthinking #algebra

#GAME2026 is happening and streaming #geometricAlgebra

https://bivector.net/game2026.html

#GAME2026 is happening and streaming #geometricAlgebra

https://bivector.net/game2026.html

I’m interested in #GeometricAlgebra (and #ExteriorAlgebra, #CliffordAlgebra, #ExteriorProduct and #WedgeProduct). I’m trying to work up an intuition for a few things. Thoughts on this welcome!

* It’s very intuitive that adding vectors means something and is useful. Join pencils end to end and now you have the pencil of their path. I have much less intuition that adding #bivectors is useful.

* In 3D, the wedge product of two vectors is an oriented area (bivector). And the wedge of an area and a vector is a volume. This makes sense. But the wedge of two areas in 3D is zero. Always (right?!). Is it even a well typed operation?

* If I have an area (say, some solar panels) and a direction (say incident sunlight), I think I can wedge them to get collected light (as a pseudo scalar). How do I know to wedge here instead of dot product? What’s the intuition for that so that the question becomes absurd?

* Is there a most simplest toy problem for playing with these to work up intuition? I think maybe solar panels (with area and orientation) and incident light (with intensity and direction) is reasonable? Because it just about makes sense to add oriented panels. And possibly even directed incident light?

Thanks!

I’m interested in #GeometricAlgebra (and #ExteriorAlgebra, #CliffordAlgebra, #ExteriorProduct and #WedgeProduct). I’m trying to work up an intuition for a few things. Thoughts on this welcome!

* It’s very intuitive that adding vectors means something and is useful. Join pencils end to end and now you have the pencil of their path. I have much less intuition that adding #bivectors is useful.

* In 3D, the wedge product of two vectors is an oriented area (bivector). And the wedge of an area and a vector is a volume. This makes sense. But the wedge of two areas in 3D is zero. Always (right?!). Is it even a well typed operation?

* If I have an area (say, some solar panels) and a direction (say incident sunlight), I think I can wedge them to get collected light (as a pseudo scalar). How do I know to wedge here instead of dot product? What’s the intuition for that so that the question becomes absurd?

* Is there a most simplest toy problem for playing with these to work up intuition? I think maybe solar panels (with area and orientation) and incident light (with intensity and direction) is reasonable? Because it just about makes sense to add oriented panels. And possibly even directed incident light?

Thanks!

https://www.youtube.com/watch?v=eY6GuTFfYpQ

An Interpretation of Relativistic Spin Entanglement Using Geometric Algebra

#geometricalgebra #physics

@gnomekat
There must be something in the air - I was thinking recently what use an infinite dimensional Clifford algebra might be put to. Not quite sure yet. Still mulling that one. LOL
#maths #GeometricAlgebra

If functions can be represented as vectors and you can do the dot product between them...

What happens if you try to use the wedge product between two functions?

This question just popped into my brain
#geometricalgebra #math

If functions can be represented as vectors and you can do the dot product between them...

What happens if you try to use the wedge product between two functions?

This question just popped into my brain
#geometricalgebra #math

Wow, a riveting tale of sunset geometry 🌅📐—because who doesn't want to spend their evening calculating the Earth's radius with a single sunset photo? 🤓🔍 Clearly, classical #trigonometry just wasn't making math nerdy enough, so let's sprinkle in some Geometric Algebra for extra pizzazz. 🎉
https://www.shapeoperator.com/2016/12/12/sunset-geometry/ #sunsetgeometry #mathnerd #GeometricAlgebra #STEMeducation #HackerNews #ngated

Wow, a riveting tale of sunset geometry 🌅📐—because who doesn't want to spend their evening calculating the Earth's radius with a single sunset photo? 🤓🔍 Clearly, classical #trigonometry just wasn't making math nerdy enough, so let's sprinkle in some Geometric Algebra for extra pizzazz. 🎉
https://www.shapeoperator.com/2016/12/12/sunset-geometry/ #sunsetgeometry #mathnerd #GeometricAlgebra #STEMeducation #HackerNews #ngated

One day, in a far future, I will understand #geometricalgebra

https://enkimute.github.io/ganja.js/examples/coffeeshop.html#pga3d_animation

One day, in a far future, I will understand #geometricalgebra

https://enkimute.github.io/ganja.js/examples/coffeeshop.html#pga3d_animation

Trying to explain to people why #geometricalgebra is so much better is really frustrating

Trying to explain to people why #geometricalgebra is so much better is really frustrating

I watched this excellent Swift Introduction to Geometric Algebra, https://youtu.be/60z_hpEAtD8?si=Gv_rHCafbNXj3WXr . I enjoyed it! Geometric algebra allows you to multiply and divide scalars, vectors, and other geometric objects. It's fascinating and helps make sense of concepts like imaginary numbers and quaternions.
#geometricAlgebra
#quaternions
#vectors

I also want to explore more #GeometricAlgebra applied to #physics and #ComputationalPhysics (not necessarily for #FluidDynamics and #CFD, but that would be preferable as it's obviously our primary topic of relevance). I can't seem to find anything that combines #SPH and GA, so it might even lead to some new interesting venues to explore.

3/n

#ComputationalFluidDynamics #SmoothedParticleHydrodynamics

I also want to explore more #GeometricAlgebra applied to #physics and #ComputationalPhysics (not necessarily for #FluidDynamics and #CFD, but that would be preferable as it's obviously our primary topic of relevance). I can't seem to find anything that combines #SPH and GA, so it might even lead to some new interesting venues to explore.

3/n

#ComputationalFluidDynamics #SmoothedParticleHydrodynamics

On a more practical note, I think #GeometricAlgebra may be useful for the redesign of the #ConstructiveSoldiGeometry subsystem that we used in #GPUSPH to allow programmatic definition of the test case geometry (where the fluid is, the shape of the boundaries, how they move, etc). We do need to redesign it, and from what I've learned about GA so far, it would really help us “do it right”, in a relatively simple way.

2/n

On a more practical note, I think #GeometricAlgebra may be useful for the redesign of the #ConstructiveSoldiGeometry subsystem that we used in #GPUSPH to allow programmatic definition of the test case geometry (where the fluid is, the shape of the boundaries, how they move, etc). We do need to redesign it, and from what I've learned about GA so far, it would really help us “do it right”, in a relatively simple way.

2/n

I'm (slowly) learning #GeometricAlgebra. There's several reasons for that. (The slowly there's only one: time. I mean several reason why I want to learn GA.)

The first is, shall we say, philosophical: I find it almost insulting that, despite by otherwise pretty solid background in mathematics, I only ever heard for the first time about the existence of GA a couple of years ago. That such a powerful mathematical framework is still largely unknown is … inappropriate.

1/n

I'm (slowly) learning #GeometricAlgebra. There's several reasons for that. (The slowly there's only one: time. I mean several reason why I want to learn GA.)

The first is, shall we say, philosophical: I find it almost insulting that, despite by otherwise pretty solid background in mathematics, I only ever heard for the first time about the existence of GA a couple of years ago. That such a powerful mathematical framework is still largely unknown is … inappropriate.

1/n

@internic
'Outer product' and 'tensor product' are soooo last millennium - all the cool kids are doing geometric algebra these days.
#maths #algebra #mathematics #GeometricAlgebra

After my initial frenzy of reading and watching content (and taking copious notes) about #GeometricAlgebra when I first heard about it, I've settled on a list of heavier resources to consume (mostly books) and haven't made much progress, as usual with books these days 😅 (1/3)

After my initial frenzy of reading and watching content (and taking copious notes) about #GeometricAlgebra when I first heard about it, I've settled on a list of heavier resources to consume (mostly books) and haven't made much progress, as usual with books these days 😅 (1/3)

ok I think I got vanilla #geometricalgebra properly learned... now time to see about the projective and conformal models .. more fun to be had I think

ᕙ⁠༼⁠◕⁠ ⁠ᴥ⁠ ⁠◕⁠༽⁠ᕗ

Welp. I guess it's time for codegen.

Today I crashed clang trying to evaluate my generic functions.

#programming #GeometricAlgebra #GameDev

Last couple days I've been working on doing rigidbody dynamics in plane-based #GeometricAlgebra which has been really cool.

I've learned about a bunch of new stuff, like lie algebra interpolators and integrators.

Turns out that PGA makes interpolating between orientations basically trivial, and it integrates rotation and translation together so I don't have to do any extra work to support torques. The only thing from angular dynamics I have to do is use an inertia tensor.

#GameDev

Today I got started doing the 3d plane-based #GeometricAlgebra library and getting it finished up with _Generic support. This mostly wraps up the math library for my game engine, at which point I can move on from this.

If I end up feeling like I want to optimize things I might eventually follow along with Look, Ma, No Matrices! and work to reduce the number of multiplies required, but honestly I think that's mostly useful for shaders. DCE and inlining work well enough for CPU side.

#GameDev

see a Michael Penn video about why the cross product doesn't exist in 4d and all the comments are about #geometricalgebra

yess yess it's spreading.. good..

As of today I think my 2d projective #GeometricAlgebra implementation in C is done!

I'll be moving on to my 3d implementation to flesh it out and add type generic functionality to it, but for now just 2d PGA is enough.

Now on to the next steps in my engine, working on the systems part of my species-based #ECS. That will include component queries, system initialization and deinitialization, and more.

#GameDev

As of today I think my 2d projective #GeometricAlgebra implementation in C is done!

I'll be moving on to my 3d implementation to flesh it out and add type generic functionality to it, but for now just 2d PGA is enough.

Now on to the next steps in my engine, working on the systems part of my species-based #ECS. That will include component queries, system initialization and deinitialization, and more.

#GameDev

In spite of me railing on _Generic in C these last few days, it's been a quite useful tool.

Today I'm getting very close to a complete implementation of a 2d type-generic plane-based #GeometricAlgebra library for the C #programming language, primarily useful for my game engine which has been the focus of my #GameDev work for the last while.

Next steps for me after this will be doing the same for my partially-complete implementation of a 3d version.

In spite of me railing on _Generic in C these last few days, it's been a quite useful tool.

Today I'm getting very close to a complete implementation of a 2d type-generic plane-based #GeometricAlgebra library for the C #programming language, primarily useful for my game engine which has been the focus of my #GameDev work for the last while.

Next steps for me after this will be doing the same for my partially-complete implementation of a 3d version.

Interesting – must dig deeper:

“Look, Ma, No Matrices!”, Steven De Keninck (https://enkimute.github.io/LookMaNoMatrices/).

Via HN: https://news.ycombinator.com/item?id=39538670

#Graphics #ComputerGraphics #Matrices #GeometricAlgebra #ProjectiveGeometricAlgebra #PGA #Mathematics #GraphicsProgramming #Programming

Interesting – must dig deeper:

“Look, Ma, No Matrices!”, Steven De Keninck (https://enkimute.github.io/LookMaNoMatrices/).

Via HN: https://news.ycombinator.com/item?id=39538670

#Graphics #ComputerGraphics #Matrices #GeometricAlgebra #ProjectiveGeometricAlgebra #PGA #Mathematics #GraphicsProgramming #Programming

I really liked this passage from a talk by @acegikmo on #GeometricAlgebra, which brilliantly demonstrates the absurdity of doing a bunch of mental gymnastics just to deal with various concepts (vectors, complex numbers, quaternions, etc.) that can actually be elegantly derived from the same basic principles: https://youtube.com/clip/Ugkx3hUwQoYj__hcOmBClCwFpA3MGQQ_eCo_?si=ptj_0uCvEb5yUteb

I really liked this passage from a talk by @acegikmo on #GeometricAlgebra, which brilliantly demonstrates the absurdity of doing a bunch of mental gymnastics just to deal with various concepts (vectors, complex numbers, quaternions, etc.) that can actually be elegantly derived from the same basic principles: https://youtube.com/clip/Ugkx3hUwQoYj__hcOmBClCwFpA3MGQQ_eCo_?si=ptj_0uCvEb5yUteb

@charliemac This actually still isn't right. For instance, look at (e1^e2)(e2^e3) = e2e3

The product is the same grade as both multiplicands and therefore the inner and outer products just don't exist/are 0.

The answer here seems to explain this better: https://math.stackexchange.com/questions/1685398/geometric-product-of-two-bivectors

Basically, it's like multiplying polynomials. You get terms at various powers (or grades, for multivectors). The inner and outer are simply terms with particular grades of j-k (inner) and j+k (outer).

#geometricalgebra

@charliemac This actually still isn't right. For instance, look at (e1^e2)(e2^e3) = e2e3

The product is the same grade as both multiplicands and therefore the inner and outer products just don't exist/are 0.

The answer here seems to explain this better: https://math.stackexchange.com/questions/1685398/geometric-product-of-two-bivectors

Basically, it's like multiplying polynomials. You get terms at various powers (or grades, for multivectors). The inner and outer are simply terms with particular grades of j-k (inner) and j+k (outer).

#geometricalgebra

@charliemac Perfect, thanks! That's exactly the conclusion I came to. The outer product combines subspaces and therefore the grade goes up. The inner product projects one subspace onto another and therefore the grade goes down (relative to the largest of the two).

But how to express that in #python #code. I guess just term-wise? I should do some examples by hand.

#GeometricAlgebra

@charliemac Perfect, thanks! That's exactly the conclusion I came to. The outer product combines subspaces and therefore the grade goes up. The inner product projects one subspace onto another and therefore the grade goes down (relative to the largest of the two).

But how to express that in #python #code. I guess just term-wise? I should do some examples by hand.

#GeometricAlgebra

Narrowing in on my #geometricalgebra confusion.

The definitions at this point:

https://youtu.be/njxg5nUwZKA?t=1320

contradict definitions 3.10 & 3.111 on page 22 here:

http://www.jaapsuter.com/geometric-algebra.pdf

Is the dot product the *sum* or the *difference*?

My #GeometricAlgebra journey has gotten mired in confusion.

It was all outer products and that was fine. Now we are into situations that have outer, inner and geometric products and I'm getting mixed up

I think the root of it is that multivector components are always either perpendicular or parallel, which means you can switch the products up a lot of the time.

It doesn't help GA enthusiasts in particular seems to suffer from a desire to blow minds re: elegance vs actual education.

Just saw a picture of a fried dumpling and was consumed (ha) with a hyperfocal desire To Know All Of Dumplings and make them at home

But I'm still perfecting the previous #hyperfocus, spaghetti. And I've made recent great strides in a long term plan to become King of #Pizza (protip: heavy on the gluten, lighter on the cheese)

Fortunately it's a three day weekend so I can do it all! But I need to buckle down and watch 14 hours of youtube dumpling tutorials. Plus work on #GeometricAlgebra

#adhd

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