Slicing bagels to learn gauge theory (Hopf fibration)

Getting the perfect bagel slice is harder than it looks.
Chapters:
0:00 Introduction
0:52 Basepoints and fibres
2:53 Three ways to slice a bagel
8:06 A taste of the mathematics of the Hopf fibration
11:38 Gauge theory
Links:
wikipedia: en.wikipedia.org/wiki/Hopf_fi...
Niles Johnson's KZbin video: • Hopf fibration -- fibe...
A nice set of notes (not mine): nilesjohnson.net/hopf-article...
Notes (mine): WIP
Picture of geisha standing in front of Hopf fibration: xahlee.org/index.html

[@logo2462]

Your Evangelion x Drake Tee is 🔥

[@nice3294]

Wow, some neat good geometric stuff for the hopf fibration. Looking forward to that gauge theory vid!

[@atimholt]

I described the Hopf fibration to my teenage nephew. It helped that I'd previously shown him the game Hyperbolica. There's a level that takes place on the surface of a 3-Sphere.

[@alexanderjenkins7929]

Thanks for the great videos, i bought a bagel. Please get yourself a good bread knife! I'd love to see a video explaining Riemann Sphere and its use in physics - maybe using profiteroles

[@7eventh]

No bagels were harmed in the making of this video

[@davidwright8432]

Unfortunately for bagels, not true! I personally would not like to be sliced. Even in the service of geometry.

[@Alkatross]

I like when the halves are equal. Then the cream cheese ratio remains infinite.

[@willywalter6366]

Tasty Bagel while watching - extraordnary great work! Thank You!

[@kristina-alexandra]

Wonderful video. Thank you 🙏

[@tanchienhao]

Awesome video!!

[@hyperduality2838]

Orbits (rotations) are dual to stabilizers (reflections) -- group theory. Two reflections give a rotation. "Always two there are" -- Yoda.

[@kmg3658]

Wonderful. Thank You.

[@cboniefbr]

Awesome vid and T-shirt

[@davidhand9721]

Is there some reason we don't visually represent the 3 sphere as a 2 sphere + its volume? The 4th coordinate (say, w) is the distance to the edge, or (1-r). The edge is w = 0, and the origin is w = 1. I guess w couldn't be negative; is that it, though? Just use a different color for -w.

[@5chr0d1nger5cat]

i am confused why the circles are tilted in 3D when you rotate them around z to obtain torus

[@SpotterVideo]

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. ==

[@marlobardo4274]

Hopf sharing a fractal bagel with Mandelbrot that Villarceau sliced in the form of two crescent shaped kifli/Kipferl with a Vesica Pisces forming at the intersection between the two squarable lunes making up one torus. On a serious note what do you call the inverse of a torus i.e. the shape in the middle? A hyperbole? Thank you for this video...

[@logo2462]

Ya lost me at the Hopf map/ projective complex geometry bit at about 10 minutes in. I would love to see more details on the last 5 minutes of the video.

[@OnePlusOneSpace]

Ty for the feedback! The explanation was quick for sure. I was thinking of writing a short set of notes to accompany the video which would go into more detail.

[@davidhand9721]

A line is an infinitely large circle. What else would it be?

[@adram3lech]

I feel like 0:35 needs explanation, I have no idea what you mean by that correspondence. Do you mean this?: we map 3d cartesian space to the surface of a 3d sphere? Why circles, is it because they give nice results or is it a requirement that comes from the operation? On a side note, you need a sturdier table :d

[@OnePlusOneSpace]

Haha you're not the first to comment about the table! To be more precise, there's a bijection (one-to-one mapping) between circles *in the specific arrangement discussed in the video* in 3d Cartesian space, and points on the sphere. By sphere I always mean the surface. The Hopf map discussed later in the video is a map from 3d space onto the sphere. It's not one-to-one, but the set of points in 3d space which map onto the same point of the sphere (say, the North Pole, for concreteness) is precisely a circle. So the circles are 'baked in' to this Hopf map.

[@xicad1533]

she hopf on my fibration

[@marcoottina654]

5:00 wait, isn't it the magnetic field of an electric current in a wire?

[@nickbros]

5:00 looks like the equipotential surfaces of 2 infinite line charges

[@AkamiChannel]

I'm just at the beginning but when you say "sphere" I don't know if you are talking about the 2-sphere or 3-sphere.

[@ewanduck55]

The metaverse really couldn't ever

[@davidwright8432]

Pumpkin spice bagel geometry! All you need ... except for some cream cheese?

[@loveistheonlything3626]

You need a sharper knife or fresher bagels. 😄 Thanks for the video.

[@tobiasthrien1]

3:28 Do you have a patreon so that I can donate you a proper table? This is horrible 😂

[@OnePlusOneSpace]

Not yet, but thank you for the offer 😂

[@chrisleon27]

Watching this video makes me want to sleep

[@michaelcharlesthearchangel]

Quantum || V is Y

[@davidtaylor3771]

Sorry but I couldn't really connect with the content. I may try again later. But you sure have got me hungry.

[@handledeeznudz]

bread

[@OnePlusOneSpace]

bread

[@handledeeznudz]

@@OnePlusOneSpace bread

[@tylerbakeman]

“A remarkable structure of nested tori”. Honestly, it’s just the same shape at different scales - we see this frequently in math. This topic couldn’t be more uninteresting. Outside of computer graphics, I don’t think fibers are useful in mathematics. Depends on what you consider to be basically a fiber though (is interpolation a fiber for example? because interpolation is necessary in calculus).

[@magicfigures]

Look please very nice animation of big magic squares on torus for big numbers - such homepage under number-galaxy eu.