Dirac's Belt Trick: Why a 2π rotation twists space but a 4π rotation fixes it
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- Опубликовано: 24 июл 2024
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When you twist your arm or a belt by 360 degrees, the hand or endpoint is back to where it started but the rest of your arm or belt is still twisted. But if you do a 720 degree twist, you can manage to untwist your arm or belt! This is known as Dirac's Belt Trick or the Balinese Cup Trick. This crazy fact is even connected to physics with spin 1/2 particles, so let's try and figure out why! We will study rotations in 2 and 3 dimensions, and specifically study them topologically as opposed to algebraically as you might have seen before with rotation matrices. For a 2D rotation this is identified with points on a circle S^1. For a 3D rotation we need both an axis or rotation and an angle of rotation and we identify this with the solid ball of radius pi where a point in the ball gives a vector from the origin to the point that is our axis of rotation and the length of this vector is the angle. There is a catch: we have a double counting along the boundary so we have to identify antipodal points as the same. If you eliminate the origin (ie no rotation) this is sometimes called the Special Orthogonal Group SO(3) which is topologically the same as 3D Real Projective Space RP(3). A belt is then a path and I show an explicit way I can continuously deform the 4pi rotation path back to the identity.
0:00 Dirac's Belt Trick
1:37 2D rotations and the Circle
2:35 Axis and angle of 3D rotations
3:24 Modelling rotations using a solid ball
5:31 The double counting problem and identifying antipodes
6:43 Paths of Rotations
9:49 Deforming the 4π path
12:49 Thanks to Brilliant.org/TreforBazett
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I am going to admit that I immediately grabbed a nearby cup and tried the 4pi rotation after watching the first 24 seconds. 😀
Just a thought. But I have never seen any evidence that time is more than a compact dimension. nobody moves around in time. even if it was scalar it would destroy the resolution of the universe. energy could bleed in temporal as well as physical dimensions.
@@KaliFissure I think waves that have frequency instead of amplitude as an energy signature are doing just that but how exactly does this video help with this line of reasoning can you please explain?
@@fabiopilnik827 I don't understand this idea of frequency without amplitude.
@@KaliFissure The distinction is which one quantifies the energy of the wave and not if they are inseparable.
@@fabiopilnik827 the universe is a physical place. It isn't math. You can't pull apart parts of a wave. What is amplitude without frequency? Static charge. And frequency without amplitude is a flat straight line.
The belt trick reminds me of a lecture I saw by Eugen Merzbacher about 50 years ago. He was lecturing to a bunch of physics teachers at the Baltimore meeting of the APS. In his magic trick, he took an 8" cardboard square with holes punched in the four corners. A six-foot string was tied to each corner and he had four physics teachers in the audience hold the ends of the four strings He then rotated the cardboard by 2 pi around a horizontal axis, twisting two of the strings together on two opposite sides of the square. He asserted that the strings couldn't be untwisted without changing the orientation of the cardboard (or looping the strings over the bodies of the physics teachers). A few people tried it, but soon gave up. Then Merzbacher gave the cardboard another 2-pi rotation along the same axis, in the same direction, and showed that the strings could now be untangled by a couple of deft moves, lifting one string over the cardboard and another string underneath without changing the orientation of the cardboard. He cautioned the physics teachers to practice this before performing it for their classes. I remember being amazed, and wondering if this could have some connection to Fermion wave functions, which flip signs after a 2π coordinate rotation, but are unchanged by a 4π rotation. I never was good enough at theory to figure it out.
Wow
I've independently showed the cup trick and tried to explain loops in the space of rotations to my math-inclined friends, but now I have a video I can point them to, thanks! (And it's a lot clearer than the physics-slanted videos on the same topic seem to be.)
This is why I like unit quaternions, which Hamilton discovered more than half a century before Dirac rediscovered them. They have one real scalar and a vector along 3 imaginary axes and are a natural extension of complex numbers. For orientations, they're normalized so the absolute value is 1. It all works out nicely, and as a bonus, you don't have to remember the right-hand rule, because already _ij=k,_ _jk=i,_ and _ki=j._ They also have no axis bias or gimbal lock so are ideal to represent orientations in computer graphics, and special relativity is more elegant using them.
Clifford be like: eh, imma eat quaternions and gibbs like vectors; and make one giant clifford vector ;)
@@yash1152 Nom nom nom!
But maybe it's a Lie.
I use this in threejs for orienting objects towards the central viewer
Thanks again to Brilliant for sponsoring todays video! ► brilliant.org/TreforBazett/ A few notes that didn't make it in the video:
1) When you think about SD rotations, you might think about the "Special Orthogonal Group", represented as matrices where AA^T=A^TA=I and det(A)=1, specifically the 3x3 case called SO(3)
2) Note carefully that the zero matrix is not an element of SO(3) (what is the axis, exactly) so we are thinking of "no rotation" as not being a rotation at all.
3) SO(3) is in bijective correspondence with the space I talk about in the video IF you remove the origin. I.e. a ball with radius 0
I think there's a small confusion here. It's true that the zero matrix is not in SO(3), but that isn't the "do nothing" operation (it shrinks everything to zero which is definitely not allowed). The do nothing operation is given by the identity matrix, which IS in SO(3). So you should be able to identify the boundary-glued ball with SO(3) without having to remove any points from it first.
On a different note, the video was fantastic!
This is really interesting!! I feel like this could apply to DNA/RNA and protein folding.
I mean it certainly does, and it's reasonably likely that someone has, or will after seeing this video :P. However I think mathematicians are often specifically involved in protein folding software. It shows up also in the electron "spin" where "twists" are made and then undone in the field, I don't think it is literally the case since spin isn't meant to be treated as literal, but it does help illuminate the ways that the field lines process. I'm just regurgitating PBS Spacetime, you'd need to watch their video on spin if you actually wanted to understand it because I don't fully. Still really cool.
i just finished my first year in math courses, but from what i hear from smarter people, is topology, specifically regarding knots, and group theory are super important for understanding protein folding.
It is
so from what I learned from basic algebraic topology, it is the manifestation of the fact that RP3 (or just RP2 from the CW structure of building RPn inductively from attaching a cell to Rpn-1 and the fact that we are working with the fundamental group) has the fundamental group Z/2Z and the generator of the group is the 2pi rotation
Exactly!
Yeah, the fact that this video helped me visualize that was definitely my favorite part!
Thank you so much Sir.I was really having hard time for a day to imagine the fundamnental group of SO3 and u gave a crystal clear intuition for it without pen and paper 😢😢❤
Around 7:30, are your belt rotations correct? When you say pi/2, I see pi. And when you say pi, I see 2pi.
Oh quite right! Yes, overlayed the wrong pair of images. Hopefully it still makes sense:/
The 2π-twisted belt is a homotopically nontrivial loop γ in SO(3).
The 4π-twisted belt is the square γ^2 of γ and it can "clearly" be homotoped to a "straight belt", so it's homotopically trivial.
Equivalently, the belt is a nonclosed path η from the identity 1 to the element -1 in SU(2)=Spin(3)=Sp(1)=U(1, *H* ), the universal cover of SO(3), which happens to be a double cover. η is the lift of γ from SO(3) to it's universal cover SU(2).
In turn, the lift of γ^2 is a loop in SU(2) (necessarily homotopically trivial cause SU(2) is simply connected).
Yeah exactly what I was going to say
Yup exactly. Good job at stating the obvious
If he'd done that cup trick a few centuries ago he would have been burned at the stake.
This is interesting because it clearly shows the link between 3D space and the projective plane. One day, I built a model of the projective plane in 3D and figured out there are actually 2 non-superimposable versions of it ("enantiomers"). Is that right?
5:15 minor correction: the direction of rotation should be the opposite to it (same as the pi example at 5:04)
Oh thanks! Actually the mistake is the 5:04 example at pi (I think I played the slider backwards oops). That is, our convention is to do a counterclockwise rotation looking down the axis from the origin.
@@DrTrefor Why not the Right Hand Thumb rule convention? I see other examples at 5:31 which also follow that.
The correspondence could be clearer, but it looks like v is the axis and ||v|| is the angle of rotation (angle must be assumed in right-handed orientation or something like that). Most of the examples were unit vectors and numbers are not provided for the other direction.
The pictures on the shirt is great! 😊
"a full rotation disturbs space but 2 rotations fixes it"
is this why matter particles have a spin 1/2?
Sir, Can you please explain how subtables can be formed in the overleaf?
When you rotate the cup clockwise (from the viewer's perspective) by 2pi, you twist your arm, and when you then rotate the cup counter-clockwise, you untwist your arm. Mind-blowing. Decent job on the concealing part with the big movements though. Most people in a live audience might fall for that.
Both of my rotations are in the same orientations, I promise!
@@DrTrefor No, once you move it below your arm, once above your arm.
Something to do with surface tension when you have a little heavy ball on top of three floating. Empty space is a falling surface. The first falling creates the next.
In the book visualizing quaternions by andrew hanson explains this dirac belt trick (originally performed to explain electron spin).
Are quaternions related to electron spin.
I have still not understood how quaternions rotate.
They are indeed! And to the group called “special unitary” which is related to what I talk about here but slightly different
They absolutely are. They are related to sl(2,C) which is the lie algebra that dictates how spinors (like electrons) transform under Lorentz transformations.
With 2D rotations you do have an axis of rotation (except it's always perpendicular to your plane), and what you should really do to make this analogy work is: in 2D, use a vector perpendicular to the plane at the origin, with length between π and -π; in 3D, use a vector at the origin, with length between π and -π. To avoid double-counting π-rotations, you have to apply the same gluing in both cases. Now the two cases are really analogous. And at that point, you see that 2D rotations are 1:1 to the interval [-π, π] - gluing = (-π, π] in the same way that 3D rotations are 1:1 to the sphere with radius π - gluing. The fact that the interval (-π, π] is 1:1 with a circle is an unnecessary complication here IMO.
Love it 😍 very nice 👍
I hope your arm is OK after the first 2*pi rotation.
I just finished my vector calculus final exam. I want to thank you for everything you have taught me, but I don’t want anything to do with calculus till next semester. I’m I’m gonna have to unsubscribe for now, but I will be back. Once again thank you
haha fair enough!
Both Matter and Energy described as "Quanta" of Spatial Curvature. (A string is revealed to be a twisted cord when viewed up close.)
Is there an alternative interpretation of "Asymptotic Freedom"? What if Quarks are actually made up of twisted tubes which become physically entangled with two other twisted tubes to produce a proton? Instead of the Strong Force being mediated by the constant exchange of gluons, it would be mediated by the physical entanglement of these twisted tubes. When only two twisted tubules are entangled, a meson is produced which is unstable and rapidly unwinds (decays) into something else. A proton would be analogous to three twisted rubber bands becoming entangled and the "Quarks" would be the places where the tubes are tangled together. The behavior would be the same as rubber balls (representing the Quarks) connected with twisted rubber bands being separated from each other or placed closer together producing the exact same phenomenon as "Asymptotic Freedom" in protons and neutrons. The force would become greater as the balls are separated, but the force would become less if the balls were placed closer together. Therefore, the gluon is a synthetic particle (zero mass, zero charge) invented to explain the Strong Force. An artificial Christmas tree can hold the ornaments in place, but it is not a real tree.
String Theory was not a waste of time, because Geometry is the key to Math and Physics. However, can we describe Standard Model interactions using only one extra spatial dimension? What did some of the old clockmakers use to store the energy to power the clock? Was it a string or was it a spring?
What if we describe subatomic particles as spatial curvature, instead of trying to describe General Relativity as being mediated by particles? Fixing the Standard Model with more particles is like trying to mend a torn fishing net with small rubber balls, instead of a piece of twisted twine.
Quantum Entangled Twisted Tubules:
“We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct.” Neils Bohr
(lecture on a theory of elementary particles given by Wolfgang Pauli in New York, c. 1957-8, in Scientific American vol. 199, no. 3, 1958)
The following is meant to be a generalized framework for an extension of Kaluza-Klein Theory. Does it agree with some aspects of the “Twistor Theory” of Roger Penrose, and the work of Eric Weinstein on “Geometric Unity”, and the work of Dr. Lisa Randall on the possibility of one extra spatial dimension? During the early history of mankind, the twisting of fibers was used to produce thread, and this thread was used to produce fabrics. The twist of the thread is locked up within these fabrics. Is matter made up of twisted 3D-4D structures which store spatial curvature that we describe as “particles"? Are the twist cycles the "quanta" of Quantum Mechanics?
When we draw a sine wave on a blackboard, we are representing spatial curvature. Does a photon transfer spatial curvature from one location to another? Wrap a piece of wire around a pencil and it can produce a 3D coil of wire, much like a spring. When viewed from the side it can look like a two-dimensional sine wave. You could coil the wire with either a right-hand twist, or with a left-hand twist. Could Planck's Constant be proportional to the twist cycles. A photon with a higher frequency has more energy. ( E=hf, More spatial curvature as the frequency increases = more Energy ). What if Quark/Gluons are actually made up of these twisted tubes which become entangled with other tubes to produce quarks where the tubes are entangled? (In the same way twisted electrical extension cords can become entangled.) Therefore, the gluons are a part of the quarks. Quarks cannot exist without gluons, and vice-versa. Mesons are made up of two entangled tubes (Quarks/Gluons), while protons and neutrons would be made up of three entangled tubes. (Quarks/Gluons) The "Color Charge" would be related to the XYZ coordinates (orientation) of entanglement. "Asymptotic Freedom", and "flux tubes" are logically based on this concept. The Dirac “belt trick” also reveals the concept of twist in the ½ spin of subatomic particles. If each twist cycle is proportional to h, we have identified the source of Quantum Mechanics as a consequence twist cycle geometry.
Modern physicists say the Strong Force is mediated by a constant exchange of Gluons. The diagrams produced by some modern physicists actually represent the Strong Force like a spring connecting the two quarks. Asymptotic Freedom acts like real springs. Their drawing is actually more correct than their theory and matches perfectly to what I am saying in this model. You cannot separate the Gluons from the Quarks because they are a part of the same thing. The Quarks are the places where the Gluons are entangled with each other.
Neutrinos would be made up of a twisted torus (like a twisted donut) within this model. The twist in the torus can either be Right-Hand or Left-Hand. Some twisted donuts can be larger than others, which can produce three different types of neutrinos. If a twisted tube winds up on one end and unwinds on the other end as it moves through space, this would help explain the “spin” of normal particles, and perhaps also the “Higgs Field”. However, if the end of the twisted tube joins to the other end of the twisted tube forming a twisted torus (neutrino), would this help explain “Parity Symmetry” violation in Beta Decay? Could the conversion of twist cycles to writhe cycles through the process of supercoiling help explain “neutrino oscillations”? Spatial curvature (mass) would be conserved, but the structure could change.
=====================
Gravity is a result of a very small curvature imbalance within atoms. (This is why the force of gravity is so small.) Instead of attempting to explain matter as "particles", this concept attempts to explain matter more in the manner of our current understanding of the space-time curvature of gravity. If an electron has qualities of both a particle and a wave, it cannot be either one. It must be something else. Therefore, a "particle" is actually a structure which stores spatial curvature. Can an electron-positron pair (which are made up of opposite directions of twist) annihilate each other by unwinding into each other producing Gamma Ray photons?
Does an electron travel through space like a threaded nut traveling down a threaded rod, with each twist cycle proportional to Planck’s Constant? Does it wind up on one end, while unwinding on the other end? Is this related to the Higgs field? Does this help explain the strange ½ spin of many subatomic particles? Does the 720 degree rotation of a 1/2 spin particle require at least one extra dimension?
Alpha decay occurs when the two protons and two neutrons (which are bound together by entangled tubes), become un-entangled from the rest of the nucleons
. Beta decay occurs when the tube of a down quark/gluon in a neutron becomes overtwisted and breaks producing a twisted torus (neutrino) and an up quark, and the ejected electron. The production of the torus may help explain the “Symmetry Violation” in Beta Decay, because one end of the broken tube section is connected to the other end of the tube produced, like a snake eating its tail. The phenomenon of Supercoiling involving twist and writhe cycles may reveal how overtwisted quarks can produce these new particles. The conversion of twists into writhes, and vice-versa, is an interesting process, which is also found in DNA molecules. Could the production of multiple writhe cycles help explain the three generations of quarks and neutrinos? If the twist cycles increase, the writhe cycles would also have a tendency to increase.
Gamma photons are produced when a tube unwinds producing electromagnetic waves. ( Mass=1/Length )
The “Electric Charge” of electrons or positrons would be the result of one twist cycle being displayed at the 3D-4D surface interface of the particle. The physical entanglement of twisted tubes in quarks within protons and neutrons and mesons displays an overall external surface charge of an integer number. Because the neutrinos do not have open tube ends, (They are a twisted torus.) they have no overall electric charge.
Within this model a black hole could represent a quantum of gravity, because it is one cycle of spatial gravitational curvature. Therefore, instead of a graviton being a subatomic particle it could be considered to be a black hole. The overall gravitational attraction would be caused by a very tiny curvature imbalance within atoms.
In this model Alpha equals the compactification ratio within the twistor cone, which is approximately 1/137.
1= Hypertubule diameter at 4D interface
137= Cone’s larger end diameter at 3D interface where the photons are absorbed or emitted.
The 4D twisted Hypertubule gets longer or shorter as twisting or untwisting occurs. (720 degrees per twist cycle.)
How many neutrinos are left over from the Big Bang? They have a small mass, but they could be very large in number. Could this help explain Dark Matter?
Why did Paul Dirac use the twist in a belt to help explain particle spin? Is Dirac’s belt trick related to this model? Is the “Quantum” unit based on twist cycles?
I started out imagining a subatomic Einstein-Rosen Bridge whose internal surface is twisted with either a Right-Hand twist, or a Left-Hand twist producing a twisted 3D/4D membrane. This topological Soliton model grew out of that simple idea. I was also trying to imagine a way to stuff the curvature of a 3 D sine wave into subatomic particles.
.==
i much prefer to think about rotations in a plane rather than around an axis. it makes three-dimensional rotations much simpler in calculation (using geometric algebra). the axis and the plane give the same information (the axis is just the geometric algebra dual of the plane of the rotation) but the plane is much more intuitive to me
Will you be creating a topology playlist on this channel ?
I've been wondering about this. I've been doing a lot of topology/geometry videos recently and just putting them in my "Cool Math" playlist, but I should probably make a dedicated topology playlist at some point that starts with some gentle introductions to the ideas:D
@@DrTrefor Wonder no more .
Make it so.
:))))
@@DrTrefor sir please solve this
Σnx^(n-1), x>1 for convergence
Ok my imagination is trapped in a gordian knot now. Thanks
haha awesome:D
I'm confused why you can remove those 4pi paths from the edge of the circle? I understood everything up until that moment.
Imagine going the other way: start with a loop entirely within the circle and pull it across the boundary. If you pull it up across the top of the circle, the loop would reappear from the bottom of the circle. So now you have a loop which goes up through the top, appears on the bottom, turns around and then goes back through the bottom back to where it came from. If you pinch the points that lie on the boundary together (one set on the top, and one set on the bottom), you have reconstructed the original 4π path.
didn't get anything, need to rewatch it again
What are the applications in physics for example?
Check out Spin 1/2 particles, same basic phenomena!
There is an anti twisting mechanism invented by Dale Adams in 1971 that is very much related to the Belt Trick. I built a model of this mechanism. Take a look if you like.
I have a question, couldn't you associate every 3d rotacion with a single point on the sphere? I mean grab an arbitrary point (like the north pole) and then for every point on the sphere, isn't there one and just one rotation that moves the arbitrary point to the chosen point?
> _"isn't there one and just one rotation that moves the arbitrary point to the chosen point?"_
umh, nooo, i guess? 'cz then you are not bound to the shortest path... u can go in curves about it. it's fixing the axis which only leaves you with 2 options (CW or CCW)
Wouldnt tht somehow means 1=0.5 and 2=1 at the same time..?
Would that make 0, nil, null, non existent? Wouldnt nil exist if it has to be expressed..? That would make sense for 0 not to exist to some extent....
electron half spin is an artifact of the fact that time is a compact dimension only one Planck second in size. through Kuramoto synchrony this evolves a temporal hyperplane. one side matter the other antimatter. an electron does one orbit on this side and then does an internal orbit on other side of temporal membrane. 2 orbits to make one orbit.
Like wouldn't enegy dissipate forward and backward in time as well if time was scalar?
The universe would be amorphous.
@@KaliFissure Time is not a scalar in relativistic QFT.
@@zray2937 what is it then?
@@KaliFissure It's a component of a Minkowski 4-vector.
@@zray2937 to parrot something is not to explain it or understand it. I'm sorry but just saying what you were taught to do and say in school isn't sufficient. St Einstein " if you can't explain it to a child you probably don't understand it yourself"
Idk if you go into this in the video, but since rotation is a change from axus to another, could there exist some higher dimensional rotation that goes from plane to plane instead of just axis to axis?
At the final step of the deformation of the figure eight to a point, i got a little confused because the end points of the path changed, didn't it ? 🤔
I believe not! The endpoints are the start of the blue and the end of the red, and I think those always stay at the origin, this is true even if other "middle" points that are at the origin also leave it.
@@DrTrefor Ohhh now i see it! Thanks ^^ Your videos are always so helpful, your game-theory / Lagrange multiplier playlist helped me a lot to get me through my master degree at the beginning. Your videos are very important, hope you always keep doing them :)
I don't understand the connection between continuously deforming those curves in the ball, and your ability to untwist an object by translating its endpoint. It doesn't seem like it is guaranteed that translation in particular will have the ability to perform the equivalent transformation on the belt.
Interesting...now tell me how to untangle a 100' garden hose😉
A trip to the hardware store for a new one:D
That’s no problem knot theory couldn’t solve!
7:08 that looks like the proof of pythagorean theorem
This is so cool!
Though not as cool as my dad's belt trick.
🔥🔥🔥
Almost broke my arm trying that lol
11:20 wait, why does the bottom circle shrink? doesn't the start and end point of the curves move?
esto lo de la topología tiene muchos errores de una mala interpretación del espacio los fenómenos que suceden con la forma que se le da en el espacio y cuando uno hace un recorrido sobre su superficie hacé este sea cortando la o haciendo una línea o otra cosa
una cinta de papel nunca deja de tener dos superficies y el concepto de dimensiones está mal pues el espacio solo es analizando con la idea del concepto mental de la línea recta
Y les digo esto para que analize bien lo que sucede con la correa
Wouldn't it have been better for you to add a negative rotation vector at the origin so that both antipodes would be covered? Showing the vector to just one antipus and not the other can baffle the viewer.
What are you trying to say?
@@fullfungo Double-ended vectors.
@@cpiantes but they would only apply on the boundary. All other points are unique and not related to the points on the other side.
I don’t think double sided vectors would be a useful visual here.
Imagine a vector tracing a path; then suddenly it turns into a two-sided vector and a moment later snaps back to being a single vector on the other side. I’m not sure if this would be useful
1D, 2D, 3D are spatial.
4D, 5D, 6D are temporal.
ruclips.net/video/EgsUDby0X1M/видео.html Isn't hat rotarion by 2 pi?
Topology was never my strongest, lulz. The part with the four path segments, it's a bit important to track that they stay in order as a loop, since segment 1 is not allowed to unlink and hook up with 3, or the start of 3 cannot hook up with the start of 2. Often numbers and direction arrows may be used. Looks correct though.
0.15 Whos feeling their arms?
the arm isn´t actually turning 4pi. it is going 2pi and then 2pi back.
The cup actually is going 4pi!
@@DrTrefor that´s correct. but as you can see, i was referring to the arm, which is twisted in one direction first and then untwisted by going the other direction and not going the full 4pi as the cup. the rotation of the arm and the cup are not the same.
Μοbious
Revolution is not the same rotation.
Theoretical physicists are losing their shit right now!
7:24 you claim the rotation is pi/2 but it's pi
7:32 you claim the rotation is pi but it's 2pi
Hy my ideal sir
Sir how i can buy your T shirts like this which you wear in this video
I am from Pakistan
I speak for all programmers when I say I hate quaternions 😡
e vince un panettone ripieno
1st comment >.<
yay!
Please stop using pi here. Use tau. It’s a 2 tau rotation.
hahahaah its a trick... do it without taking your hand
So . . . am I missing something, or did dude forget to explain what this has to do with anything scientific?
Interesting. What's the point, though?
What even twists space? What are you even talking about?
Or, is it for the attention?
I'm sorry, but I think this video introduces more confusion than it does clarify the basic concepts. Also, I think the host glosses over and ultimately hand waves many of the (often gratuitous) concepts he introduces them (eg antipodall vector attenuating to 0). The unsatisfying explanations are not helped by the fact that the Graphics are sparse, poorly labelled and not instructive, not to mention the fact that there is a curious absence of any mathematical notation, formulae etc.
Although the is well-intentioned and has good energy, in terms of graphics, rigour and simplicity/elegance of explanation his videos are not yet within the same league as analogous Math channels such as 3blue1brown, numberphile etc.
You didn't snap your arm for this video... unsubscribed.
Lol
A child of two can see that you twisted your arm In different ways. The first time your hand went under the arpit, the second time over. I respect you greatly but don't promote nonsense. Btw I don't believe Dorac had any time for this.
ruclips.net/video/EgsUDby0X1M/видео.html Isn't hat rotarion by 2 pi?