Niles Johnson: Visualizations of the Hopf fibration 

Thank you for making my medical licensing exam seem easy. This is the motivation I needed!

This is blowing my mind right now. The geometry of how 3 dimensions can be inverted without intersecting itself.

Here from Grand Illusions! Thanks Tim!

What a great teacher

The best 4d space explanation video on RU-vid

JRE brought me some good fibrations

What a privilege to view this, thanks to all that made this possible and to Professor Johnson for its clarity!

This fellow enjoys his occupation.

I wish I'd known about this video earlier...it makes fibrations so much easier to understand! Thanks a lot Mr. Johnson!

This is the craziest coolest stuff, thanks for making these videos!!!

Very clear and helpful talk! Thank you for uploading!

This was remarkably easy to understand and quite enjoyable, thanks!

This is fantastic. Very well explained!

This has got me revisiting all the knot theory I did for my MSc.

Someone, somewhere, has passed the great opportunity to call Brougham Bridge as The Quaternion Bridge. Just saying. Neat lecture by the way. Congrats.

Conservation of Spatial Curvature: 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. .-----

Great respect for teachers.

very good talk!

I absolutely love it and I have no idea why.

This is so helpful, thank you!

Great visualization for those who can understand the math

Well, it made sense the first 7 minutes or so. It's clear that to comprehend the whole thing I need to watch more Rick and Morty.

It made more sense the 2nd or 3rd time through. Clear as it can get, the rest is my fault.

I thought it went a little fast. Probably from trying to squeeze the whole thing into 50 minutes. I’ll prob just watch it again and pause every so often. Thanks! Loved the animation at the end!

Can you please make a video on topology verify the hopf bundles are in fact bundles

Outstqanding lesson. Thanks for posting.

Muchas gracias...

great teacher. thanks

Can you trace out a shape on the 3-sphere that would not leave a circle in the stereographic projection? I'm guessing so. If this is correct then why don't we see any different shapes than circles in the (amazing) video at the end?... I'm also wondering about the line (the one circle that doesn't resemble a circle), what point on the 3-sphere is that projected from? -Surely if it's the point "at the top" (where the projecting light rays come from) then it should be out of sight infinitely far away? I'm hoping I'm making sense and that I don't sound horribly confused. Thanks!

Thank you for your wonderful lecture. I have a question about difference between Hopf band and (S^1)x(I). Obviously(?) they are homeomorphic. But do they have different fiber structure?

thank you, much better than in physics lesson on medington high

this is damn better than the "dimension" video!!!

Really nice clear talk. One tiny nitpicky comment is that when you are talking about fibrations, really you are just talking about fibre bundles - fibrations are a little more general.

Thank You Well said.

Where does one learn stuff like these?

Isn't an easier way to envision S3 to think of the 4th dimension as time and then S3 "starts" as a small version of S2 (actually a point) and this sphere grows to actual S2, then gets smaller again to a point? Well the "growing-velocity" is cosinoid, is what needs to be added.

These might be the same question, 1. Where in the 3-ball is the 2-sphere? 2. Why are the projected circles linked?

The quaternion q = W+iX+jY+kW can be expressed as a 2×2 matrix Q₂ₓ₂ = [ Q₁₁ , Q₁₂ ; Q₂₁ , Q₂₂ ] = [ W + iX , Y + iZ ; −Y + iZ , W − iX ] The determinant of Q is 1 e.g. det Q = W² + X² + Y² + Z² ≡ 1 hence Q⁻¹ = [ W − iX , −Y − iZ ; Y − iZ , W + iX ] The projection η = qkq⁻¹ in matrix form is : QKQ⁻¹ ≡ H where K is considered a quaternion in matrix form as: K = [ 0 , i ; i , 0 ] since η = qkq⁻¹ = q' = ( W' , X' , Y' , Z' ) similarly H = QKQ⁻¹ = Q' = [ W' + iX' , Y' + iZ' ; −Y' + iZ' , W' − iX' ] Then, H₁₁ = (QKQ⁻¹)₁₁ = 2i(WY + XZ) = W' + iX' which means W' = 0 and X' = 2WY + 2XZ. Also, H₁₂ = (QKQ⁻¹)₁₂ = 2YZ − 2WX + i(W² − X² − Y² + Z²) = Y' + iZ' thus Y' = 2YZ − 2WX and Z' = W² − X² − Y² + Z² The determinant of H = det (QKQ⁻¹) = det Q ⋅ det K ⋅ det Q⁻¹ = (1)(1)(1) = 1 = ( 2WY + 2XZ )² + ( 2YZ − 2WX )² + ( W² − X² − Y² + Z² )² which is a unit 2-sphere.

@jamma246 Ah, of course you're right! I should have made that comment during the talk :)

3.35 separated by what? Planck length?

excellent...............................

I don't speak this language. I need to learn mathematics.

28:47 so where did the S¹ come from in this fibration?

how is the result obtained at 28.22, why is w=0

What is the meaning of having an inverse of q?

the difficulties in expressing irrational coordinates

Hamilton's original carvings are gone, but there's a plaque.

Why wouldn't R x S1 be an infinite flat circle, like R x R with rounded corners? And why wouldn't S1 x S1 be an infinite sphere? Wouldn't a torus be better described by S2 x a hollow S1? Or couldn't that infinite cylinder be described by R x S2? Is there an assumption I missed?

22:10 Brougham Bridge in Dublin!

In most important part of lecture you mostly see only lecturer's back. He is writing something on the board but it's almost impossible to see what he is writing

9:00 isn't that a definition of a fibre bundle and not fibration?

I thought I would never understand what s vibration was but this is very simple

He has a roll of toilet paper handy if some student shits themselves trying to understand this.

Are you fucking serious. THAT'S ALL A FIBRE IS?? WHY the HELL could NO ONE explain that before in terms that were anything less than absolutely mystifying?? Thank you so much for this lol

#hopffibration

Got here because of the JRE. I found this explanation far more comprehensible than Eric Weinstein’s.

Damn, I feel stupid now

I got lost at the Cartesian product stuff oop-

JRE brought me here...

Erases the board. *eraser marks*... Looks about right.

oh eric, what have you done to by brain

H is for Hamiltonians -- which are a whole other sidebar 😂😂😂

hmmmmmm donuts

A chalk board lil

Wake up camerman.

This teacher is toochaotic, comes out as insecure of himself.