Glad you're interested! The basic idea is not that magical though: Each frame is a 3d view, so I made a Sage object which could compute and store the relevant data, and produce the image. Then I wrote a bunch of helper functions which would create a sequence of objects with the appropriate input parameters (positions on the 2-sphere)! I have more production notes and source code linked in the video description.
Amazing animation. I was having a very hard time trying to make sense of the usual pictures of the Hopf fibration. This animation with the corresponding pts on the sphere shown on the side finally made it click. Thank you for the hard work you put into making this ❤️
I must have watched this 5 or 6 times, just mesmerised, until I actually started paying attention. Mathematically, conceptually and aesthetically delightful
I think this was the most useful video I’ve found so far for understanding this concept. I’ve been looking for a paper of some sort that talks about generalizations of it, but have yet to find what I’m looking for. I’d also really like some videos about the places it can be applied in physics. I first heard about the notion of a fiber bundle a couple years ago, as my friend tried to explain to me what it would mean for a circle to be mapped onto every point in space. What made some degree of intuitive sense was that such a fiber bundle could “naturalize” the space for wave functions…. But when I discovered the Hopf Fibration I just felt an immediate sense of “Oh! This might deepen my notion of this a lot.” Anyway, rad video. If anyone’s got some good references to help me expand this idea, I’m interested.
@@SaulBedMan A 2-sphere in topology is the surface of a 3-d globe (it's called that because if you're an ant on the surface of a globe, you have 2 free directions of movement - locally, it's like a 2d plane. For the same reason, a circle is usually called a 1 sphere.)
Conservation of Spatial Curvature (both Matter and Energy described as "Quanta" of Spatial Curvature) 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 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. ------------------------ 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 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 the “Twistor Theory” of Roger Penrose, and the work of Eric Weinstein on “Geometric Unity”? 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 gluons are actually made up of these twisted tubes which become entangled with other tubes to produce quarks. (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 Mesons. 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. Gamma photons are produced when a tube unwinds producing electromagnetic waves. >>>>>>>>>>>>>>>>>>>>>> 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. We know there is an unequal distribution of electrical charge within each atom because the positive charge is concentrated within the nucleus, even though the overall electrical charge of the atom is balanced by equal positive and negative charge. >>>>>>>>>>>>>>>>>>>>>> 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. The 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. .
p.s. The thread idea isn't explicitly shown in the movie, but thinking about why this thread trick shows that the Hopf link is fibered might help you see how the video does show that fact.
could you make a video series how you get from a point in the spheres to one of those fascinating animations, I'm starting my first years of college and considering switching to math, this is wonderful!
Glad you enjoyed it! There are a few links for further information in the "description" section above. There are also these slides from a more recent talk, but they probably don't make as much sense until after you've seen the earlier stuff: nilesjohnson.net/notes/Hopf-fib-vis.pdf
If you think you've got it, then you probably do :) The clearest demonstration of this in the video is certainly quite symmetric, and meant to be quite explicit. And this particular demonstration relies on the structure of the Hopf fibration; perhaps that's what you mean by "more natural" than the circle action for a generic Hopf link.
Hey Niles, This is great stuff mate. I'm addicted to this vide, I've seen it dozens of times and I don't think I'll stop soon :) I'm a beginner to differential and algebriac geometry and it'd be great if you can share your thoughts about steps to learn the math behind this. Thanks again for sharing. Have a nice day. Ahmed
Glad you're enjoying it :) I was fascinated by the Hopf fibration as a beginning topology student too. I put links to a couple of expository articles at nilesjohnson.net/hopf.html Hopefully those will help get you started.
That's a great question -- let me try to give a hint instead of a short answer. Another way to see that the Hopf link is fibered is to start with a Seifert surface, then imagine a piece of thread along one side of the surface, with each end tied to a different component of the link. Now isotop (continuously move) the string from one side of the Seifert surface to the other, without untying the ends.
Quick question: Do the colors in this represent any specific characteristics or qualities? or do they just represent a path to help visualize the motion? I ask because as a musician, who never really studied this type of thing, I'm pondering how I might represent different parts of this fibration video into musical elements/representation or audio visualization if you will. Might take awhile!
Each loop shown in the big ball (S^3) is the fiber over a specific point in the smaller sphere (S^2). The meaning of the colors is to match which loop above goes with each point below.
@@kellyryanobrien1 Been looking into this. I think the answer is no. What it reminded me of is the geocentric model, and the amount of space covered by of all the planets.
Hello there, I think you could set a vibrating laser beam to reflect off of a metal, reflective sphere, and using different frequencies of sound, fine tune the shape until you get a circle refracting onto the wall. Add more lasers to create more of the shape. I know this doesn't answer your question, but is what I'll be be doing to represent this shape irl using sound
I thought I understood stereographic projection. In my understanding, a locus of points near the "north pole" is mapped to a great distance away. But I don't see that happening, hence my confusion. Are you doing some kind of rescaling to keep the rendering contained within some bounds on screen?
I just remembered you all might like the Modular Fibration too. It's the same kind of thing, but the fibers are trefoil knots! Comes from some number theory (modular forms). th-cam.com/video/eqeqbjec97w/w-d-xo.html This is what I call a *branched* fibration, because there are a couple of points where the fibers are degenerate. But the way they degenerate is neat! More details in the description there.
I think I got it. Would you agree that the Hopf link for which it is demonstrated is a bit "non-generic" or more symmetric than an arbitrary Hopf link (perhaps for clarity of the visualization?), or that the circle action for a generic Hopf link wouldn't look as natural? (I'm having a hard time trying to make myself clear without giving it all away. :) )
Just started listening to The Portal where he is talking to Roger Penrose. I stopped and looked Up Hopf and here I is. Good at arithmetics but got lost after geometry. I still try to understand it conceptually as best I can.
How do we know that each circle (mapping to a point) interlocks in a hopf link with every other circle? I definitely am missing the deeper reasoning. What calls for the existence of these links? What types of shape have them and where do they form?
This reminds me of many animations by Nassim Haramein (madman or not?) many years ago. Then, it didn't have a name IIRC but --now-- it has a name? Is it that mainstream physics is edging towards what the mystics --purport-- to have known perhaps? We have Wolfram Physics now talking about rullial multiway graphs, Klee Irwin-s QuantumGravityResearch research into tetrahedrons and the E8 lattice, Eric Weinstein with his Geometric Unity theory and the Hopf fibration, is there an impending convergence and realisation of who or what we are and how we came to be here? Exciting times.
Yes, there is. There is a one-dimensional version of the Hopf fibration where the total space is a circle, where the fibres are 0-spheres (i.e., two points) and the base is also a circle. The projection is just the doubling map.
This is what Weinstein and Rogan should have shown. The build up from one point explains how the image is built. Well done! Now I have to figure out what it means.
I could visually follow and understand the rings correspond to the dots on the sphere until about the 1:13 mark then my brain melted. I must study this more and understand this.
Thank you for putting this up as an excellent source to begin studying probably the most abstract geometrical object out there. You were ahead of the curve so to speak ha - 10 years since you put this up is amazing. Eric Weinstein only brought this into pop culture a few years ago and most of us still cannot wrap our heads around this "the most important object in the universe" as he claims. I've recently completed special and general relativity as the main classical base to Quantum Field Theory. Is this Hopf fibration related to the 720 degree spin of the electron ( or more fundamentally quarks)? I am in awe of how Heinz Hopf and Paul Dirac were able to visualize this thing in the 1930s!
I don't know your pre- knowledge, but for someone like me who is new to fiber bundles I doubt that, no matter how good of a teacher he is, trying to explain this in 3 minutes would only add Confusion... I prefer the nice music and some good literature
Each fiber is linked with each other fiber exactly once. This is the property that first attracted attention to the Hopf fibration, and a pair of circles in this configuration is called a Hopf link. The collection of fibers over a circle in S2 is a torus (doughnut shape), S1×S1, and each such pair of tori are linked exactly once. The collection of fibers over an arc form an annulus whose boundary circles are linked. This is known as a Hopf band; it is a Seifert surface for the Hopf link.
@@cappucino7908 Evidently, from the most helpful responses in this thread, this can be explained decently well in a hundred words or a little more. Why be a prick? You give STEM people a bad name.
A very nice animation, it's beautiful, and I've learnt a lot by watching your talk/animation and reading a bit on the Hopf fibration! And a question: It's not clear to me where can we see that the Hopf link is a fibered link. Can someone tell me what time should I look at?
this is Great! with a giant spliff and enough time, it feels like ,instead of studying it from rigorous textbooks, i can see a combined visualisation of all force fields diagrams in polar cordinates, particles in standard model, etc presented in an abstract scenario ...the key is smoking enough spliff :P
So the errie thing was that the first time I encountered this geometry (not this video) I was tripping on high dose of LSD staring into the recursive nature of consciousness; didn't know it was called the “Hopf fibration” back then. Didn't understand why I was experiencing a 3 dimensional visuals of some kind of rotating self-decoupling torus. And then a few days later I smoked some weed before heading bed and saw this again when I closed my eyes. It was pretty scary tbh. This thing was in much higher resolution than reality itself.
This was what inspired me to consider Homotopy type theory as the building block for a framework to study the nature of consciousness in a cybernetic programming language theory setting where a corresponce to topology is retained (e.g. continuation-passing-style as smoothness)
So very long ago; very interesting/enlightening. I noticed that all of your torii are slanted to the left... there's another matched direction that should be slanted to the right... There's this toy - flowtoys.com/toroflux which is single spring wire wound around a circle and linked through itself at every loop... But you can wind it a different direction, and there's no translation that can happen to make one into the other.
my ketamine trip brought me here i was that thing it was first like a puzzle once i cracked the code it lost control and spun out like a turbo from a car hitting light speed
Meanwhile, it sound good place to brag. I just proved Collatz conjecture. Just by one sentence. 2 sentences, if I will include the famous "proof is trivial and left as an exercise for the reader" to honour long dead geniuses. :) And it is trivial.
this is all a massive troll. That guy Weinstein guy on JRE had a couple amateur video animations of slinky's made and now here we all are thinking we are unlocking secrets of quantum space-time.
Glad you're interested! The basic idea is not that magical though: Each frame is a 3d view, so I made a Sage object which could compute and store the relevant data, and produce the image. Then I wrote a bunch of helper functions which would create a sequence of objects with the appropriate input parameters (positions on the 2-sphere)!
I have more production notes and source code linked in the video description.
Amazing animation. I was having a very hard time trying to make sense of the usual pictures of the Hopf fibration. This animation with the corresponding pts on the sphere shown on the side finally made it click. Thank you for the hard work you put into making this ❤️
I must have watched this 5 or 6 times, just mesmerised, until I actually started paying attention. Mathematically, conceptually and aesthetically delightful
I think this was the most useful video I’ve found so far for understanding this concept. I’ve been looking for a paper of some sort that talks about generalizations of it, but have yet to find what I’m looking for. I’d also really like some videos about the places it can be applied in physics. I first heard about the notion of a fiber bundle a couple years ago, as my friend tried to explain to me what it would mean for a circle to be mapped onto every point in space. What made some degree of intuitive sense was that such a fiber bundle could “naturalize” the space for wave functions…. But when I discovered the Hopf Fibration I just felt an immediate sense of “Oh! This might deepen my notion of this a lot.” Anyway, rad video. If anyone’s got some good references to help me expand this idea, I’m interested.
Jaw - dropping. Brings concept of fibrations to life
Good work on this Niles
So in a Hopf fibration, each point on the 2-sphere's surface corresponds to the circles on the 3-sphere?
Exactly, and the circles are all disjoint and vary continuously as the point moves around on S².
But whats a 2 sphere? Isnt that a circle? Or is it like all the point within a range from the centre?
@@SaulBedMan A 2-sphere in topology is the surface of a 3-d globe (it's called that because if you're an ant on the surface of a globe, you have 2 free directions of movement - locally, it's like a 2d plane. For the same reason, a circle is usually called a 1 sphere.)
And each pair of the circles forms a Hopf link. I found this part the most amazing.
Excellent video. Very interesting, informative and worthwhile video.
The secrets of the universe for free here, ty well done
Conservation of Spatial Curvature (both Matter and Energy described as "Quanta" of Spatial Curvature)
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 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.
------------------------
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 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 the “Twistor Theory” of Roger Penrose, and the work of Eric Weinstein on “Geometric Unity”? 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 gluons are actually made up of these twisted tubes which become entangled with other tubes to produce quarks. (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 Mesons. 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.
Gamma photons are produced when a tube unwinds producing electromagnetic waves.
>>>>>>>>>>>>>>>>>>>>>>
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. We know there is an unequal distribution of electrical charge within each atom because the positive charge is concentrated within the nucleus, even though the overall electrical charge of the atom is balanced by equal positive and negative charge.
>>>>>>>>>>>>>>>>>>>>>>
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. The 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.
.
This is truly beautiful.
That is amazing! May I share this animation to other video website? I wish more people could see that.
p.s. The thread idea isn't explicitly shown in the movie, but thinking about why this thread trick shows that the Hopf link is fibered might help you see how the video does show that fact.
Brilliant work. Congratulations!
POWERFUL JRE
I feel like that episode is going to have some very large ripples for a lot of people. Great stuff lol
Yes!
could you make a video series how you get from a point in the spheres to one of those fascinating animations, I'm starting my first years of college and considering switching to math, this is wonderful!
Glad you enjoyed it! There are a few links for further information in the "description" section above. There are also these slides from a more recent talk, but they probably don't make as much sense until after you've seen the earlier stuff:
nilesjohnson.net/notes/Hopf-fib-vis.pdf
If you think you've got it, then you probably do :) The clearest demonstration of this in the video is certainly quite symmetric, and meant to be quite explicit. And this particular demonstration relies on the structure of the Hopf fibration; perhaps that's what you mean by "more natural" than the circle action for a generic Hopf link.
I feel like this is the key to life
If this video does anything more than trip your mind out, you’re a damn super genius…
Muy bien desarrollado, muchas gracias...
Hey Niles, This is great stuff mate. I'm addicted to this vide, I've seen it dozens of times and I don't think I'll stop soon :)
I'm a beginner to differential and algebriac geometry and it'd be great if you can share your thoughts about steps to learn the math behind this. Thanks again for sharing.
Have a nice day.
Ahmed
Glad you're enjoying it :) I was fascinated by the Hopf fibration as a beginning topology student too. I put links to a couple of expository articles at nilesjohnson.net/hopf.html Hopefully those will help get you started.
Just saw the articles after the comment :)
Keep it up Niles.
Quality art. Thank you for sharing this.
The Magic Slinkey!
Just looking at the soul of an electromagnetic knot! So cool, men. Thanks.
That's a great question -- let me try to give a hint instead of a short answer. Another way to see that the Hopf link is fibered is to start with a Seifert surface, then imagine a piece of thread along one side of the surface, with each end tied to a different component of the link. Now isotop (continuously move) the string from one side of the Seifert surface to the other, without untying the ends.
Quick question: Do the colors in this represent any specific characteristics or qualities? or do they just represent a path to help visualize the motion? I ask because as a musician, who never really studied this type of thing, I'm pondering how I might represent different parts of this fibration video into musical elements/representation or audio visualization if you will. Might take awhile!
Each loop shown in the big ball (S^3) is the fiber over a specific point in the smaller sphere (S^2). The meaning of the colors is to match which loop above goes with each point below.
@@NilesJohnson can you correlate this to the cycles of influence (days) and the orbit of the planets ?
@@kellyryanobrien1 Been looking into this. I think the answer is no. What it reminded me of is the geocentric model, and the amount of space covered by of all the planets.
Hello there, I think you could set a vibrating laser beam to reflect off of a metal, reflective sphere, and using different frequencies of sound, fine tune the shape until you get a circle refracting onto the wall. Add more lasers to create more of the shape.
I know this doesn't answer your question, but is what I'll be be doing to represent this shape irl using sound
What an intellectual roller coaster!
Look at all those mobious strips
I thought I understood stereographic projection. In my understanding, a locus of points near the "north pole" is mapped to a great distance away. But I don't see that happening, hence my confusion. Are you doing some kind of rescaling to keep the rendering contained within some bounds on screen?
Song name:
"Mandelbrot set" by Johnathan Coulton
I now have a better understanding of a hypersphere than just "changing size"
I just remembered you all might like the Modular Fibration too. It's the same kind of thing, but the fibers are trefoil knots! Comes from some number theory (modular forms).
th-cam.com/video/eqeqbjec97w/w-d-xo.html
This is what I call a *branched* fibration, because there are a couple of points where the fibers are degenerate. But the way they degenerate is neat! More details in the description there.
Nancy Hocking brought me here. Astonishing!
Gracias x estas bellas creaciones y x utilizar la forma toroidal
#epic #soulful Thanks for sharing
What's the use of this fibre
I think I got it. Would you agree that the Hopf link for which it is demonstrated is a bit "non-generic" or more symmetric than an arbitrary Hopf link (perhaps for clarity of the visualization?), or that the circle action for a generic Hopf link wouldn't look as natural? (I'm having a hard time trying to make myself clear without giving it all away. :) )
Joe Rogan and Eric Weinstein! What have you done to me???😩😩
at what points in the video are the parameters of the "scavenger hunt" from the lecture met?
Whats the song here?
This video represent how I feel on the inside
Inspired music choice. Took me two chords to recognize, then started singing along
What is it?
"Mandelbrot Set" by Jonathan Coulton
@@charlesmartin1972 Thank you!
Do 4D shapes appear to turn back in on themselves and knot together because we can't see the space properly?
ISTM that Robinson congruences and magnetic fields have shapes that are included in Hopt fibrations
Whats the relation between the coordinate system on the right and the lines?
Niles,
Could you teach me hot to develop such animations?
If it is done in SAGE, will be great to learn.
Thanks & Regards,
Vijay
no voice over?
Can you make this a screen saver
onsmoko on your hackingtosh
I come back to this for the song
Song?
Just started listening to The Portal where he is talking to Roger Penrose. I stopped and looked Up Hopf and here I is. Good at arithmetics but got lost after geometry. I still try to understand it conceptually as best I can.
How do we know that each circle (mapping to a point) interlocks in a hopf link with every other circle? I definitely am missing the deeper reasoning. What calls for the existence of these links? What types of shape have them and where do they form?
en.wikipedia.org/wiki/Hopf_fibration#Geometry_and_applications actually explains it nicely, for anyone who was wondering!
Joe Rogan brought me here
he left me here
@Braxton Heath me too bro good find
How? Did he ever mention this?
Same
What is figure at 1:16 called? Has it got a name?
It's three nested/linked tori (there is no difference between nested tori and linked tori in the 3-sphere).
This reminds me of many animations by Nassim Haramein (madman or not?) many years ago. Then, it didn't have a name IIRC but --now-- it has a name? Is it that mainstream physics is edging towards what the mystics --purport-- to have known perhaps? We have Wolfram Physics now talking about rullial multiway graphs, Klee Irwin-s QuantumGravityResearch research into tetrahedrons and the E8 lattice, Eric Weinstein with his Geometric Unity theory and the Hopf fibration, is there an impending convergence and realisation of who or what we are and how we came to be here? Exciting times.
looks like my childhood slinky after i used it for 5 minutes
See Ken Wheeler on magnetism.
Some of those images made me think of the sun and all the magnetic arcs
Can someone please tell me what the bigger picture in this video represents? Is there an analogy in lower dimensions?
Thanks =)
Yes, there is. There is a one-dimensional version of the Hopf fibration where the total space is a circle, where the fibres are 0-spheres (i.e., two points) and the base is also a circle. The projection is just the doubling map.
looks like a Star Trek warpfieldbubble
What that guy was claiming at Joe Rogan is that what we see as point like particles can be dfferent objects in a higher dimension.
Espectacular!!!!
I enjoyed your music and now I am playing it with Flying by the Beatles
Ooooh... "FIBRation"... I thought they were saying "vibration"
I searched "Hopp Vibration" but still made my way here lol
It's the Mandelbrot Set song!
This is what Weinstein and Rogan should have shown. The build up from one point explains how the image is built. Well done! Now I have to figure out what it means.
I could visually follow and understand the rings correspond to the dots on the sphere until about the 1:13 mark then my brain melted. I must study this more and understand this.
This shape speaks to me 🤔
Awesome.
you are a genius !!!!!
Thank you for putting this up as an excellent source to begin studying probably the most abstract geometrical object out there. You were ahead of the curve so to speak ha - 10 years since you put this up is amazing. Eric Weinstein only brought this into pop culture a few years ago and most of us still cannot wrap our heads around this "the most important object in the universe" as he claims. I've recently completed special and general relativity as the main classical base to Quantum Field Theory. Is this Hopf fibration related to the 720 degree spin of the electron ( or more fundamentally quarks)? I am in awe of how Heinz Hopf and Paul Dirac were able to visualize this thing in the 1930s!
メッチャ綺麗やん！
How about a voice over talking about what we're looking at instead of the music.
How about picking up a book and then watching?
I don't know your pre- knowledge, but for someone like me who is new to fiber bundles I doubt that, no matter how good of a teacher he is, trying to explain this in 3 minutes would only add Confusion... I prefer the nice music and some good literature
Each fiber is linked with each other fiber exactly once. This is the property that first attracted attention to the Hopf fibration, and a pair of circles in this configuration is called a Hopf link.
The collection of fibers over a circle in S2 is a torus (doughnut shape), S1×S1, and each such pair of tori are linked exactly once.
The collection of fibers over an arc form an annulus whose boundary circles are linked. This is known as a Hopf band; it is a Seifert surface for the Hopf link.
@@cappucino7908 Evidently, from the most helpful responses in this thread, this can be explained decently well in a hundred words or a little more. Why be a prick? You give STEM people a bad name.
Cappu Cino how about commentating it so people might want to read a book about it? The tunes are not helpful.
A very nice animation, it's beautiful, and I've learnt a lot by watching your talk/animation and reading a bit on the Hopf fibration!
And a question: It's not clear to me where can we see that the Hopf link is a fibered link. Can someone tell me what time should I look at?
ty😃
this is Great! with a giant spliff and enough time, it feels like ,instead of studying it from rigorous textbooks, i can see a combined visualisation of all force fields diagrams in polar cordinates, particles in standard model, etc presented in an abstract scenario ...the key is smoking enough spliff :P
0:56
nice
Well done!!
Anyone who wants the missing subtitles:
"Dunn dunn dunn du (now you understand)" (repeat).
muted
nice animation! thanks.
need the soundtrack, please. :)
NONTRIVIAL CIRCLE BUNDLE OVER THE SPHERE!
More psilocybin please 🥰✌️👍
AHAHAHAH did anyone else first search for the "hop vibration" ? But I got here!
This could be done in Mathematica so easily. But this is still a great animation.
So the errie thing was that the first time I encountered this geometry (not this video) I was tripping on high dose of LSD staring into the recursive nature of consciousness; didn't know it was called the “Hopf fibration” back then. Didn't understand why I was experiencing a 3 dimensional visuals of some kind of rotating self-decoupling torus.
And then a few days later I smoked some weed before heading bed and saw this again when I closed my eyes. It was pretty scary tbh. This thing was in much higher resolution than reality itself.
This was what inspired me to consider Homotopy type theory as the building block for a framework to study the nature of consciousness in a cybernetic programming language theory setting where a corresponce to topology is retained (e.g. continuation-passing-style as smoothness)
So very long ago; very interesting/enlightening. I noticed that all of your torii are slanted to the left... there's another matched direction that should be slanted to the right... There's this toy - flowtoys.com/toroflux which is single spring wire wound around a circle and linked through itself at every loop... But you can wind it a different direction, and there's no translation that can happen to make one into the other.
i came here to say this is god
Wow, this is so... mesmerizing. Like a quantic brain dreaming of technicolor sheeps :okno:
Who here wants to conquer it then ruins any hope of understanding it.
Calvin Steinberg I know almost all of it
Sounds like music from sonic
my ketamine trip brought me here i was that thing
it was first like a puzzle once i cracked the code
it lost control and spun out like a turbo from a car hitting light speed
so how do we build one in real life and implement it into something to benefit mankind 105
we are not super intelligent we are just VERY curious
e=mc squared.
Electromagnetic toroidal field
Elemental my dear watson
Or Macro my dear black hole.
DMT brought me here. And, Joe Rogan.
Tom Bilyeu almost brought me here....
Meanwhile, it sound good place to brag.
I just proved Collatz conjecture. Just by one sentence.
2 sentences, if I will include the famous "proof is trivial and left as an exercise for the reader" to honour long dead geniuses. :)
And it is trivial.
this is all a massive troll. That guy Weinstein guy on JRE had a couple amateur video animations of slinky's made and now here we all are thinking we are unlocking secrets of quantum space-time.
thanks joe rogan and eric weinstein