Taking it further, you could argue that any computation can be broken down into linear algebra over finite fields. The real magic is in how we choose to interpret those reflections and transformations.
Wow@@Lucky9_9 - are you like a true information expert? Everything *_is_* information including thought patterns, waves (and their combinations), vectors and their properties and of course clocks and their "ticking" 🖖
@@digbysirchickentf2315 or you might think that any straight motion is a rotation about a point at infinity (a revolution by a circle of enormous radius)
Nice you just won a sub. I was worried if the video was going to be a crackpot, but instead, it turned out you are not only totally sane, but brilliant and love geometric algebra! Hooray! And you actually taught me something new about GA, I look forward to applying it! :) Thank you
@@sbasu4748 I wouldn't know as much about that, my background is in physics and mathematics. I think this can be used for efficient computations of motion, but don't quote me on it.
@@sbasu4748Y'all need chatGPT 😂 The concept of hyperplane reflection, while abstract, can be applied in various engineering fields, particularly in computational methods and optimization problems. Here are some potential application areas: 1. **Robotics and Motion Planning**: Hyperplane reflections can be utilized to optimize paths and motions in robotic systems, ensuring smooth and efficient navigation through space by reflecting trajectories across hyperplanes to avoid obstacles. 2. **Computer Graphics**: In graphics rendering, hyperplane reflection algorithms can enhance image processing techniques, especially in ray tracing where reflections and refractions are simulated for realistic visuals. 3. **Control Systems**: In control theory, hyperplane reflection can be used in the design of controllers that handle dynamic systems, particularly in managing stability and performance through feedback mechanisms. 4. **Signal Processing**: For filtering and analyzing signals, hyperplane reflections can be applied to transform and process multidimensional data in ways that reduce noise and enhance signal clarity. 5. **Optimization Algorithms**: In engineering optimization problems, reflecting across hyperplanes can be part of iterative methods to find minima or maxima, especially in multi-objective optimization scenarios. 6. **Structural Engineering**: In analyzing stresses and strains in structures, hyperplane reflection methods can simplify the modeling of symmetrical structures and reduce the computational complexity in simulations. These applications leverage the mathematical properties of hyperplane reflection to solve complex problems efficiently, making them valuable in many advanced engineering tasks.
The notion that any bivector can be represented as the geometric product of the volume element and a vector is extremely important. Its a natural way to encode the topology of a 3-sphere in terms of vectors, as it is the group of all unit quaternions (bivectors), and it allows to construct a counterexample to bell's theorem. The choice of handedness (the sign of the volume element) is not merely a convention, it represents two entirely different tangent bundles on the manifold, and acts as a de-facto hidden variable. Just like a left hand in a mirror looks like a right hand, so do particles and photons: this is why it looks like to us that bertlmann's socks are both left handed or both right handed! Ultimately, bell failed because he assumed that the elements of reality in an EPR experiment commute, which is just not true on a 3-sphere.
@@ivocanevo :) It's also very interesting that everything seems to have angular momentum, the quantum world is governed by Planck's constant, with units of angular momentum... Another property of a 3-sphere is that it is parallelizable, i.e. you can construct 3 linearly independent vector fields on the surface (unlike the 2-sphere).... The result of the parallelism though is that torsion is everywhere non-vanishing!
@@carlpanzram7081 it does sound like it doesn't it? Unfortunately I can cram only so much stuff in a single comment. If you want more, look up Joy Christian's papers on the Arxiv. I also suggest the channel PathfinderPhysics for a simple analysis, although the relevant series of videos is not yet fully available
Good explanation, good visuals, great topic. Instant subscription and plan to watch the rest of your work! Thank you for the way that you covered this.
Used heavily in Quantum Physics. Though, i wish, QM would delve deeper into using matrix and graphical representations of the vectors, than just use them abstractly.
I can't admit I understood half of that, but I'm curious about practical implications like say in a classical rigid body collision simulation? I've been considering making a Vampire survivors style game from scratch and was thinking about the physics simulation so I could have like 10k enemies on screen without slowdown, some data structures use hyperplanes for optimization (like kd-trees) and help with pruning collision candidates but I've been procrastinating on starting the project.
This was interesting. Any resources for further reading and to get a better intuitive understanding? I love the visualisations btw. Helped a lot to understand the physical representation
@@goodfortunetoyou scheme theory goes crazy hard in algebraic geometry. go check out the definition of the spectrum of a ring (like a good polynomial ring C[x,y]) and then look up what an affine scheme on it is, changed my life
If i was living in quake and my world was truely 3d and one dimension of time wouldnt some objects hyperplane be the record of movements from start to end of time
Reflection is key. 🐲✨🐲✨🐲✨ "Before I start, I must see my end. Destination known, my mind's journey now begins. Upon my chariot, heart and soul's fate revealed. In time, all points converge; hope's strength resteeled. But to earn final peace at the Universe's endless refrain, we must see all in nothingness... before we start again." 🐲✨🐲✨🐲✨ -- Diamond Dragons (series)
It's kind of like in 3d software, a polygon has a normal, and this reflection is just flipping the normal. (Pretending the polygon is infinite in its planar surface, obviously, to theorize it into a hyperplane)
I’m not sure if this is a misinterpretation, but does this mean that “all motion stems from the phenomenon of reflections”? If so, I’m having trouble understanding how vibrations and fluctuations could arise from nothing but pure reflective activity.
Good question! It does, effectively! Vibrations and fluctuations could be thought of as the spontaneous creation of reflective hyperplanes or the change of their relative separation. I'm sure there's a more elegant way to put it, but I hope that helps!
Good question! Yes, although the math for a rotation/translation changing in time and for ones that don't is exactly the same! Effectively, there would just be change in the relative distance/separation of the hyperplanes over time. Hope this helps!
@@EccentricTuber This might be a language barrier thing, but I feel like now you're just saying that motion of a geometrically complex thing is motion of geometrically simpler things, which doesn't really explain motion. It just kicks the can further down the road.
Tensors are related to GA, but they're typically just the components of multivectors. As for Special Unitary, the compositions of reflections are elements of Pin(n), which double cover O(n). The reflections that compose to give rotations and translations are elements of Spin(p,q,1) which double cover SO(p,q,1), so we're not far from Special unitary symmetries at all! Note that for n=3, Spin(3)=SU(2).
Never seen a spinorial representation of motion. I know that it can be used for the Ising model solution but idk what would be moving there, hypothetically its all just spins flipping on a 2D grid. Maybe projective geometric algebra could be used in statistical physics? P.D: I took a peek at the paper and I see a block matrix decomposition which is something you also do with the transfer matrix of the Ising model. Gotta brush up on the solution but maybe then I'll go into the discord and ask a few questions
Please correct me if I am wrong (which I could be!): I think that an isometric transformation can actually modify the shape of an object, because you can have metrics that distort shapes. An example is length contraction for relativity. Distances remain the same, but lengths, and therefore shapes warp based on speed. Am I correct here?
Good point! Yes, Lorentz boosts indeed are isometric transformations in the sense that hyperbolic angles are conserved, and distances satisfy the Minkowski metric rather than the Euclidean metric. So the definition I gave technically includes it, given that the definition of "shape" is modified to work with Minkowski space. That's actually why I mentioned spacetime at the end of the video: You can do relativity with this formalism (and it's actually pretty powerful)
@EccentricTuber sorry I meant 3d hyperplanes eg. Spinors..I understood that for glide symmetry you need an odd number of dimensions so 2d would need additional 1d to translate the object. Is that right?
I can't believe I've watched this whole 15 minute video and understood nothing 😂 What exactly does it mean that you can express motion as a reflection? How does this theory relate to practical reality? Does it have a use?
It's actually very useful for studying spinors and pointors! My latest video uses reflections to describe spinors. It's also good from a Comp. Graphics perspective, I'm told.
is it just me, or is this a sort of convoluted way of extrapolating motion kind of like a fourier transform of linear motion., Are the trajectories of objects in hyperspace able to be quantized to the smallest level? becuase it didn't seem like it could in your video. (probably because it went way over my head)
Good question! So this is neither like a fourier transform, nor exactly analogous to "quantization". It's just a way to break motion into invariant reflections that can be performed in arbitrary order. It's conceptually similar to the prime factorization of an integer! Hope this helps!
@@EccentricTuber yeah I know, but it just made me think of a discrete moment of space in time, in some world where the universe worked in discrete time
Could you please elaborate on your perspective? I believe we could gain valuable insights from your thoughts. Therefore, I would appreciate it if you could share in detail how many of us might be mistaken.
@@saftheartist6137 ok well, you’re off the mark on a lot here. Let me give you some points: Point #1: We live in the context. Point #2: We beat Medicare Point #3: Obamna! Point #4: I couldn’t give you a verse from the Bible, all the parts are so good, it’s a great book Point #5: America can be defined in a single word: assahhghhh wsaagjjn Don’t recognize these? Here you see wisdom from America’s most prominent politicians (I can point you to their sources if you want to watch them in their full glory). Reflect on this wisdom and you will understand
@@longextinct Oh, so you provide humor? Well, that’s fine. There’s a time and place for that too. However, the category of this video is education; your humor would be more appreciated in videos categorized under comedy and then in the category of entertainment.
@longextinct I'm here after finishing the Bodhidharma any% speedrun (cave wall staring) to confirm that your belligerent stance is correct. Also I learned everything I need to know about representation theory from listening to Rubberband Man by the Spinors on repeat. Seven years of that and I was primed for the first TikTok watermelon video I could find. I lost all respect for Gallagher once I was shown the potential of rubber bands to destroy fruit. A mallet is such a barbaric implement--and the word hardly sounds like geometric terminology--case closed. Enough of this stream of consciousness business. Let me get to the point. Yes, following decades of brinksmanship on both sides a shooting war has broken out between me and a crack commando unit of elementary school teachers. Wanted by the federal government, they survive as soldiers of fortune and are currently based out of Los Angeles. If you have a problem, if no one else can help, if you can find them... maybe you can hire the A-Team. I say "you" because I definitely cannot hire the A-Team. There is as you rightly guessed too much enmity on the part of my fifth grade teacher.
@@damienasmodeus928 Don't worry, you're not dumb! It's a tough subject if you don't have a background in geometry or Geometric Algebra. The idea is that any form of motion can be decomposed into successive reflections. So a rotation is just two reflections, and so is a translation. But they can also be composed, so a motion that describes a rotation AND a simultaneous translation can still be described by reflections. If you need to learn more, I recommend we continue this conversation in the Bivector discord! It's in the description 🙂
@@daolinh108 Don't worry, you're not dumb! It's a tough subject if you don't have a background in geometry or Geometric Algebra. The idea is that any form of motion can be decomposed into successive reflections. So a rotation is just two reflections, and so is a translation. But they can also be composed, so a motion that describes a rotation AND a simultaneous translation can still be described by reflections. Does that help?
Broski, imagine that you reflect the world along some spatial axis, but every causal relationship stays invariant, would you be able to tell the difference if i did this to you and the world you are in? Trippy thing to think about, with respect to what relationships are intrinsically identical in experience given some postulated physical state of the world, you should do some physics, there is a lot of cool stuff to do, and a lot of silly misconceptions floating around that have their root in different forms of representation taken to seriously as physical theories, that is, a lot of folks, professional and kook alike just say words and cling to beliefs that are not well motivated, because they fail to understand that what is physically meaningful is that which is independent of representation. For example, in the context of relativistic effects you can define lenght contraction to be physical and only give real lenghts in one coordinate system, without changing any of the physics what so ever, this in itself os an arbitrary choice of representation, whether it has any further physical meaning depends on the extensions of the physics as it appears in nature that we have yet to discover, this would be a simple representation of the physics in euclidean space, and it works just the same as normal conventional views of special relativity. The other more conventional view is that spatial and temporal relations depend on relative motion, and it is mathematically identical, in terms of observables, the former view is just what you get if yoj start with your physics in one frame, and then use a Galilean transform to see what is going on in a different reference frame, it is not what you would see in such a reference frame, but it is the same physics transformed by some linear transformation that does not preserve the form of the equations, not very hard to grasp if you ask me but people cling to these ancient views that we discovered the structure pf spacetime and so on, or that the ether was disproven and bla bla blah, and all it really amounts to is a misconception of what is known vs what is possible to postulate in the form of a specific representation of the intrinsic behaviour that matches experience/experiment. We need ypung smart people to grow up and ignore dogma and nonsense like that, saying thatbone representation is different in physical significance than the other is just humbug, they make the exact same predictions, the physics is identical, the only difference is what kinds of extensions look natural to explore to the intuition of a person that knows the subject well. We cannot make inferences about which option reflects nature in its full detail best off the back of fancies about what representations have been proven or historically won out, there is no proof that two identical things are different and history is irrelevant, there is no logic beyond guessing what to extend and what to leave as fundamental. A lot of physicists are confused about things like that, the field is there for the taking if you work hard, you should give it a go. There is a lot of basic logic and foundational mathematical work to do in physics, both with respect to methods of inferences, which has never been satisfactorily handled, and with respect to concrete mathematical foundations, ofc the mathematical foundations of physics should not be viewed as something closed, that is to be finished up neatly and printed in a pamphlet, it is open ended, but there is a lot that can be done that has never been dreamt of before, it doesn't look that open if you look sten to people talk about the frontiers, but they are somewhat ignorant of what is possible, ofc it doesn't apply universally, but if you are not that familiar woth the broader literature, then it is easy to get the impression that the mathematical foundations are very restricted, and that is nonsense, there are basic kinds of foundational progress left on the table, that could in principle have been done by newton, or even Archimedes in the sands of the Sicilian beaches. Classical mechanics never reached maturity before it was supplanted by a kind of silly and contrived framework of quantum mechanics, which has its merits and is useful, but is really only a way to define a statistical theory, that is not in a category of its own, destinct from classical mechanics, to me there is only mechanics, and quantum mechanics is part of the space of mechanics that can have random variables coupled or not as solutions. The divides we get fed by history are frivolous and meaningless, only the mathematical details and the physical inferences habe any real meaming to be thought about, but we humans have no shortage of desire for fables and nonsense as reasons for discrimination theoretically. Be ware, here there be morons. Anyway, got a bit dramatic there, i just wanted to say my piece because you seem like a smart young lad, and i think it is best if you ignore the reasons other people have for believing what they do, other than as curiosities to be understood and analysed, only that which is provable is true, and only that which is conserved by a shift in perspective is physically significant. And when it comes to buildong something new, a guess is indespencible, mathematical derivations are valid, but the reason for picking one possible extension to a framework over another never comes from proof unless it is forced upon you, do well not to fall into the trap of thinking something is forced because someone tells you so, and seek out the details that matter for determining what to think and what to pursue, do not listen to decrees, but charish suggestions offered openly.
You could use 3 shadows to show different projections of a flat 3D hyperplane in 4D space, right? Since you only need a straight line and a normal to define a 2D plane, you just need a symmetrical 2D shape and some mapping to define a 3D hyperplane, right?
I mean it works wonders, but it's totally up to every person to use what they like most! (And if you really think about it, hyperplanes' extrinsic orientations are just the traditional vector direction ☺)
There's a quiz attached to this video to check retention/understanding: quizwithit.com/start_thequiz/1723262855188x209768946150408200
This is the equivalent of writing all logic statements in terms of exclusively NAND gates.
which is actually commonly done when you are doing cmos logic
so
Actually, logic is really just reflections in vector spaces over Z/2Z
I'm more of an XOR mind but I like your style 😎
Taking it further, you could argue that any computation can be broken down into linear algebra over finite fields. The real magic is in how we choose to interpret those reflections and transformations.
Wow@@Lucky9_9 - are you like a true information expert? Everything *_is_* information including thought patterns, waves (and their combinations), vectors and their properties and of course clocks and their "ticking" 🖖
Damn, i never thought about the fact that any transformation is a rotatiom followed by a reflection and that you can do both in GA. Very cool!
No there is very little rotation overall. At small scales everything travels in nearly straight lines or very shallow curves, just zoom in far enough.
@@digbysirchickentf2315 or you might think that any straight motion is a rotation about a point at infinity (a revolution by a circle of enormous radius)
this is the geometric equivalent of compiling everything to mov statements
Nice you just won a sub. I was worried if the video was going to be a crackpot, but instead, it turned out you are not only totally sane, but brilliant and love geometric algebra! Hooray! And you actually taught me something new about GA, I look forward to applying it! :) Thank you
@@Sanchuniathon384 That's the best: Make people interested because they think I'm crazy, but then deliver! I'm glad it delivered :)
@@EccentricTuber Hi, excellent video. I liked your work. Wanted to know what are the application areas of this concept in terms of engineering.
@@sbasu4748 I wouldn't know as much about that, my background is in physics and mathematics. I think this can be used for efficient computations of motion, but don't quote me on it.
I came here thinking that it might have implications for the fundamentals of physics... But I'm not disappointed.
@@sbasu4748Y'all need chatGPT 😂
The concept of hyperplane reflection, while abstract, can be applied in various engineering fields, particularly in computational methods and optimization problems. Here are some potential application areas:
1. **Robotics and Motion Planning**: Hyperplane reflections can be utilized to optimize paths and motions in robotic systems, ensuring smooth and efficient navigation through space by reflecting trajectories across hyperplanes to avoid obstacles.
2. **Computer Graphics**: In graphics rendering, hyperplane reflection algorithms can enhance image processing techniques, especially in ray tracing where reflections and refractions are simulated for realistic visuals.
3. **Control Systems**: In control theory, hyperplane reflection can be used in the design of controllers that handle dynamic systems, particularly in managing stability and performance through feedback mechanisms.
4. **Signal Processing**: For filtering and analyzing signals, hyperplane reflections can be applied to transform and process multidimensional data in ways that reduce noise and enhance signal clarity.
5. **Optimization Algorithms**: In engineering optimization problems, reflecting across hyperplanes can be part of iterative methods to find minima or maxima, especially in multi-objective optimization scenarios.
6. **Structural Engineering**: In analyzing stresses and strains in structures, hyperplane reflection methods can simplify the modeling of symmetrical structures and reduce the computational complexity in simulations.
These applications leverage the mathematical properties of hyperplane reflection to solve complex problems efficiently, making them valuable in many advanced engineering tasks.
'A 3 dimensional hyperplane in a 4 dimensional space is beyond human cognition'
4d golf on steam: exist 😂
@@timmygilbert4102 Never in my life did I expect to be proven wrong by golf 😭😭
Actually, a 3 dimensional hyperplane is a static slice of our world in each moment.
@CodeParade is legit
exactly @@true_xander
Im gonna follow the founders of thermodynamic statistics soon.
@@EstParum Please stay alive 😊
Reflecting into this comment section
I laughed out loud of the hyper plane by chat gpt
Thank you
The notion that any bivector can be represented as the geometric product of the volume element and a vector is extremely important. Its a natural way to encode the topology of a 3-sphere in terms of vectors, as it is the group of all unit quaternions (bivectors), and it allows to construct a counterexample to bell's theorem. The choice of handedness (the sign of the volume element) is not merely a convention, it represents two entirely different tangent bundles on the manifold, and acts as a de-facto hidden variable. Just like a left hand in a mirror looks like a right hand, so do particles and photons: this is why it looks like to us that bertlmann's socks are both left handed or both right handed!
Ultimately, bell failed because he assumed that the elements of reality in an EPR experiment commute, which is just not true on a 3-sphere.
To me this is jargon poetry, and I'm here for it.
@@ivocanevo :) It's also very interesting that everything seems to have angular momentum, the quantum world is governed by Planck's constant, with units of angular momentum... Another property of a 3-sphere is that it is parallelizable, i.e. you can construct 3 linearly independent vector fields on the surface (unlike the 2-sphere).... The result of the parallelism though is that torsion is everywhere non-vanishing!
Is this complete gobeldigoog?
@@carlpanzram7081 it does sound like it doesn't it? Unfortunately I can cram only so much stuff in a single comment. If you want more, look up Joy Christian's papers on the Arxiv. I also suggest the channel PathfinderPhysics for a simple analysis, although the relevant series of videos is not yet fully available
Disney has been aware of it all along.
What do you mean?
@@GU-jt5fe their animation history
Reflections are just 180 rotations thru a higher dimention.
@@5hape5hift3r I'd argue this view doesn't help with intuition, although interesting on its own 😊
That kept me thinking, what about point reflections?
I think this is only true for 2D objects, not 3D objects like my hands or my feet.
@@linuxp00 point reflections in even dimentions are rotations.
But odd dimentions are rotations with a rotation thru an additional dimention.
@@pronounjow 3d point reflections are rotation with an additional higher dimensional rotation.
Really well made video, ive personally never taken a geometric algebra course before so it was nice to have the ideas in this video so well explained.
@@Jiff8484 Thanks, I'm super glad you liked it! Means a lot
Great work! Would love to see this tie into spinors. 👍
You'll like my next video, then! ☺
Yes!
This is like describing every particle with a lone quark.
Good explanation, good visuals, great topic. Instant subscription and plan to watch the rest of your work! Thank you for the way that you covered this.
Thank you for not being a crackpot video 🙏🙏🙏🙏
Used heavily in Quantum Physics. Though, i wish, QM would delve deeper into using matrix and graphical representations of the vectors, than just use them abstractly.
Thank you for sharing this. I also appreciate your clear and well considered appropriate you used in this video.
Way over my head but very interesting, will check out the other videos, thank you
I can't admit I understood half of that, but I'm curious about practical implications like say in a classical rigid body collision simulation?
I've been considering making a Vampire survivors style game from scratch and was thinking about the physics simulation so I could have like 10k enemies on screen without slowdown, some data structures use hyperplanes for optimization (like kd-trees) and help with pruning collision candidates but I've been procrastinating on starting the project.
This was interesting. Any resources for further reading and to get a better intuitive understanding? I love the visualisations btw. Helped a lot to understand the physical representation
I have a video on resources for GA, and there's also the Bivector.net website and Discord server that will answer any questions you have!
I don't know anything about algebraic geometry, but this looks cool.
this is geometric algebra, which is (confusingly) not the same thing as algebraic geometry. algebraic geometry is great too
@connemignonne lol, I guess I need to read up on my geobraic algometry.
@@goodfortunetoyou scheme theory goes crazy hard in algebraic geometry. go check out the definition of the spectrum of a ring (like a good polynomial ring C[x,y]) and then look up what an affine scheme on it is, changed my life
that's a clever take ... keep this sh*t going ☺
Will do, thanks for the encouragement!
If i was living in quake and my world was truely 3d and one dimension of time wouldnt some objects hyperplane be the record of movements from start to end of time
Worldlines can be records of movement of inertial (constant velocity) frames, but in general hyperplanes are not records of movement.
@@EccentricTuber hmmm
Reflection is key.
🐲✨🐲✨🐲✨
"Before I start, I must see my end. Destination known, my mind's journey now begins. Upon my chariot, heart and soul's fate revealed. In time, all points converge; hope's strength resteeled. But to earn final peace at the Universe's endless refrain, we must see all in nothingness... before we start again."
🐲✨🐲✨🐲✨
-- Diamond Dragons (series)
It's kind of like in 3d software, a polygon has a normal, and this reflection is just flipping the normal. (Pretending the polygon is infinite in its planar surface, obviously, to theorize it into a hyperplane)
I’m not sure if this is a misinterpretation, but does this mean that “all motion stems from the phenomenon of reflections”?
If so, I’m having trouble understanding how vibrations and fluctuations could arise from nothing but pure reflective activity.
Good question! It does, effectively! Vibrations and fluctuations could be thought of as the spontaneous creation of reflective hyperplanes or the change of their relative separation. I'm sure there's a more elegant way to put it, but I hope that helps!
Vector = hyperplane graphic killed me lmao
Isn't motion defined as change with respect to time? What does the topic of this video have to do with "moving"?
I’m confused about how the relationship between movement and reflection is linked.
I also understand that evolution is change over time.
Good question! Yes, although the math for a rotation/translation changing in time and for ones that don't is exactly the same! Effectively, there would just be change in the relative distance/separation of the hyperplanes over time. Hope this helps!
@@EccentricTuber This might be a language barrier thing, but I feel like now you're just saying that motion of a geometrically complex thing is motion of geometrically simpler things, which doesn't really explain motion. It just kicks the can further down the road.
Thank you. I tough exactly the same while watching this video.
@@lumipakkanen3510 We can continue talking about this over Discord. If you go to the Bivector server, ask for me and someone'll ping me.
I am going to tell my kids this proves light has asynchronous two way speeds.
@@815TypeSirius As in the speed of light would vary for two observers in a co-inertial frame?
@EccentricTuber basically no one knows the one way speed of light. Wheee.
How far are we from multi-linear transformations - tensors - and Special Orthogonal group of symmetries? Special unitary?
Tensors are related to GA, but they're typically just the components of multivectors. As for Special Unitary, the compositions of reflections are elements of Pin(n), which double cover O(n). The reflections that compose to give rotations and translations are elements of Spin(p,q,1) which double cover SO(p,q,1), so we're not far from Special unitary symmetries at all! Note that for n=3, Spin(3)=SU(2).
Never seen a spinorial representation of motion. I know that it can be used for the Ising model solution but idk what would be moving there, hypothetically its all just spins flipping on a 2D grid. Maybe projective geometric algebra could be used in statistical physics?
P.D: I took a peek at the paper and I see a block matrix decomposition which is something you also do with the transfer matrix of the Ising model. Gotta brush up on the solution but maybe then I'll go into the discord and ask a few questions
Thanks for another reminder to start studying GA more carefully.
In nature what would the shapes reflect off of the vaccuum?
I am so out of my depth here 😵💫
(1)2:34 it was just an Hyper Duper AIOplane :)
Please correct me if I am wrong (which I could be!): I think that an isometric transformation can actually modify the shape of an object, because you can have metrics that distort shapes. An example is length contraction for relativity. Distances remain the same, but lengths, and therefore shapes warp based on speed. Am I correct here?
Good point! Yes, Lorentz boosts indeed are isometric transformations in the sense that hyperbolic angles are conserved, and distances satisfy the Minkowski metric rather than the Euclidean metric. So the definition I gave technically includes it, given that the definition of "shape" is modified to work with Minkowski space. That's actually why I mentioned spacetime at the end of the video: You can do relativity with this formalism (and it's actually pretty powerful)
So, gravity is a hyperplane within the time-space-space
This seems like a way of saying all straight lines can be represented as a linear combination of sinusoids (taylor series or otherwise)
Unfortunately, I don't think that's a valid application of the idea. It would be interesting to see the proper application though, maybe someday...
#Deep
reflecting into her DMs
@@katie-ampersand 😭😭😭
Might this imply that spacetime and the mass and movement of bodies is derived from 2d hyperplanes eg a spinor field
It warrants more investigation, although I'm not sure myself as I've yet to explore this. Also, it's not a given they'd be 2D hyperplanes (lines).
@EccentricTuber sorry I meant 3d hyperplanes eg. Spinors..I understood that for glide symmetry you need an odd number of dimensions so 2d would need additional 1d to translate the object. Is that right?
I can't believe I've watched this whole 15 minute video and understood nothing 😂
What exactly does it mean that you can express motion as a reflection? How does this theory relate to practical reality? Does it have a use?
It's actually very useful for studying spinors and pointors! My latest video uses reflections to describe spinors. It's also good from a Comp. Graphics perspective, I'm told.
is it just me, or is this a sort of convoluted way of extrapolating motion kind of like a fourier transform of linear motion., Are the trajectories of objects in hyperspace able to be quantized to the smallest level? becuase it didn't seem like it could in your video. (probably because it went way over my head)
Good question! So this is neither like a fourier transform, nor exactly analogous to "quantization". It's just a way to break motion into invariant reflections that can be performed in arbitrary order. It's conceptually similar to the prime factorization of an integer! Hope this helps!
@@EccentricTuber That makes sense! thanks a lot man!
A 3d hyperplane is literally just space, isnt it?
Not exactly, it is a 3-dimensional object embedded in a 4-dimensional space!
@@EccentricTuber yeah I know, but it just made me think of a discrete moment of space in time, in some world where the universe worked in discrete time
Geobegra 3:00 lol
Parmenides!!!
I will not be keeping it civil in this comment section, all of you are wrong and your fifth grade teacher probably still hates you
Could you please elaborate on your perspective? I believe we could gain valuable insights from your thoughts. Therefore, I would appreciate it if you could share in detail how many of us might be mistaken.
@@saftheartist6137 ok well, you’re off the mark on a lot here. Let me give you some points:
Point #1: We live in the context.
Point #2: We beat Medicare
Point #3: Obamna!
Point #4: I couldn’t give you a verse from the Bible, all the parts are so good, it’s a great book
Point #5: America can be defined in a single word: assahhghhh wsaagjjn
Don’t recognize these? Here you see wisdom from America’s most prominent politicians (I can point you to their sources if you want to watch them in their full glory). Reflect on this wisdom and you will understand
@@longextinct I'm crying 😭😭😭 you sound like my friend
@@longextinct Oh, so you provide humor? Well, that’s fine. There’s a time and place for that too. However, the category of this video is education; your humor would be more appreciated in videos categorized under comedy and then in the category of entertainment.
@longextinct I'm here after finishing the Bodhidharma any% speedrun (cave wall staring) to confirm that your belligerent stance is correct. Also I learned everything I need to know about representation theory from listening to Rubberband Man by the Spinors on repeat. Seven years of that and I was primed for the first TikTok watermelon video I could find. I lost all respect for Gallagher once I was shown the potential of rubber bands to destroy fruit. A mallet is such a barbaric implement--and the word hardly sounds like geometric terminology--case closed. Enough of this stream of consciousness business. Let me get to the point.
Yes, following decades of brinksmanship on both sides a shooting war has broken out between me and a crack commando unit of elementary school teachers. Wanted by the federal government, they survive as soldiers of fortune and are currently based out of Los Angeles. If you have a problem, if no one else can help, if you can find them... maybe you can hire the A-Team. I say "you" because I definitely cannot hire the A-Team. There is as you rightly guessed too much enmity on the part of my fifth grade teacher.
I think I'm dumb. I watched your video but I still don't understand how Motion is Reflection. Can't you explain it in simpler terms?
@@damienasmodeus928 Don't worry, you're not dumb! It's a tough subject if you don't have a background in geometry or Geometric Algebra. The idea is that any form of motion can be decomposed into successive reflections. So a rotation is just two reflections, and so is a translation. But they can also be composed, so a motion that describes a rotation AND a simultaneous translation can still be described by reflections. If you need to learn more, I recommend we continue this conversation in the Bivector discord! It's in the description 🙂
Yes.
I think I'm dumb or brain rotted bc i gave up at 3:52
@@daolinh108 Don't worry, you're not dumb! It's a tough subject if you don't have a background in geometry or Geometric Algebra. The idea is that any form of motion can be decomposed into successive reflections. So a rotation is just two reflections, and so is a translation. But they can also be composed, so a motion that describes a rotation AND a simultaneous translation can still be described by reflections. Does that help?
Broski, imagine that you reflect the world along some spatial axis, but every causal relationship stays invariant, would you be able to tell the difference if i did this to you and the world you are in? Trippy thing to think about, with respect to what relationships are intrinsically identical in experience given some postulated physical state of the world, you should do some physics, there is a lot of cool stuff to do, and a lot of silly misconceptions floating around that have their root in different forms of representation taken to seriously as physical theories, that is, a lot of folks, professional and kook alike just say words and cling to beliefs that are not well motivated, because they fail to understand that what is physically meaningful is that which is independent of representation. For example, in the context of relativistic effects you can define lenght contraction to be physical and only give real lenghts in one coordinate system, without changing any of the physics what so ever, this in itself os an arbitrary choice of representation, whether it has any further physical meaning depends on the extensions of the physics as it appears in nature that we have yet to discover, this would be a simple representation of the physics in euclidean space, and it works just the same as normal conventional views of special relativity. The other more conventional view is that spatial and temporal relations depend on relative motion, and it is mathematically identical, in terms of observables, the former view is just what you get if yoj start with your physics in one frame, and then use a Galilean transform to see what is going on in a different reference frame, it is not what you would see in such a reference frame, but it is the same physics transformed by some linear transformation that does not preserve the form of the equations, not very hard to grasp if you ask me but people cling to these ancient views that we discovered the structure pf spacetime and so on, or that the ether was disproven and bla bla blah, and all it really amounts to is a misconception of what is known vs what is possible to postulate in the form of a specific representation of the intrinsic behaviour that matches experience/experiment. We need ypung smart people to grow up and ignore dogma and nonsense like that, saying thatbone representation is different in physical significance than the other is just humbug, they make the exact same predictions, the physics is identical, the only difference is what kinds of extensions look natural to explore to the intuition of a person that knows the subject well. We cannot make inferences about which option reflects nature in its full detail best off the back of fancies about what representations have been proven or historically won out, there is no proof that two identical things are different and history is irrelevant, there is no logic beyond guessing what to extend and what to leave as fundamental. A lot of physicists are confused about things like that, the field is there for the taking if you work hard, you should give it a go. There is a lot of basic logic and foundational mathematical work to do in physics, both with respect to methods of inferences, which has never been satisfactorily handled, and with respect to concrete mathematical foundations, ofc the mathematical foundations of physics should not be viewed as something closed, that is to be finished up neatly and printed in a pamphlet, it is open ended, but there is a lot that can be done that has never been dreamt of before, it doesn't look that open if you look sten to people talk about the frontiers, but they are somewhat ignorant of what is possible, ofc it doesn't apply universally, but if you are not that familiar woth the broader literature, then it is easy to get the impression that the mathematical foundations are very restricted, and that is nonsense, there are basic kinds of foundational progress left on the table, that could in principle have been done by newton, or even Archimedes in the sands of the Sicilian beaches. Classical mechanics never reached maturity before it was supplanted by a kind of silly and contrived framework of quantum mechanics, which has its merits and is useful, but is really only a way to define a statistical theory, that is not in a category of its own, destinct from classical mechanics, to me there is only mechanics, and quantum mechanics is part of the space of mechanics that can have random variables coupled or not as solutions. The divides we get fed by history are frivolous and meaningless, only the mathematical details and the physical inferences habe any real meaming to be thought about, but we humans have no shortage of desire for fables and nonsense as reasons for discrimination theoretically. Be ware, here there be morons. Anyway, got a bit dramatic there, i just wanted to say my piece because you seem like a smart young lad, and i think it is best if you ignore the reasons other people have for believing what they do, other than as curiosities to be understood and analysed, only that which is provable is true, and only that which is conserved by a shift in perspective is physically significant. And when it comes to buildong something new, a guess is indespencible, mathematical derivations are valid, but the reason for picking one possible extension to a framework over another never comes from proof unless it is forced upon you, do well not to fall into the trap of thinking something is forced because someone tells you so, and seek out the details that matter for determining what to think and what to pursue, do not listen to decrees, but charish suggestions offered openly.
You could use 3 shadows to show different projections of a flat 3D hyperplane in 4D space, right? Since you only need a straight line and a normal to define a 2D plane, you just need a symmetrical 2D shape and some mapping to define a 3D hyperplane, right?
NAAHHHH why is everyone using the dual representation? That's totally backwards!
I mean it works wonders, but it's totally up to every person to use what they like most!
(And if you really think about it, hyperplanes' extrinsic orientations are just the traditional vector direction ☺)
Your mom is just a reflection
Interested to see if this covers the topic of motors
Guys what if the real hyperplane is the one ai tried to tell us and this guy is just disinfo, just sayin
This video just confirms that motion is: ceasing to exist at one spot and staring to exist in a neighboring spot.
You can not draw this conclusion from this video.