Відео

My area of expertise is definitely not software: I'm more of a pen and paper guy! But there's something called ganja.js made by Enki. If you join the Bivector discord, there are channels for discussing this too :)

Eh bien ouais, c'est ce que nous appelons un clickbait

@@EccentricTuber si oh! 😃

Yeah, I am planning to do it. It may be a few months til I finish the video.

@@EccentricTuber Space/time symmetries (rotations, boosts) are dual to Mobius maps -- stereographic projection. Particles are dual to antiparticles, spin up is dual to spin down -- the Dirac equation. Space is dual to time -- Einstein. "Always two there are" -- Yoda. The magnetic field only exists for moving particles with a velocity! Electro is dual to magnetic -- photons are dual.

Representations can also be used in a non-matrix context. And the APS is a common name in literature. Also, presenting matrices in the video is to show that those who use matrices don't have to. I have reasons for showing things in my video.

The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics. Lie groups are dual to Lie algebras (tangent plane). Space is dual to time -- Einstein. Space/time symmetries are dual to Mobius maps -- stereographic projection. The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Injection is dual to surjection synthesizes bijection or isomorphism. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Categories (form, syntax) are dual to sets (substance, semantics). "Always two there are" -- Yoda.

That's a good question! It's because paravectors experience Lorentz contractions in their direction of motion whereas biparavectors experience Lorentz contractions perpendicular to their direction of motion! The dagger allows the Lorentz rotor to anticommute with orthogonal directions to motion and thereby cancel out, while for biparavectors the Clifford conjugate allows the Lorentz rotor to cancel out in directions of motion but stay in orthogonal directions!

I definitely didn't choose a good color scheme for that video! I'm glad you enjoyed the content regardless of its poor formatting :)

@@EccentricTuber best wishes for you and your family) sharing knowledge and teaching are very worthy deeds.

You're actually noticing a very important pattern! The APS, the Algebra I'm using, is isomorphic to the even subalgebra of the STA. The section you read is likely on Spacetime Splits, which is a method within the STA and is related to the APS. So it's actually not an inconsistency, but that author was likely referring to the isomorphism between the STA and the APS!

Also, I had included the APS wiki page in the description for, I'll call it, the aggressively curious! The citations and references there go back to Baylis' papers.

The APS is isomorphic to the even subalgebra of the STA. And as for deriving it, I did do so. But I did that using commonly known facts about GA in general, but I applied them in this context. Read up on Baylis' paravector approach to relativity. As for being more intuitive, I'd argue the average person would find the APS more intuitive that the STA because it works in 3D space, modeling time as a scalar not a vector. Overall, the STA is better. But it's not the best choice for someone who doesn't need all the bells and whistles of the STA. The audience is then intended for those familiar with the APS, and I hope to show them that the APS can also be used for relativistic problems. Anything presented poorly is more a reflection of my skills in teaching, not the worthiness of the algebra itself.

Space/time symmetries (rotations, boosts) are dual to Mobius maps -- stereographic projection. It is an equivalent or dual description of space/time. Space is dual to time -- Einstein. "Always two there are" -- Yoda. Sense is dual to nonsense.

@@EccentricTuber What I mean is that, just by deriving the basis you need to describe events in the STA, the entirety of SR can be derived trivially. In the APS, you need to make a lot of assertions that actually come from the STA. For example, why are scalars in APS treated effectively as quantities of time? What motivates the use of the sigma vectors as rotors if they're just vectors when all other rotors are formed from exponentiating bivectors? There are no mysteries like that in the STA. Everything makes perfect sense and couldn't be any other way.

@@hyperduality2838 You have to move on and learn other things. Duality is not the only concept. It's a coincidence that many systems can be described in (sets of) binary distinctions. Compliments and inverses are a common feature necessitated by logic, but it doesn't mean anything deep about nature. It only means that mathematics is logical. Indeed, not all mathematical operations can be seen as dualities. Projections, ideals, idempotents, modular rings, and countless others have relationships that prohibit a one-to-one mapping that could be called a duality. By obsessively relating everything you learn back to duality, no matter how forced the comparison, you are robbing yourself of the depth of understanding you obviously could have of math and science if you just stopped. One-to-one relationships exist, but they are not special. They don't underlie the universe. Please think about opening your mind to other concepts. I'm not calling you names, I'm not intending to embarrass or demean you, I'm trying to help you take your interest and intelligence in the right direction. Think about it.

To each their own. I usually prefer a conversation over a declarative "I don't like this MEHH", but even for behavior, to each their own.

Glad I could be of help :)

@@EccentricTuber Sine is dual to cosine or dual sine -- the word co means mutual and implies duality. Sinh is dual to Cosh -- hyperbolic functions. Space/time symmetries are dual to Mobius maps -- stereographic projection. Space is dual to time -- Einstein. "Always two there are" -- Yoda.

Glad you liked the video! That's a good point. I'd been thinking along the lines of the addition within the exponential as being different than the normal addition outside of an exponential. So while they use the same + symbol, being within the exponential implies non-general commutativity additionwise. But now that I look at the definition of a Lie algebra I'm thinking I should've written as you said: exp(g1 + g2) =/= exp(g2 + g1). Well, at least the audience will understand the concept I'm trying to convey!

Thanks dude, you made my day!

You hit the nail on the head, I think :)

Well, I recorded this at a concert. The song name is in the video title, though.

OH nvm i get it

Thank you for the feedback! It always is such a great feeling when the math just works! While I'd like to take credit for the ideas in this video, I sadly didn't discover them. I first read about them in "Matrix Gateway to Geometric Algebra, Spacetime and Spinors" by Garret Sobczyk (Chapter 3). As for their construction, roughly speaking it comes from longterm experience with matrix mechanics: you just kinda know that's how it's done. I know, it's not a good explanation for a mathematician lol. The book discusses nilpotent constructions for the Spacetime Algebra (I'm pretty sure), but they're drastically less introductory compared to the (1+1)D spacetime. Hope this helps!

Chirality is dual to Helicity. Lie groups are dual to Lie algebras (tangent plane). Space is dual to time -- Einstein. Space/time symmetries are dual to Mobius maps -- stereographic projection. The inner product is dual to the outer product (wedge product) synthesizes the geometric product -- Clifford Algebra. Spin up is dual to spin down, particles are dual to anti-particles -- the Dirac equation. Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases. Injection is dual to surjection synthesizes bijection or isomorphism. All numbers fall within the complex plane. Real is dual to imaginary -- complex numbers are dual. All numbers are dual. The integers are self dual as they are their own conjugates. Syntax is dual to semantics -- languages or communication. If mathematics is a language then it is dual. Categories (form, syntax) are dual to sets (substance, semantics). "Always two there are" -- Yoda.

I agree with this. The book gives you all the necessary Geometric Algebra concepts you need to apply to problems (Super Engineering friendly). The author also gives you examples problems solved with and without Geometric Algebra to get an idea of how to apply it. Currently reading which supplements my Electrical Engineering classes. The concept of phasors, and the complex plane became so intuitive. Also, will come in handy for Electromagnetics class.

Not for this series specifically. I may someday restart the series though! I do have other GA and physics videos that are much better and more recent, however. I also talk about video/channel plans in the community tab if you're interested!

This made my day :)

Noncommutative arguments are one of the initially most confusing topics in my opinion too!

​@@EccentricTuber Specifically: define d[] over "+","*","^", and "log" binary operators. For implicit differentiation d[a+b] = da + db d[a*b] = da b + a db d[a^b] = (b/a)(a^b) da + log_e[a] (a^b) db d[log_a[b]] = ... (its complicated) For addition and multiplication, it is clear that a and b can be multi-vectors. I have no idea what d[] is for ^ and log though. It is THIS that confuses me when I see the gradient operator applied to an object.

Very true, I should've included it as well!

Yeah, the topics in more advanced rotational physics were hard at first because I didn't learn much about that in my undergrad. Lots of supplemental sidework was needed for sure! I honestly don't know 😂 Recording while scripted is so weird! I'd guess that it clashes with what we'd like to say if we were just freeflowing.

I couldn’t find any solutions to the problems at the end of each chapter. Does anyone know where I can find them?

The name sounds somewhat familiar, but I couldn't name any of his works. I'll look into him!

Hopefully someday I'll get around to it!

I think that's a good idea