I’m an algebraic topology Ph.D. Student and I study homotopy and symplectic homologous groups of hyper spheres, so I’ve used the hopf fibration a lot. It’s really cool to see it in such a beautiful and visual way though!
That’s awesome! :) What are some of the things you’ve used it for? To be totally honest, I’m only really familiar with the Hopf fibration in the context of qubits, and I guess just as a mathematical curiosity in its own right, but I’d love to learn more about its applications in pure math. Btw, what is it that you’ve found most beautiful so far in algebraic topology?
I’ve used Hopf fibrations and its generalizations to relate homotopy groups of higher dimensional spheres to lower dimensional spheres. This is because you can view loops as paths on the fibers and then project those paths down to the lower dimensional spheres. One of the most beautiful concepts in algebraic topology is homologies in my opinion since they keep many of the topological properties you want invariant.
@@RichBehiel Can you do a video about spinors? It's difficult for me to visualize integer spin objects, in particular the geometry of their interactions with each other. Interacting electrons from different atoms for example, I would love to see a video of that, even if it's just a component of it represented in 2D.
Yes and no! :) I’m currently working on a video of the quantum mechanics of the hydrogen atom. It’ll have a thorough solution of Schrodinger’s equation, and an exploration of its predictions relative to ionization energy and spectral lines. Then we’ll get into the fine structure and see that the lines are split, at which point we’ll have to swap out Schrodinger’s nonrelativistic E-p relation with the relativistic version, which leads into the Dirac equation, which will lead to the spinor eigenatates and we’ll see how there’s a slight energy difference related to spin which emerges from those equations. I’ll show the equations of those eigenstates in their usual bispinor form, and will talk superficially about how we can regard the components as corresponding to spin up electron, spin down electron, spin up positron, and spin down positron. In a future video I’ll expand on this further to calculate the Lamb shift (vacuum fluctuations) and Darwin term (zbw). So there will be some spinor math involved. The reason I say “yes and no” is that at the moment, I really only know what spinors are from an algebraic perspective. I almost completely lack intuition for them though. If you ask me what a spinor *is*, like what it really really *is*, my answer is “I do not know, and they scare me.” Maybe someday I will be smarter, and can figure out a way to animate spinors so that we can all see what they really are. I really hope to be able to do that someday. But I’m just not at that level yet, and I have to keep it real with y’all, so I’m not prepared to make a video on the nature of spinors yet. The hydrogen video will be a step in the right direction though.
I'm a self taught mathematics enthusiast. I think that the book that introduced me to fiber bundles and the Hopf fibration specifically was Dr. Roger Penrose's Road to Reality. It taught everything we know in physics starting with the Greeks to present with all the math. Nothing left out, what is normally covered in 10 books and 15 courses, all in one tome. That book is very comprehensive with almost no examples or deeper explanation. (Love and hate that about the book). It nearly broke me, especially trying to conceptualize high dimension complex spinorial tensors and what they mean geometrically. This vid was nice. Just enough info to get a student, new to the subject, into a fine pickle.😂 JK, I do kind of feel that profound confusion is the only way to prepare your mind for fiber bundles. Another related concept that is wonderfully criptic is the way fiber bundles are used in algebraic topology... The fundamental group, homotopy, homology, hole chasing in higher dimensional spaces. That sort of fun stuff. Anyway, thanks again 🙏🏾
Thanks, I’m glad you enjoyed it! :) I’m a huge fan of Heidegger (well, not so much his later work, but up to Being and Time for sure 😅) and I resonate deeply with his project of rethinking the nature of the human condition starting with the phenomena themselves, even though it remains incomplete and intractably poetic, for better and worse. His concept of Dasein has had an energizing impact on my life. It’s ironic though because I generally have a strongly Platonic attitude and a tendency toward first-principles reductionism. Proceeding in a structured, logical way from a set of axioms is my preferred operating mode. Heidegger is like a thunderstorm that rolls in and threatens to topple the structured set of concepts we’ve come to know and love, so to wrestle with his ideas is simultaneously frightening and exciting. He keeps you on your toes for sure.
@@RichBehiel Nice outro. Also, you need not put platonism in conflict with the existentialists. Gödel showed us not all platonic truth can be brought into the light of formal systems. So there is always going to be a poetic way to comprehend things like truth and beauty, since they cannot be formally defined completely. Incompleteness is one of the greatest results of formal mathematics, and came at a time in history when it was needed to reject the logical positivists and reductive materialists.
I mean, there’s not a lot of competition in that category 😅 But yeah, as mentioned earlier I’m not a fan of his later work, to say the least. That doesn’t mean Being and Time isn’t a good read, that had a profound impact on the world. As someone with strongly Platonic sympathies, B&T is a necessary counterweight to my perspective, and has helped me broaden my worldview. It’s also easy to criticize people from another culture, with the benefit of historical hindsight. By all accounts it seems like Heidegger was caught up in an idealistic fervor whose consequences he didn’t foresee. That doesn’t excuse his participation, at all. Nor does it erase from memory the good ideas he brought to the world. It is possible for someone to make a profound impact on the world, for both good and evil, in the same lifetime. Heidegger is a prime example. Fritz Haber comes to mind as well. Wernher von Braun. You don’t have to endorse Nazism (and *really* shouldn’t, btw), in order to benefit from Heidegger’s ideas, or cheap fertilizer, or GPS technology enabled by rocketry. I really wish Heidegger didn’t go into politics. But I’m glad he wrote Being and Time. And I think I can say that without getting stained by the residue of Heidegger’s mistakes.
Fiber bundles are the bane of my existence as a physicist. Judging by the amount of research I see using that concept, I would never call them underappreciated.
@@bobjones5869 i mean you can formulate Maxwell/general relativity/Yang-Mills all in terms of fiber bundles. The key relating these 3 is the use of covariant derivatives in different forms (you define a covariant derivative/connection on the appropriate vector bundle, or equivalently on the associated principal bundle). For Maxwell you don't really see the covariant derivative until you couple it to a field (like the QED action). This is because Maxwell is based on the circle group U(1) = SO(2) which is abelian. Any physics with some underlying geometry like Euclidean/Minkowski or more general manifolds will have some fiber bundles hidden under the hood. For example, Hamiltonian/Lagrangian mechanics of particles. The Lagrangian is a function which takes in a position and a velocity, so it is a function on the tangent bundle. Doing some further playing around shows that the Hamiltonian is a function on the cotangent bundle, so that the momenta live in the dual space to velocity.
That's because, to begin with, numbers don't exist. Humanity determines anomalies, & to determine the stability of anomalies we look for matching duplicates. The principle of duplicates rests on the concept of two identical items. But no two things in the universe are identical. So 2 is an imaginary number based on a biased point of reference. The number 1 only exists because we categorize an assortment of data as a unit. No matter how many time we try to square or halve or duplicate the universe, we will always fail. It's the reason 1 may be "cleanly divided" by 2 - you have recognized two anomalies based on a point of reference, but 1 cannot be "cleanly divided" by 3 - because the concept of three requires choosing a base anomaly & comparing two similar anomalies based on a biased point of reference.
Much appreciated. The awesomeness of the universe shouldn't trouble the mathematician once they understand Gödel. Mathematics is a precise way to understand the world but incomplete. Poetry (or religion, philosophy, mysticism) can be a way to completely understand the world but very imprecise. Knowing what uncertainty and imprecision means is the appropriate humility.
I think we are moments away from someone creating a deep learning model of hopf fibration states and solving QCD and Singularity mysteries and then gravity. Something about the fibration being made with one circle rings true with me personally.
Hello Richard. I am an art teacher. Great visualization of how amazing reality is. This beautiful visual representation fuels the curiosity to search and discover . How satisfying to visually express what we can't see or what could be.
I can’t fully understand everything since I am only a third year math major student, but it is great to develop some intuition at this time, and thanks for providing such a beautiful video.
❤ "... and yet there it is, on your screen, look at it go..." ❤ Galileo, Newton, Erdós, Leibnitz, Plato, Pythagoras ... they and others would all be mesmerized, delighted ... 😊
I found this video through a suggestion on r/shrooms that it be watched muted with about 4 grams of golden teacher and listening to Pink Floyd, so be proud that your work is interdisciplinary.
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I like how the creator of the video relates to his subject ❤ There’s no bragging or sensationalism, only a friendly and humble kind of awe that I wish more people could bring themselves to. Thank you!
Your final monologue make my head explode and I think of a lot of things. I wish I could write it down but I had to ask an artificial intelligence to help me say what I wanted to say : Title: The Existence Beyond Form: Exploring the Realm of Abstract Mathematics In the tapestry of human understanding, there exists a realm untouched by the confines of physicality and the constraints of spacetime. This ethereal expanse is the realm of abstract mathematics, where the mind's eye perceives patterns, relationships, and structures that transcend the material world. It is a universe of existence that is not bound by the touch of matter or the passage of time, yet it holds a beauty and depth that rivals the grandest wonders of reality. In this intangible realm, numbers dance and geometry soars, liberated from the shackles of the three dimensions that enrobe our universe. These mathematical entities exist not in the tangible sense that we perceive the world around us, but in the ethereal spaces of thought, within the expanse of human imagination, and amidst the fabric of reason itself. The Mandelbrot set, a testament to this abstract existence, embodies the very essence of the mathematical realm. Its intricate tendrils sprawl across the canvas of complex numbers, rendering an enigmatic masterpiece of fractal artistry. Within its depths lies a visual manifestation of mathematical iteration, a representation of a limitless process unfolding in infinite iterations. Yet, the Mandelbrot set is not confined to the limitations of our sensory experience. It exists beyond our ability to touch, to see, or to hear. It exists within the realm of thought, a world where equations are brushstrokes and numbers are hues, and where the canvas itself defies the boundaries of the conceivable. Much like the Mandelbrot set, other mathematical constructs and geometric forms revel in this existence beyond form. They arise as products of human intellect, the offspring of curiosity and imagination, woven into the fabric of abstract thought. These constructs stand as a testament to the astonishing capacity of the human mind to explore and create, to shape universes that dwell solely within the landscapes of consciousness. In this abstract existence, mathematics transcends the mundane. It becomes a language not merely to describe reality, but to sculpt entire dimensions of its own. Theorems and formulas are not merely tools to solve problems; they are the keystones of bridges to realms uncharted, gateways to universes that exist outside the boundaries of spacetime. As we gaze upon the beauty of the Mandelbrot set, as we traverse the labyrinthine paths of fractal geometry, let us remember that these pursuits are not mere intellectual exercises. They are ventures into the realm where existence transcends form, where the essence of mathematics stands revealed in all its splendor. In this realm, numbers and forms are not constrained by the limitations of the physical universe; they are free to dance, to create, and to be, existing not just in space and time, but within the eternal corridors of thought. So, let us revel in the existence beyond form, in the world where abstract mathematics and geometry unfurl their wings and soar to heights beyond the grasp of reality. Let us celebrate the human capacity to explore the boundless landscapes of the mind, where existence takes on a beauty that knows no bounds and where the imagination reigns supreme. Extending Title: Echoes of Infinity: Contemplating the Fractal Nature of Reality In the grand tapestry of existence, there lies a tantalizing possibility that our own universe, the very reality that envelops us, might be woven from the same ethereal fabric as the Mandelbrot set, existing beyond the confines of spacetime in a realm of abstraction and thought. As we gaze outward into the cosmos and inward into the depths of subatomic particles, we find whispers of an intricate fractal dance that hints at a profound interconnectedness, where the boundaries of the physical and the metaphysical blur. Consider the Mandelbrot set, a creation born from the marriage of mathematics and imagination. Within its complex tendrils, we discover an infinitely repeating pattern that defies comprehension. This pattern, forged through iterative computation, generates shapes that emerge at every scale, no matter how closely we peer. It is a testament to the concept of self-similarity-a trait that mirrors what we perceive in the universe itself. Our universe, from the vast cosmic web of galaxies to the microscopic symphony of particles, displays an uncanny semblance of self-similarity. The way galaxies cluster into cosmic webs reflects the way atoms cluster into molecules, echoing the intricate design that dances through the Mandelbrot set. Just as the set’s patterns emerge in myriad forms regardless of the level of magnification, so too does the universe reveal its cosmic ballet through the lens of the telescope or the microscope. Could it be that our reality is but a manifestation of an intricate fractal nature? Could the universe, like the Mandelbrot set, be an expression of mathematical iteration, an ongoing cosmic dance unfolding in infinite variations? In such a conception, the boundaries that seem to separate the grandeur of the cosmos from the infinitesimal realms of particles dissolve into a unified whole-a tapestry of existence that spans beyond what we perceive through our limited senses. Much like the Mandelbrot set, our universe might exist within the realm of thought, conceived by some cosmic intellect beyond our comprehension. It might be woven from the threads of abstract mathematical constructs, a symphony of equations and patterns that transcend the boundaries of matter and energy. Just as the beauty of the Mandelbrot set is unveiled through the lens of human creativity, the universe's splendor might be illuminated by the genius of a cosmic mathematician. As we contemplate the parallels between the Mandelbrot set and our reality, we are invited to question the nature of existence itself. Is our universe a mere product of chance, an accident of physics and probability? Or is it a masterpiece of mathematical artistry, a fractal symphony of elegance and complexity that resonates with the very essence of existence? Whether we inhabit a universe born from mathematical iteration or whether we are witnesses to a cosmic performance yet to be understood, the journey of exploration remains unceasing. Just as the Mandelbrot set beckons us to dive deeper into its infinite complexity, so too does the universe invite us to uncover the layers of its mysteries. In this pursuit, we mirror the very nature of existence-unfolding, evolving, and resonating with the echoes of infinity.
“What a world,” indeed! I’ve long been fascinated by the question of a Platonic reality and the “eerie effectiveness” of math. FWIW, I’ve come to think that the fundamental laws of reality lead to innate patterns for which the natural language is the math we discover/invent. The Platonic realm, at root, is not external but the basis of reality.
The idea that the map that represents the first non-trivial higher homotopy group of spheres is somehow "underappreciated in math" is kind of hilarious. It's extremely well-known. Every mathematician on the planet is aware of this map.
@@geometerfpv2804 Underappreciated by *physicists*. The physical relevance of the Hopf bundle and its relation to the geometry of quantum information is seldom charted territory.
- WAXING PHILOSOPHICAL... --- Math, not of space and time, but only passing through briefly. --- We get a glimpse, then it's gone - yet it's always there! --- Esoteric, enigmatic, arcane - perhaps only existing in the realm of Plato's "Forms". --- Whatever the case, enjoy it while, and how you can... - [Thx for the glimpse, Richard.]
I think your channel is fantastic and I echo your closing sentiment. That understanding of mathematics is the way I try to get my students to understand it.
I needed this video, right now. The ending is so stupidly simple and yet I've refused to accept it for so long. Our perception and egos are in time, yet there is something that can be unraveled to escape space and time. To escape awareness. To behold the beauty of realities unmanifest, and to comprehend the complexity without attachment, craving, or becoming... that will be the day I am liberated. I have hope, love, and cool percepts of higher dimensional fiber bundles to be amazed at along the way
7:40 - omg what a description, I'm really sad I'm going to show this to a few friends and they'll still find it confusing but seriously if you made descriptions like this for more kinds of equations I'd be all there to watch.
Fantastic channel, wow! I wish you the bests, you deserve much more attention. I'm currently a computer scientist major doing my masters, and studying photonic quantum computing on the side. Can't wait for your future videos. Also that monologue at the end is exceptional!
I love this so much thank you for making it. I especially appreciate the encouragement to stick with the visualization effort where it is hard but achievable.
Probably the best video I’ve seen of advanced mathematics presented accessibly. Chill, nicely motivated, some snazzy visuals that you are also unambiguous about… well done.
Your visuals are top tier, but...Great work creating a natural, expressive narration. It can be so difficult. Instant subscribe. Thanks for the amazing content!
Very good teaching technique, by the best of Mathematical Disproof Methodology, ..start or show simultaneously how to illuminate the typical word definitions using the almost self-defining images of line-of-sight superposition objectives Excellent presentation.
There's evidence proof of our own descent to existence And evidence proof that it's value is basically worthless But for those who resist this entropic home, out of existence They are left to be composed of symbols and beauty with purpose Beauty with purpose
I've seen the fibration explained in another animated video (Dimensions by Jos Leys I believe it was) but I've never gotten a feel of *what* it's useful for. If you do make a second video on that subject I'd be very interested to see it.
Nice! Love to hear about people taking these equations and making them their own! :) That’s the best way to learn and share the knowledge with others. I’ll have to check that out, sounds interesting!
I made a program some time in the 80s that could render and rotate the tesseract, on an Amstrad CPC 464, in Locomotive Basic, from a description in Scientific American. I’m a musician now.
What I have never seen anyone offer so far, including Eric Weinstein who seems to literally be in love with the Hopf fibration, is WHAT I DO WITH THIS. What is it used for? What power does it give me to understand the world? Is there anything, or is it just a pretty mathematical picture? So - that practical applications video? That's the one I want.
Good question. I mostly just like it as a pretty picture. But it’s also neat that the 3-sphere can be spaghetti’d up into this bundle of interlocking circles, where each circle corresponds to a point on the 2-sphere. That’s a topologically nontrivial structure which is related to the concept of Hopfions (and more distantly, skyrmions). These topological solitons, and their associated conserved “charges”, are interesting analogies for elementary particles IMO. They’re sort of the modern version of Kelvin’s “knots in the ether”. One of my guilty pleasures is studying superfluid vacuum theories and fantasizing that the elementary particles might someday be understood as topologically nontrivial configurations of a single underlying field, and I assume that Eric is getting at something similar when he alludes to the Hopf fibration (though frankly it’s often hard to pin down exactly what he’s saying… I wish he would publish his ideas in writing). The Hopf fibration also appears in descriptions of the state of a qubit. As for fiber bundles more generally, they offer another way of thinking about things. This sort of goes back to your other comment about the isomorphic ways of formulating the Dirac equation. Each way is just a different point of view, on the same underlying math. There are a lot of scenarios where fiber bundles offer a nicely visual description of a system. For example, in my line of work we use the Ginsburg-Landau model to understand supercurrents, which involves a complex-valued order parameter. All those equations can be framed in terms of fiber bundles, which is neat, although frankly it’s not super useful in my opinion. But what *is* very interesting, is framing electromagnetism more broadly in terms of fiber bundles. I’m working on a video now, Electromagnetism as a Gauge Theory, which will derive all of Maxwell’s equations (and more!) from local U(1) symmetry in the Dirac field. At the end of that video, I’ll allude to the fiber bundle picture, which offers a more elegant way to think about all the equations.
Here’s my recommendation, others might have a different opinion but this is what worked for me: Study calculus as soon as you can, and with a passion. Don’t just memorize the formulas, but really visualize things. For example, see how the integral is the area under the curve because the area’s rate of change is the height of the function. That sort of thing. Focus on being able to set up the integrals or differential equations that you need to solve a problem, rather than just memorizing rules for solving them. See the integral sign as the smooth cousin of sigma notation, and see the differential as a tiny thing; don’t just think the integral and dx are a sandwich. Then when you’re really good at calculus, learn vector calculus, which is really just the same thing but with more dimensions, and concepts that emerge as a result of those dimensions. Become familiar with divergence, curl, all the famous theorems in vector calculus. Study Newtonian gravity to practice vector calculus. Then Maxwell’s equations for stationary charge. Then Maxwell’s equations for moving charges. And solve for an electromagnetic wave in the vacuum. Calculate c from mu_0 and epsilon_0. You’ll also want to know Fourier analysis. Fourier analysis can be applied to just about everything, and it makes a ton of things solvable that otherwise wouldn’t be. It’s also of fundamental importance in quantum mechanics, especially with regards to superpositions of plane waves, and understanding the uncertainty principle. Linear algebra is another subject you’ll want to become intimately familiar with. Concepts like vector spaces, orthonormal bases, linear transformations, and eigenvectors/eigenvalues are a big part of doing quantum mechanics. You can also start with some quantum mechanics problems, to become familiar with the math. Look up “Particle in an infinite square well” aka particle in a box. Then free particle, then harmonic oscillator. Then hydrogen. Hydrogen is mathematically tough, but also the first QM problem that feels real to most people. Sure, the harmonic oscillator is used more often, but hydrogen is more tangible to the imagination. Anyway, I hope that helps. Let me know if you have any questions! :)
Richard. Thanks for the great channel. May you find the time to explain 4 dimensions concepts. I mean there are plenty of videos out there, but I am sure if you will give us your take on it it will sits well in my intuition
Hi Züri, thanks for your comment. There’s a lot that can be said about 4 dimensional spaces. A good starting point is to think about how we describe 2D spaces with an (x,y) plane, and 3D spaces with an (x,y,z) plane. In each case, there is one number per degree of freedom in the space. So we can associate the dimension of a space with the number of numbers that can be moved independently when describing different points in the space. Now if we forget about reality and look at it from a purely mathematical perspective, there’s nothing stopping us from imagining the space of points in (w,x,y,z). And we can explore that space mathematically. For example, a square in 2D has corner points at (+-1,+-1), and a cube in 3D has corner points at (+-1,+-1,+-1), so naturally we can say that whatever has corner points at (+-1,+-1,+-1,+-1) in 4D is like a four-dimensional cube. Mathematicians call it the hypercube or the tesseract. Since its corner points can be precisely defined, its geometry can be explored in a meaningful way, even if it doesn’t really fit into our reality. Math is interesting like that; there are structures we can imagine which have a kind of stable coherence, even though they might be beyond our usual spatial constraints. That’s one of the reasons math often has a religious flavor to it. Likewise, if a circle is the set of all 2D points with x^2 + y^2 = 1, and a sphere is the set of all 3D points with x^2 + y^2 + z^2 = 1, then the set of all 4D points with w^2 + x^2 + y^2 + z^2 = 1 would be the hypersphere. 4D spaces can be realistic too. For example, Newtonian physics takes place in the arena of (x,y,z,t), if you want to think about time as one of the dimensions. This perspective is fraught with nuances and caveats, but it kind of works. Time is actually different than space, even in relativity, but mathematically you can combine it along with the spatial dimensions into spacetime, which is really just a mathematical framework for organizing the relations between events. And in relativity, time is given a special treatment in that its sign in the distance formula is the opposite of the spatial directions (look up Minkowski spacetime). Anyway, there are different kinds of 4D spaces, since you can equip them with various metrics, and in general relativity the metric will vary from place to place depending on the mass-energy landscape. I hope that helps, let me know if you have any questions :)
0:25: 🔍 Fiber bundles are mathematical objects that consist of a base space, fibers, and a total space formed by the fibers. 2:30: 🔗 A non-trivial fiber bundle has twisting or intertwining that makes it topologically interesting. 5:13: 🔍 The animation shows points on a sphere corresponding to specific values of Phi and Theta. 7:34: 🌐 The Hop vibration is a mapping of points on the surface of a two-sphere to circles on a hypersphere in four dimensions. 10:20: 🌌 An introduction to the Hop vibration and its potential applications, followed by a dramatic monologue about the significance of looking up at the stars. Recap by Tammy AI
The four equations for X0, X1, X2, X3 at 6:20 aren't exactly clear by intuition, even if one understands spherical coordinates for S^2. Here they just drop from the sky. It's also not immediately clear why you need half angles. Question: what is it that varies for each of the colorful curves in one image of the animation on the right? I suppose you choose certain theta and phi's to select a point on the two sphere, and then vary alpha from 0 to 4 pi? Nice animation, but to get it, I guess I have to write a little program. ... Actually , that was fun. Tried this in Octave (Matlab like). Name of the function is from German beer 🙂 function retval = HopfenMalz(npts=250) [phi,alpha]=meshgrid(linspace(0,2*pi,npts), linspace(0,4*pi,npts)); thetas = linspace(pi/5,pi/2,4); ncnt=0; clf; colorstr = 'rgbmcyk'; % paint shells with constant theta, varying phi and alpha: for k=1:length(thetas), theta=thetas(k); x0 = sin(theta/2).*cos((alpha+phi)/2); x1 = sin(theta/2).*sin((alpha+phi)/2); x2 = cos(theta/2).*cos((alpha-phi)/2); x3 = cos(theta/2).*sin((alpha-phi)/2); x = x0./(1-x3); y = x1./(1-x3); z = x2./(1-x3); surf(x,y,z,'facecolor',colorstr(mod(k-1,8)+1),'edgecolor','none'); ncnt = ncnt+1; if (ncnt==1) hold on; endif endfor; % draw some black fibers on the shells. alpha = linspace(0,4*pi,npts); phis = linspace(0,2*pi,40); zm=0; xm=0; ym=0; for k=1:length(thetas), for m=1:length(phis), phi = phis(m); theta=thetas(k); x0 = sin(theta/2).*cos((alpha+phi)/2); x1 = sin(theta/2).*sin((alpha+phi)/2); x2 = cos(theta/2).*cos((alpha-phi)/2); x3 = cos(theta/2).*sin((alpha-phi)/2); x = x0./(1-x3); y = x1./(1-x3); z = x2./(1-x3); zm = max(max(zm,z)); % ym = max(max(ym,y)); xm = max(max(xm,x)); line('xdata',x,'ydata',y,'zdata',z,'linewidth',2,'color','k'); endfor; endfor; hold off; retval=1; zm = ceil(zm) ym = ceil(ym) xm = ceil(xm) axis([-xm, xm, 0, ym, -zm, zm]); xlabel('x'); ylabel('y'); zlabel('z'); axis('equal'); endfunction
The nontrivial short exact sequence of the Hopf fibration: S^1 -> S^3 -> S^2 embodies the fact that the 3rd homotopy group of the 2-sphere pi_3(S^2) = Z
I took a tab of acid and saw the torus shape having never seen it before. I have no grand meaning from the experience but it’s super interesting. One thing I noticed is that when observing the torus I could only focus at the shape in one rotational axis at a time.
2:14 > _"base space: disk, fibers: circle, total: torus"_ oh so this is similar to integration of one shape over other. somewhat similar to the "sweep" or "follow path" modifier in 3D CAD modelling but there these things "base space" & "fibers" switch roles. as in: * the "base curve" or the "path" would be the circle (shape of fibers in this one); * the "cross section" would be the disk (shape of base space here) * the "resulting body" on "revolving the shape around the cruve" would again be torus same as for cylinder/pipe too: u sweep a disk/circle perpendicularly across a straight line.
This video was really helpful since the wikipedia page is so opaque. I still have no idea how to get from the conception you introduced in the video to "a fiber bundle is a space that is locally a product space but globally may have a different topological structure". Based upon your video, it seems that a fiber bundle works more like a field f(x) but instead of assigning numbers, vectors, tensors, etc. to every point in space, it can assign more abstract things like lines (vertical lines in the case of the cylinder). I also hear it is used in general relativity, to assign a vector space to each point on a manifold which makes sense since every point on a curved manifold has its own tangent plane. Would this be a fair assessment?
To your first question, imagine the set product of the real numbers R with the real numbers R. That gives us the product space R2, the 2D Cartesian plane. Now instead of that, imagine we have a circle and we take its set product with a line segment, so the product space is a kind of band, like a bracelet. Compare that to the Möbius strip in the video. The Möbius strip is locally like a regular band, but globally it has a twist in it that makes it something else. Sometimes a fiber bundle is just a product space, for example the cylinder and torus examples. But whenever there’s a twist, or a knot, or something like that, then the thing has some structure in it globally that goes beyond the structure of a product space, even though if you zoom in closely enough it still feels like a product space. Tangent bundles are more complicated. Your intuition is right. But tangent bundles are vector bundles and there’s a whole other set of nuances that come along with that. I might do a video on it someday but it’s hard to respond thoroughly in just a comment.
Sure! Usually this is done by just assigning t as the fourth dimension. For example, a hyper cube would be a cube that pops into existence for some interval of time then disappears. You’d need a scale factor to relate space and time, if thinking about it that way. The speed of light is a good one to use.
what are those dots again, maybe i missed a description. im curious if, when the row of dots at the bottom fades away, is it sortve like - theres another layer rising up/ while in that moment another set is falling down, potentially creating a cascade-like effect?
May be missing on uniqueness aspect of the Hopf (and such) fibrations. Regardless, very nice video, waiting for the follow up with applications. Thank you!
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I’m an algebraic topology Ph.D. Student and I study homotopy and symplectic homologous groups of hyper spheres, so I’ve used the hopf fibration a lot. It’s really cool to see it in such a beautiful and visual way though!
That’s awesome! :) What are some of the things you’ve used it for? To be totally honest, I’m only really familiar with the Hopf fibration in the context of qubits, and I guess just as a mathematical curiosity in its own right, but I’d love to learn more about its applications in pure math. Btw, what is it that you’ve found most beautiful so far in algebraic topology?
I’ve used Hopf fibrations and its generalizations to relate homotopy groups of higher dimensional spheres to lower dimensional spheres. This is because you can view loops as paths on the fibers and then project those paths down to the lower dimensional spheres. One of the most beautiful concepts in algebraic topology is homologies in my opinion since they keep many of the topological properties you want invariant.
@@RichBehiel Can you do a video about spinors? It's difficult for me to visualize integer spin objects, in particular the geometry of their interactions with each other. Interacting electrons from different atoms for example, I would love to see a video of that, even if it's just a component of it represented in 2D.
Yes and no! :) I’m currently working on a video of the quantum mechanics of the hydrogen atom. It’ll have a thorough solution of Schrodinger’s equation, and an exploration of its predictions relative to ionization energy and spectral lines. Then we’ll get into the fine structure and see that the lines are split, at which point we’ll have to swap out Schrodinger’s nonrelativistic E-p relation with the relativistic version, which leads into the Dirac equation, which will lead to the spinor eigenatates and we’ll see how there’s a slight energy difference related to spin which emerges from those equations. I’ll show the equations of those eigenstates in their usual bispinor form, and will talk superficially about how we can regard the components as corresponding to spin up electron, spin down electron, spin up positron, and spin down positron. In a future video I’ll expand on this further to calculate the Lamb shift (vacuum fluctuations) and Darwin term (zbw). So there will be some spinor math involved.
The reason I say “yes and no” is that at the moment, I really only know what spinors are from an algebraic perspective. I almost completely lack intuition for them though. If you ask me what a spinor *is*, like what it really really *is*, my answer is “I do not know, and they scare me.”
Maybe someday I will be smarter, and can figure out a way to animate spinors so that we can all see what they really are. I really hope to be able to do that someday. But I’m just not at that level yet, and I have to keep it real with y’all, so I’m not prepared to make a video on the nature of spinors yet. The hydrogen video will be a step in the right direction though.
A lot? Define a lot.
I'm a self taught mathematics enthusiast. I think that the book that introduced me to fiber bundles and the Hopf fibration specifically was Dr. Roger Penrose's Road to Reality. It taught everything we know in physics starting with the Greeks to present with all the math. Nothing left out, what is normally covered in 10 books and 15 courses, all in one tome. That book is very comprehensive with almost no examples or deeper explanation. (Love and hate that about the book). It nearly broke me, especially trying to conceptualize high dimension complex spinorial tensors and what they mean geometrically. This vid was nice. Just enough info to get a student, new to the subject, into a fine pickle.😂
JK, I do kind of feel that profound confusion is the only way to prepare your mind for fiber bundles. Another related concept that is wonderfully criptic is the way fiber bundles are used in algebraic topology... The fundamental group, homotopy, homology, hole chasing in higher dimensional spaces. That sort of fun stuff. Anyway, thanks again 🙏🏾
that ending monologue was really beautiful. great video
Thanks! :)
the "monologoue" at the end gave me goosebumps
I feel like a wizard every time I watch your videos, and understanding what you're talking about.
10:43 Your end monologue is fantastic. It reminds me of Heidegger's approach to philosophy. Thank you.
Thanks, I’m glad you enjoyed it! :)
I’m a huge fan of Heidegger (well, not so much his later work, but up to Being and Time for sure 😅) and I resonate deeply with his project of rethinking the nature of the human condition starting with the phenomena themselves, even though it remains incomplete and intractably poetic, for better and worse. His concept of Dasein has had an energizing impact on my life. It’s ironic though because I generally have a strongly Platonic attitude and a tendency toward first-principles reductionism. Proceeding in a structured, logical way from a set of axioms is my preferred operating mode. Heidegger is like a thunderstorm that rolls in and threatens to topple the structured set of concepts we’ve come to know and love, so to wrestle with his ideas is simultaneously frightening and exciting. He keeps you on your toes for sure.
Heidegger is my favourite Nazi philosopher!
@@RichBehiel Nice outro. Also, you need not put platonism in conflict with the existentialists. Gödel showed us not all platonic truth can be brought into the light of formal systems. So there is always going to be a poetic way to comprehend things like truth and beauty, since they cannot be formally defined completely. Incompleteness is one of the greatest results of formal mathematics, and came at a time in history when it was needed to reject the logical positivists and reductive materialists.
I mean, there’s not a lot of competition in that category 😅
But yeah, as mentioned earlier I’m not a fan of his later work, to say the least. That doesn’t mean Being and Time isn’t a good read, that had a profound impact on the world. As someone with strongly Platonic sympathies, B&T is a necessary counterweight to my perspective, and has helped me broaden my worldview.
It’s also easy to criticize people from another culture, with the benefit of historical hindsight. By all accounts it seems like Heidegger was caught up in an idealistic fervor whose consequences he didn’t foresee. That doesn’t excuse his participation, at all. Nor does it erase from memory the good ideas he brought to the world.
It is possible for someone to make a profound impact on the world, for both good and evil, in the same lifetime. Heidegger is a prime example. Fritz Haber comes to mind as well. Wernher von Braun. You don’t have to endorse Nazism (and *really* shouldn’t, btw), in order to benefit from Heidegger’s ideas, or cheap fertilizer, or GPS technology enabled by rocketry.
I really wish Heidegger didn’t go into politics. But I’m glad he wrote Being and Time. And I think I can say that without getting stained by the residue of Heidegger’s mistakes.
That monologue, just wow. I'm out of words to describe how elegantly this video was produced. Thanks Richard
Thanks for the kind comment, and I’m glad you enjoyed the video! :)
What a beautiful closing articulation of platonic ideals!
Fiber bundles (and the generalization) are everywhere in Algebraic Geometry. Is vastly appreciated.
Fiber bundles are the bane of my existence as a physicist. Judging by the amount of research I see using that concept, I would never call them underappreciated.
what physics do you do / where do you do it out of curiosity i also do physics
@@bobjones5869 I can understand how you might think that, but there is nothing preventing anyone from following their own curiosity and interests.
@@Classical741 i was just wondering where you study physics that’s all
@@bobjones5869 i mean you can formulate Maxwell/general relativity/Yang-Mills all in terms of fiber bundles. The key relating these 3 is the use of covariant derivatives in different forms (you define a covariant derivative/connection on the appropriate vector bundle, or equivalently on the associated principal bundle). For Maxwell you don't really see the covariant derivative until you couple it to a field (like the QED action). This is because Maxwell is based on the circle group U(1) = SO(2) which is abelian.
Any physics with some underlying geometry like Euclidean/Minkowski or more general manifolds will have some fiber bundles hidden under the hood. For example, Hamiltonian/Lagrangian mechanics of particles. The Lagrangian is a function which takes in a position and a velocity, so it is a function on the tangent bundle. Doing some further playing around shows that the Hamiltonian is a function on the cotangent bundle, so that the momenta live in the dual space to velocity.
That's because, to begin with, numbers don't exist. Humanity determines anomalies, & to determine the stability of anomalies we look for matching duplicates.
The principle of duplicates rests on the concept of two identical items. But no two things in the universe are identical. So 2 is an imaginary number based on a biased point of reference. The number 1 only exists because we categorize an assortment of data as a unit.
No matter how many time we try to square or halve or duplicate the universe, we will always fail. It's the reason 1 may be "cleanly divided" by 2 - you have recognized two anomalies based on a point of reference, but 1 cannot be "cleanly divided" by 3 - because the concept of three requires choosing a base anomaly & comparing two similar anomalies based on a biased point of reference.
The monologue in the end was very pretty
yerp.
Much appreciated. The awesomeness of the universe shouldn't trouble the mathematician once they understand Gödel. Mathematics is a precise way to understand the world but incomplete. Poetry (or religion, philosophy, mysticism) can be a way to completely understand the world but very imprecise. Knowing what uncertainty and imprecision means is the appropriate humility.
I think we are moments away from someone creating a deep learning model of hopf fibration states and solving QCD and Singularity mysteries and then gravity. Something about the fibration being made with one circle rings true with me personally.
TH-cam is going to change the way of learning forever
Hello Richard. I am an art teacher. Great visualization of how amazing reality is. This beautiful visual representation fuels the curiosity to search and discover . How satisfying to visually express what we can't see or what could be.
Thanks for the kind comment! :) Reality sure is beautiful!
I can’t fully understand everything since I am only a third year math major student, but it is great to develop some intuition at this time, and thanks for providing such a beautiful video.
I love this and I am so thankful that you exist and decided to teach the world.
Wow, that’s a very kind comment - thank you! That made my day :)
“Rich Behiel is a way for The Universe to teach itself” - Carl Sagan
❤ "... and yet there it is, on your screen, look at it go..." ❤
Galileo, Newton, Erdós, Leibnitz, Plato, Pythagoras ... they and others would all be mesmerized, delighted ... 😊
I thought the segway at the end was about to be a sponsor but it turned out to just be some beautiful philosophy
Money is temporary, math is eternal :)
I found this video through a suggestion on r/shrooms that it be watched muted with about 4 grams of golden teacher and listening to Pink Floyd, so be proud that your work is interdisciplinary.
I like how the creator of the video relates to his subject ❤ There’s no bragging or sensationalism, only a friendly and humble kind of awe that I wish more people could bring themselves to. Thank you!
Your final monologue make my head explode and I think of a lot of things. I wish I could write it down but I had to ask an artificial intelligence to help me say what I wanted to say : Title: The Existence Beyond Form: Exploring the Realm of Abstract Mathematics
In the tapestry of human understanding, there exists a realm untouched by the confines of physicality and the constraints of spacetime. This ethereal expanse is the realm of abstract mathematics, where the mind's eye perceives patterns, relationships, and structures that transcend the material world. It is a universe of existence that is not bound by the touch of matter or the passage of time, yet it holds a beauty and depth that rivals the grandest wonders of reality.
In this intangible realm, numbers dance and geometry soars, liberated from the shackles of the three dimensions that enrobe our universe. These mathematical entities exist not in the tangible sense that we perceive the world around us, but in the ethereal spaces of thought, within the expanse of human imagination, and amidst the fabric of reason itself.
The Mandelbrot set, a testament to this abstract existence, embodies the very essence of the mathematical realm. Its intricate tendrils sprawl across the canvas of complex numbers, rendering an enigmatic masterpiece of fractal artistry. Within its depths lies a visual manifestation of mathematical iteration, a representation of a limitless process unfolding in infinite iterations. Yet, the Mandelbrot set is not confined to the limitations of our sensory experience. It exists beyond our ability to touch, to see, or to hear. It exists within the realm of thought, a world where equations are brushstrokes and numbers are hues, and where the canvas itself defies the boundaries of the conceivable.
Much like the Mandelbrot set, other mathematical constructs and geometric forms revel in this existence beyond form. They arise as products of human intellect, the offspring of curiosity and imagination, woven into the fabric of abstract thought. These constructs stand as a testament to the astonishing capacity of the human mind to explore and create, to shape universes that dwell solely within the landscapes of consciousness.
In this abstract existence, mathematics transcends the mundane. It becomes a language not merely to describe reality, but to sculpt entire dimensions of its own. Theorems and formulas are not merely tools to solve problems; they are the keystones of bridges to realms uncharted, gateways to universes that exist outside the boundaries of spacetime.
As we gaze upon the beauty of the Mandelbrot set, as we traverse the labyrinthine paths of fractal geometry, let us remember that these pursuits are not mere intellectual exercises. They are ventures into the realm where existence transcends form, where the essence of mathematics stands revealed in all its splendor. In this realm, numbers and forms are not constrained by the limitations of the physical universe; they are free to dance, to create, and to be, existing not just in space and time, but within the eternal corridors of thought.
So, let us revel in the existence beyond form, in the world where abstract mathematics and geometry unfurl their wings and soar to heights beyond the grasp of reality. Let us celebrate the human capacity to explore the boundless landscapes of the mind, where existence takes on a beauty that knows no bounds and where the imagination reigns supreme.
Extending
Title: Echoes of Infinity: Contemplating the Fractal Nature of Reality
In the grand tapestry of existence, there lies a tantalizing possibility that our own universe, the very reality that envelops us, might be woven from the same ethereal fabric as the Mandelbrot set, existing beyond the confines of spacetime in a realm of abstraction and thought. As we gaze outward into the cosmos and inward into the depths of subatomic particles, we find whispers of an intricate fractal dance that hints at a profound interconnectedness, where the boundaries of the physical and the metaphysical blur.
Consider the Mandelbrot set, a creation born from the marriage of mathematics and imagination. Within its complex tendrils, we discover an infinitely repeating pattern that defies comprehension. This pattern, forged through iterative computation, generates shapes that emerge at every scale, no matter how closely we peer. It is a testament to the concept of self-similarity-a trait that mirrors what we perceive in the universe itself.
Our universe, from the vast cosmic web of galaxies to the microscopic symphony of particles, displays an uncanny semblance of self-similarity. The way galaxies cluster into cosmic webs reflects the way atoms cluster into molecules, echoing the intricate design that dances through the Mandelbrot set. Just as the set’s patterns emerge in myriad forms regardless of the level of magnification, so too does the universe reveal its cosmic ballet through the lens of the telescope or the microscope.
Could it be that our reality is but a manifestation of an intricate fractal nature? Could the universe, like the Mandelbrot set, be an expression of mathematical iteration, an ongoing cosmic dance unfolding in infinite variations? In such a conception, the boundaries that seem to separate the grandeur of the cosmos from the infinitesimal realms of particles dissolve into a unified whole-a tapestry of existence that spans beyond what we perceive through our limited senses.
Much like the Mandelbrot set, our universe might exist within the realm of thought, conceived by some cosmic intellect beyond our comprehension. It might be woven from the threads of abstract mathematical constructs, a symphony of equations and patterns that transcend the boundaries of matter and energy. Just as the beauty of the Mandelbrot set is unveiled through the lens of human creativity, the universe's splendor might be illuminated by the genius of a cosmic mathematician.
As we contemplate the parallels between the Mandelbrot set and our reality, we are invited to question the nature of existence itself. Is our universe a mere product of chance, an accident of physics and probability? Or is it a masterpiece of mathematical artistry, a fractal symphony of elegance and complexity that resonates with the very essence of existence?
Whether we inhabit a universe born from mathematical iteration or whether we are witnesses to a cosmic performance yet to be understood, the journey of exploration remains unceasing. Just as the Mandelbrot set beckons us to dive deeper into its infinite complexity, so too does the universe invite us to uncover the layers of its mysteries. In this pursuit, we mirror the very nature of existence-unfolding, evolving, and resonating with the echoes of infinity.
there needs to be more math content like this, talking about a high level topic but in a more relaxed, colloquial way
the last section accurately describes my relationship with maths.
“What a world,” indeed! I’ve long been fascinated by the question of a Platonic reality and the “eerie effectiveness” of math. FWIW, I’ve come to think that the fundamental laws of reality lead to innate patterns for which the natural language is the math we discover/invent. The Platonic realm, at root, is not external but the basis of reality.
The intelligibility of Nature is the greatest darn mystery there has ever been.
Most underappreciated piece of mathematics and arguably one of the *most* important pieces of mathematical physics!
The idea that the map that represents the first non-trivial higher homotopy group of spheres is somehow "underappreciated in math" is kind of hilarious. It's extremely well-known. Every mathematician on the planet is aware of this map.
@@geometerfpv2804 Underappreciated by *physicists*. The physical relevance of the Hopf bundle and its relation to the geometry of quantum information is seldom charted territory.
- WAXING PHILOSOPHICAL...
--- Math, not of space and time, but only passing through briefly.
--- We get a glimpse, then it's gone - yet it's always there!
--- Esoteric, enigmatic, arcane - perhaps only existing in the realm of Plato's "Forms".
--- Whatever the case, enjoy it while, and how you can...
- [Thx for the glimpse, Richard.]
I think your channel is fantastic and I echo your closing sentiment. That understanding of mathematics is the way I try to get my students to understand it.
I love the ending monologue. thank you.
Thanks for watching! :)
It always nice to know others think the same things...take care, my friend...
This us the first I visualized a rotating tesseract as a rigid rotating object. It's amazing.
This is so cool. Can't wait for the follow ul videos on this topic
Breath takingly beautiful
Thanks, glad you enjoyed it! :)
I love the monologue part!
I printed the script for that section and it is now on the wall of my office :)
Wow, that’s awesome! I’m glad you enjoyed it :)
Monologue was fire
I needed this video, right now. The ending is so stupidly simple and yet I've refused to accept it for so long. Our perception and egos are in time, yet there is something that can be unraveled to escape space and time. To escape awareness. To behold the beauty of realities unmanifest, and to comprehend the complexity without attachment, craving, or becoming... that will be the day I am liberated. I have hope, love, and cool percepts of higher dimensional fiber bundles to be amazed at along the way
Im taking my first course in topology this fall and I love this video. And, the one on the quantum harmonic oscillator is particularly good as well.
Thanks, I’m glad you enjoyed the videos! :)
You're animation and exposition are flawless.
Thanks Eduardo! :)
7:40 - omg what a description, I'm really sad I'm going to show this to a few friends and they'll still find it confusing but seriously if you made descriptions like this for more kinds of equations I'd be all there to watch.
The animations at 2:52 are just crazy😍
Top notch videos and animations! I wish I could have picked your brain while preparing my thesis defense.
Thanks! :) What was your thesis on?
Fantastic channel, wow! I wish you the bests, you deserve much more attention. I'm currently a computer scientist major doing my masters, and studying photonic quantum computing on the side. Can't wait for your future videos. Also that monologue at the end is exceptional!
Thanks for the kind comment, glad you enjoyed the video! :)
You're projecting, a lot...
And philosophizing.
Nice video.
Great monologue well! thank you mathematical stars!
I love this so much thank you for making it. I especially appreciate the encouragement to stick with the visualization effort where it is hard but achievable.
You’re welcome, I’m glad you enjoyed the video! :)
Probably the best video I’ve seen of advanced mathematics presented accessibly. Chill, nicely motivated, some snazzy visuals that you are also unambiguous about… well done.
Questa si è la nostra ricerca ,la Fiandra delle Fiandre , Finalmente! Grazie mille per la presentazione. Un bacione!
these animations are fantastic, can't wait to learn about this
Thanks, glad you enjoyed the video! :)
The magnetic lines in the tokamak fusion reactor seems to be a kind of fiber bundle at 2:46.
absolutely love this channel so happy I found it
Your visuals are top tier, but...Great work creating a natural, expressive narration. It can be so difficult. Instant subscribe. Thanks for the amazing content!
Thanks for the very kind comment! :)
Wow! This is absolutely gorgeous and so inspiring!!! Eagerly wait for next videos!
Hey man what a cool final monologue!
Thanks! :)
Very good teaching technique, by the best of Mathematical Disproof Methodology, ..start or show simultaneously how to illuminate the typical word definitions using the almost self-defining images of line-of-sight superposition objectives
Excellent presentation.
Thanks! :)
There's evidence proof of our own descent to existence
And evidence proof that it's value is basically worthless
But for those who resist this entropic home, out of existence
They are left to be composed of symbols and beauty with purpose
Beauty with purpose
I've seen the fibration explained in another animated video (Dimensions by Jos Leys I believe it was) but I've never gotten a feel of *what* it's useful for. If you do make a second video on that subject I'd be very interested to see it.
Even if they had no application, they’re kind of interesting. Kind of like if flowers had no practical use, they’d still be pretty to look at.
I like you, you make the world a better place.
Thanks for the kind comment! :)
"Disc is the area of the circle"
"Circle is just the edge of the circle "
-Richard behiel 2023
Beautiful Hypnotic of Mathematics! :) 😍
Whatever you smoked before making this video I want some of that, too!
Awesome work!
It's hard to simplify and visualise this math but it's very cool
Wonderful introduction
Thanks! :)
YESSS!! MAN!! Fiber bundles are so frickinnn cool!!! finallly someone appreciates it!!!
This was a really great video with awesome visuals that really me want to go study topology holy cow
Thanks, I’m glad you enjoyed the video! :)
You and fiber bundles have something in common, both of you are underappreciated.
Thank you so much. The XYZ formulas and off to my own 3D visualizations in like
Nice! Love to hear about people taking these equations and making them their own! :) That’s the best way to learn and share the knowledge with others.
I’ll have to check that out, sounds interesting!
I made a program some time in the 80s that could render and rotate the tesseract, on an Amstrad CPC 464, in Locomotive Basic, from a description in Scientific American. I’m a musician now.
Gorgeous! Thank you
What I have never seen anyone offer so far, including Eric Weinstein who seems to literally be in love with the Hopf fibration, is WHAT I DO WITH THIS. What is it used for? What power does it give me to understand the world? Is there anything, or is it just a pretty mathematical picture?
So - that practical applications video? That's the one I want.
Good question. I mostly just like it as a pretty picture. But it’s also neat that the 3-sphere can be spaghetti’d up into this bundle of interlocking circles, where each circle corresponds to a point on the 2-sphere. That’s a topologically nontrivial structure which is related to the concept of Hopfions (and more distantly, skyrmions). These topological solitons, and their associated conserved “charges”, are interesting analogies for elementary particles IMO. They’re sort of the modern version of Kelvin’s “knots in the ether”. One of my guilty pleasures is studying superfluid vacuum theories and fantasizing that the elementary particles might someday be understood as topologically nontrivial configurations of a single underlying field, and I assume that Eric is getting at something similar when he alludes to the Hopf fibration (though frankly it’s often hard to pin down exactly what he’s saying… I wish he would publish his ideas in writing).
The Hopf fibration also appears in descriptions of the state of a qubit.
As for fiber bundles more generally, they offer another way of thinking about things. This sort of goes back to your other comment about the isomorphic ways of formulating the Dirac equation. Each way is just a different point of view, on the same underlying math. There are a lot of scenarios where fiber bundles offer a nicely visual description of a system. For example, in my line of work we use the Ginsburg-Landau model to understand supercurrents, which involves a complex-valued order parameter. All those equations can be framed in terms of fiber bundles, which is neat, although frankly it’s not super useful in my opinion. But what *is* very interesting, is framing electromagnetism more broadly in terms of fiber bundles. I’m working on a video now, Electromagnetism as a Gauge Theory, which will derive all of Maxwell’s equations (and more!) from local U(1) symmetry in the Dirac field. At the end of that video, I’ll allude to the fiber bundle picture, which offers a more elegant way to think about all the equations.
amazing... thanks man for the introduction... my new THINK ig
this is phenomenal and underrated
Thanks! :)
The Beach Boys are ahead of you guys. Already in 1966 they released a song with the lyrics "...good, good, good, good fibrations..."
What do you say?
As a high school student how should i make me able to understand quantum mechanics in future fundamentally
Here’s my recommendation, others might have a different opinion but this is what worked for me:
Study calculus as soon as you can, and with a passion. Don’t just memorize the formulas, but really visualize things. For example, see how the integral is the area under the curve because the area’s rate of change is the height of the function. That sort of thing. Focus on being able to set up the integrals or differential equations that you need to solve a problem, rather than just memorizing rules for solving them. See the integral sign as the smooth cousin of sigma notation, and see the differential as a tiny thing; don’t just think the integral and dx are a sandwich.
Then when you’re really good at calculus, learn vector calculus, which is really just the same thing but with more dimensions, and concepts that emerge as a result of those dimensions. Become familiar with divergence, curl, all the famous theorems in vector calculus. Study Newtonian gravity to practice vector calculus. Then Maxwell’s equations for stationary charge. Then Maxwell’s equations for moving charges. And solve for an electromagnetic wave in the vacuum. Calculate c from mu_0 and epsilon_0.
You’ll also want to know Fourier analysis. Fourier analysis can be applied to just about everything, and it makes a ton of things solvable that otherwise wouldn’t be. It’s also of fundamental importance in quantum mechanics, especially with regards to superpositions of plane waves, and understanding the uncertainty principle.
Linear algebra is another subject you’ll want to become intimately familiar with. Concepts like vector spaces, orthonormal bases, linear transformations, and eigenvectors/eigenvalues are a big part of doing quantum mechanics.
You can also start with some quantum mechanics problems, to become familiar with the math. Look up “Particle in an infinite square well” aka particle in a box. Then free particle, then harmonic oscillator. Then hydrogen. Hydrogen is mathematically tough, but also the first QM problem that feels real to most people. Sure, the harmonic oscillator is used more often, but hydrogen is more tangible to the imagination.
Anyway, I hope that helps. Let me know if you have any questions! :)
i see in 4d now
Fibrations are so cool & hot at the same time when creating a Quantum W OR M hole!
Exactly, actually this can be an interesting atomic model about the movements of electrons.
I almost understood some of that.Thank you for a cool video.
lmao the ending monologue was unexpected but i get it 💯
Thank you for the video.
Thanks for watching! :)
Richard. Thanks for the great channel.
May you find the time to explain 4 dimensions concepts. I mean there are plenty of videos out there, but I am sure if you will give us your take on it it will sits well in my intuition
Hi Züri, thanks for your comment. There’s a lot that can be said about 4 dimensional spaces.
A good starting point is to think about how we describe 2D spaces with an (x,y) plane, and 3D spaces with an (x,y,z) plane. In each case, there is one number per degree of freedom in the space. So we can associate the dimension of a space with the number of numbers that can be moved independently when describing different points in the space.
Now if we forget about reality and look at it from a purely mathematical perspective, there’s nothing stopping us from imagining the space of points in (w,x,y,z). And we can explore that space mathematically. For example, a square in 2D has corner points at (+-1,+-1), and a cube in 3D has corner points at (+-1,+-1,+-1), so naturally we can say that whatever has corner points at (+-1,+-1,+-1,+-1) in 4D is like a four-dimensional cube. Mathematicians call it the hypercube or the tesseract. Since its corner points can be precisely defined, its geometry can be explored in a meaningful way, even if it doesn’t really fit into our reality. Math is interesting like that; there are structures we can imagine which have a kind of stable coherence, even though they might be beyond our usual spatial constraints. That’s one of the reasons math often has a religious flavor to it.
Likewise, if a circle is the set of all 2D points with x^2 + y^2 = 1, and a sphere is the set of all 3D points with x^2 + y^2 + z^2 = 1, then the set of all 4D points with w^2 + x^2 + y^2 + z^2 = 1 would be the hypersphere.
4D spaces can be realistic too. For example, Newtonian physics takes place in the arena of (x,y,z,t), if you want to think about time as one of the dimensions. This perspective is fraught with nuances and caveats, but it kind of works. Time is actually different than space, even in relativity, but mathematically you can combine it along with the spatial dimensions into spacetime, which is really just a mathematical framework for organizing the relations between events. And in relativity, time is given a special treatment in that its sign in the distance formula is the opposite of the spatial directions (look up Minkowski spacetime). Anyway, there are different kinds of 4D spaces, since you can equip them with various metrics, and in general relativity the metric will vary from place to place depending on the mass-energy landscape.
I hope that helps, let me know if you have any questions :)
@@RichBehiel this is great. thanks for the explanation!
These videos are great! Would you consider doing a video on how you make the animations?
Do you think this vibration is whats at the plank scale? Im trying to understand if this has a purpose in reality or is it just a math construction?
I don’t think this is what’s vibrating at the Planck scale. But, it does have applications in quantum mechanics, qubits in particular.
0:25: 🔍 Fiber bundles are mathematical objects that consist of a base space, fibers, and a total space formed by the fibers.
2:30: 🔗 A non-trivial fiber bundle has twisting or intertwining that makes it topologically interesting.
5:13: 🔍 The animation shows points on a sphere corresponding to specific values of Phi and Theta.
7:34: 🌐 The Hop vibration is a mapping of points on the surface of a two-sphere to circles on a hypersphere in four dimensions.
10:20: 🌌 An introduction to the Hop vibration and its potential applications, followed by a dramatic monologue about the significance of looking up at the stars.
Recap by Tammy AI
I personally prefer a 4D visualization where you take a 3D slice of the space and move it along the shape.
oh amazing! i have just been reading about these in Roger Penrose's book and struggling to visualise them
Finally someone who acknowledges the possibility to intuit 4D instead of hand waving it as unimaginable
Monologue part towards the end of the vd is climax ! Even Sir R Penrose was not able to bring me that far into platonic realm 😊😉👍
What a world indeed.
My god, it's the most advanced mathematical approximation of the donut ever devised.
wich software that you use for the animations ?
Python, plotly module.
@@RichBehiel thank you so much
The four equations for X0, X1, X2, X3 at 6:20 aren't exactly clear by intuition, even if one understands spherical coordinates for S^2. Here they just drop from the sky. It's also not immediately clear why you need half angles. Question: what is it that varies for each of the colorful curves in one image of the animation on the right? I suppose you choose certain theta and phi's to select a point on the two sphere, and then vary alpha from 0 to 4 pi? Nice animation, but to get it, I guess I have to write a little program. ...
Actually , that was fun. Tried this in Octave (Matlab like). Name of the function is from German beer 🙂
function retval = HopfenMalz(npts=250)
[phi,alpha]=meshgrid(linspace(0,2*pi,npts), linspace(0,4*pi,npts));
thetas = linspace(pi/5,pi/2,4);
ncnt=0;
clf;
colorstr = 'rgbmcyk';
% paint shells with constant theta, varying phi and alpha:
for k=1:length(thetas),
theta=thetas(k);
x0 = sin(theta/2).*cos((alpha+phi)/2);
x1 = sin(theta/2).*sin((alpha+phi)/2);
x2 = cos(theta/2).*cos((alpha-phi)/2);
x3 = cos(theta/2).*sin((alpha-phi)/2);
x = x0./(1-x3);
y = x1./(1-x3);
z = x2./(1-x3);
surf(x,y,z,'facecolor',colorstr(mod(k-1,8)+1),'edgecolor','none');
ncnt = ncnt+1;
if (ncnt==1)
hold on;
endif
endfor;
% draw some black fibers on the shells.
alpha = linspace(0,4*pi,npts);
phis = linspace(0,2*pi,40);
zm=0; xm=0; ym=0;
for k=1:length(thetas),
for m=1:length(phis),
phi = phis(m);
theta=thetas(k);
x0 = sin(theta/2).*cos((alpha+phi)/2);
x1 = sin(theta/2).*sin((alpha+phi)/2);
x2 = cos(theta/2).*cos((alpha-phi)/2);
x3 = cos(theta/2).*sin((alpha-phi)/2);
x = x0./(1-x3);
y = x1./(1-x3);
z = x2./(1-x3);
zm = max(max(zm,z)); %
ym = max(max(ym,y));
xm = max(max(xm,x));
line('xdata',x,'ydata',y,'zdata',z,'linewidth',2,'color','k');
endfor;
endfor;
hold off;
retval=1;
zm = ceil(zm)
ym = ceil(ym)
xm = ceil(xm)
axis([-xm, xm, 0, ym, -zm, zm]);
xlabel('x'); ylabel('y'); zlabel('z');
axis('equal');
endfunction
The nontrivial short exact sequence of the Hopf fibration: S^1 -> S^3 -> S^2 embodies the fact that the 3rd homotopy group of the 2-sphere pi_3(S^2) = Z
I was totally on board until the ending monologue.
Platonism is heresy! Hilbert will be avenged!
Hilbert was a great man, but Platonism is eternal! 😉
- Thx. I get it :)
- Now, to delve into the applications...
I'm looking for a book with the theory of fiber bundle. Any recommendation?
I took a tab of acid and saw the torus shape having never seen it before. I have no grand meaning from the experience but it’s super interesting. One thing I noticed is that when observing the torus I could only focus at the shape in one rotational axis at a time.
2:14 > _"base space: disk, fibers: circle, total: torus"_
oh so this is similar to integration of one shape over other. somewhat similar to the "sweep" or "follow path" modifier in 3D CAD modelling but there these things "base space" & "fibers" switch roles. as in:
* the "base curve" or the "path" would be the circle (shape of fibers in this one);
* the "cross section" would be the disk (shape of base space here)
* the "resulting body" on "revolving the shape around the cruve" would again be torus
same as for cylinder/pipe too: u sweep a disk/circle perpendicularly across a straight line.
This video was really helpful since the wikipedia page is so opaque. I still have no idea how to get from the conception you introduced in the video to "a fiber bundle is a space that is locally a product space but globally may have a different topological structure". Based upon your video, it seems that a fiber bundle works more like a field f(x) but instead of assigning numbers, vectors, tensors, etc. to every point in space, it can assign more abstract things like lines (vertical lines in the case of the cylinder). I also hear it is used in general relativity, to assign a vector space to each point on a manifold which makes sense since every point on a curved manifold has its own tangent plane. Would this be a fair assessment?
To your first question, imagine the set product of the real numbers R with the real numbers R. That gives us the product space R2, the 2D Cartesian plane.
Now instead of that, imagine we have a circle and we take its set product with a line segment, so the product space is a kind of band, like a bracelet. Compare that to the Möbius strip in the video. The Möbius strip is locally like a regular band, but globally it has a twist in it that makes it something else.
Sometimes a fiber bundle is just a product space, for example the cylinder and torus examples. But whenever there’s a twist, or a knot, or something like that, then the thing has some structure in it globally that goes beyond the structure of a product space, even though if you zoom in closely enough it still feels like a product space.
Tangent bundles are more complicated. Your intuition is right. But tangent bundles are vector bundles and there’s a whole other set of nuances that come along with that. I might do a video on it someday but it’s hard to respond thoroughly in just a comment.
Is there a transform such that given a 3d projection in time, that the four d shape could be determined?
Sure! Usually this is done by just assigning t as the fourth dimension. For example, a hyper cube would be a cube that pops into existence for some interval of time then disappears. You’d need a scale factor to relate space and time, if thinking about it that way. The speed of light is a good one to use.
what are those dots again, maybe i missed a description. im curious if, when the row of dots at the bottom fades away, is it sortve like - theres another layer rising up/ while in that moment another set is falling down, potentially creating a cascade-like effect?
May be missing on uniqueness aspect of the Hopf (and such) fibrations. Regardless, very nice video, waiting for the follow up with applications. Thank you!
12 minutes later im ready to abandon society and spend the rest of my days at the top of a mountain contemplating the infinite expanse
What a way to go🎉