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.
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.
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.
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.
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!
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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!
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.
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.
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.
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.
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 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
- 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.]
“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.
Wow! I've been trying to get a feel for fiber bundles for ages, but now it's clear. And it's clear why it wasn't clear before! :) I love your piece about mathematical stars! Amazing animations too! Love to know how you made them.😍
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!
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!
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.
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.
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
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.
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.
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
12:02 - how does this structure in N dimensions? What would that look like? How many dimensions can you go? What is the beginning? And where is the end? Who bends these fibers to make our space time? Who decides where and when symmetry is broken and restored? All glory to the Most High.
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.
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 :)
If we take the decomposition of the 3-sphere into two 3-balls, (Or, some particular reasonable realization of that decomposition which has metric/coordinate information, not just topological info) and like, have one of those two 3-balls as how we map (half of) the 3-sphere into 3D space, should the fibration end up looking pretty similar (but with some or all of the fibers being cut in two where the resulting endpoints are on the boundary of the 3-ball) Or, would it give a significantly different impression? I guess because the horizontal circles on the 2-sphere seem to correspond to torii made from the fibers at those base-points, then, well, the same must be true for any projection.. But like, would these torii be cut in two, meeting the surface of the 3-bell along two circles? Or, I suppose this would depend on how you cut the 3-sphere into two 3-balls?
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.
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 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.
I've watched several different videos about the Hopf vibration in the last few weeks, and this is the clearest and best laid out - just brilliant. Liked and subscribed. (what are you using for the smooth animations? Manim?)
Thanks, glad you enjoyed the video! :) For the animations I use Python, mostly matplotlib, but if it’s 3D then I’ll often use plotly which takes longer to render but is better at getting the perspective right in 3D. Sometimes matplotlib will get confused with 3D plots. To make the animations smooth, I always render at 60 fps and at a super high resolution, like twice the resolution it ends up being when the final video is rendered. There are other tricks to making things look smooth, like setting the transparency alpha factor so new objects fade into view, and slowly adjusting the camera position in 3D. In general I’ve learned that it’s best to avoid having anything sit still for too long, otherwise it feels like the video is frozen, so giving the perspective a subtle sine wave adjustment helps to give a dynamic feel to objects that aren’t moving anymore, for example those twisted toroidal fiber bundles after they reached the end of their twist.
@@RichBehiel Thanks for the info! Right now I'm just creating things in Geogebra 3D, animating with sliders, and screen recording. I've dabbled with Matplotlib but still experimenting, and Plotly has been on my to-learn list.
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!
Thank you so much for making the video! The visuals are just… BEAUTIFUL. The monologue is accurate and I love it very much! May I ask for the name of the background music at the monologue btw? It’s brilliant ❤
Hey look, that's what an electron is! I'd suggest starting with "Is the electron a photon with toroidal topology?" by Williamson and van der Mark, then finish with "The Hopf Fibration and Encoding Torus Knots in Light Fields" by John Vincent for a deep dive. Wild stuff.
I’ve read the former, and I’ll check out the latter this evening. Thanks for the recommendation! :) I lean very strongly in the direction that the electron can be brought into the photon field, as some kind of fibration or knot or whatever. Problem is, I haven’t found a model yet that works. They mostly fall apart when it comes to demonstrating stability.
@@RichBehiel the second work deals exactly with the stability of the hopf fibration as a solution of Maxwell's equations, but does not make the connection.
@@RichBehiel so, was it a good read? I'm really interested in these developments because I feel like we've been stuck in the wrong pair of shoes with the standard model, but unfortunately my physics training stops at the undergraduate level.
I enjoyed it, and it’s interesting to think about potential applications with regards to information processing. Only problem is those topologically nontrivial solutions to Maxwell’s equations seem to have a continuous set of allowed energies. But we really want something with a single ground state energy (electron) which can be excited into the unstable muon and tau states. To get there, we would need some kind of topological constraint or something… and this is where these ideas usually fall short of being convincing. The devil is always in the details. The dream would be to show that a topologically nontrivial solution of Maxwell’s equations (or expansion thereof) has a calculable mass, is stable, has two unstable excited modes, and bonus points if the charge can be calculated as a consequence of its geometry. That’s an *extremely* tall order though, and might even be hopelessly misguided. But it’s still the thing that keeps me up at night. An interesting analogue in condensed matter are Abrikosov vortices in superconducting films. They have quantized magnetic flux which can be calculated from first principles based on the geometry of the Ginsburg-Landau order parameter (macroscopic wavefunction), and there’s even an emergent coulomb-like interaction which causes the vortices to repel each other, though this force actually becomes weakly attractive in the sparse vortex limit due to nonlocal effects. But anyway, these vortices are totally well understood, I work with them every day, and they are suspiciously similar to what I imagine an elementary particle ought to look like. Especially in light of the various analogies between the Higgs field and superconductivity. But superfluid/superconductor models of the vacuum have a long history, and not a whole lot of interesting progress.
as a crocheter who is merely a math enthusiast im very amused to learn about fiber bundles math fibers,,, they look like really cursed balls of yarn XD PS: what the heck did you use to make the animated graphics? theyre so cool
@@nyuh I noticed you made a visualization with blender for AoC. You are probably aware that you can script practically everything in blender using python, but I wanted to mention it in case you aren't. So you could basically use blender to "plot"/render/animate very cool mathematical shapes.
This question I ask out of pure ignorance: What would happen if have a function z=f(x,y) such that we have a domain (call it \D). I am guessing that you can define a fiber bundle such that if you have a disc of radius \epsilon this fiber will extend from f = 0 to f(x, y). What if you have a singularity in this domain? I have no idea what that would be called, how would you handle this topology? Are there similar extensions to higher dimensions? I really don't know about this, so any reference would be appreciated.
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.
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’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.
that ending monologue was really beautiful. great video
Thanks! :)
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 🙏🏾
I feel like a wizard every time I watch your videos, and understanding what you're talking about.
the "monologoue" at the end gave me goosebumps
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! :)
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
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.
What a beautiful closing articulation of platonic ideals!
Fiber bundles (and the generalization) are everywhere in Algebraic Geometry. Is vastly appreciated.
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.
The monologue in the end was very pretty
yerp.
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.
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.
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 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!
The magnetic lines in the tokamak fusion reactor seems to be a kind of fiber bundle at 2:46.
This is so cool. Can't wait for the follow ul videos on this topic
Questa si è la nostra ricerca ,la Fiandra delle Fiandre , Finalmente! Grazie mille per la presentazione. Un bacione!
You're animation and exposition are flawless.
Thanks Eduardo! :)
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.
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.
I love how you're like genuinely super enthralled by how cool math is
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.
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.
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 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
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 :)
- 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.]
Wow! This is absolutely gorgeous and so inspiring!!! Eagerly wait for next videos!
“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.
Breath takingly beautiful
Thanks, glad you enjoyed it! :)
Wow! I've been trying to get a feel for fiber bundles for ages, but now it's clear. And it's clear why it wasn't clear before! :) I love your piece about mathematical stars! Amazing animations too! Love to know how you made them.😍
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! :)
I love the ending monologue. thank you.
Thanks for watching! :)
It always nice to know others think the same things...take care, my friend...
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! :)
absolutely love this channel so happy I found it
these animations are fantastic, can't wait to learn about this
Thanks, glad you enjoyed the video! :)
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 :)
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! :)
Whatever you smoked before making this video I want some of that, too!
Awesome work!
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.
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
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! :)
Hey man what a cool final monologue!
Thanks! :)
Great monologue well! thank you mathematical stars!
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.
Top notch videos and animations! I wish I could have picked your brain while preparing my thesis defense.
Thanks! :) What was your thesis on?
YESSS!! MAN!! Fiber bundles are so frickinnn cool!!! finallly someone appreciates it!!!
the last section accurately describes my relationship with maths.
Monologue was fire
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! :)
Beautiful Hypnotic of Mathematics! :) 😍
This us the first I visualized a rotating tesseract as a rigid rotating object. It's amazing.
there needs to be more math content like this, talking about a high level topic but in a more relaxed, colloquial way
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.
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
You're projecting, a lot...
And philosophizing.
Nice video.
The animations at 2:52 are just crazy😍
12:02 - how does this structure in N dimensions? What would that look like? How many dimensions can you go? What is the beginning? And where is the end? Who bends these fibers to make our space time? Who decides where and when symmetry is broken and restored? All glory to the Most High.
Gorgeous! Thank you
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! :)
this is phenomenal and underrated
Thanks! :)
"Disc is the area of the circle"
"Circle is just the edge of the circle "
-Richard behiel 2023
Wonderful introduction
Thanks! :)
Thank you for the video.
Thanks for watching! :)
I think this concept is used to describe some topological defects in spintronics
I like you, you make the world a better place.
Thanks for the kind comment! :)
These videos are great! Would you consider doing a video on how you make the animations?
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!
This is so great! Looking forward to more videos
Thanks! :)
If we take the decomposition of the 3-sphere into two 3-balls,
(Or, some particular reasonable realization of that decomposition which has metric/coordinate information, not just topological info)
and like, have one of those two 3-balls as how we map (half of) the 3-sphere into 3D space,
should the fibration end up looking pretty similar (but with some or all of the fibers being cut in two where the resulting endpoints are on the boundary of the 3-ball)
Or, would it give a significantly different impression?
I guess because the horizontal circles on the 2-sphere seem to correspond to torii made from the fibers at those base-points, then, well, the same must be true for any projection..
But like, would these torii be cut in two, meeting the surface of the 3-bell along two circles?
Or, I suppose this would depend on how you cut the 3-sphere into two 3-balls?
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.
This was ethereal!
Well done, superb video! What software did you use to make the animation?
Thanks! :) Python, plotly module.
@@RichBehiel
Any chance that you could share some of the code?
I almost understood some of that.Thank you for a cool video.
really good narration of this topic, looking forward to what other things you have to say about mathematics in the future
You and fiber bundles have something in common, both of you are underappreciated.
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 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.
Spectacular video
wich software that you use for the animations ?
Python, plotly module.
@@RichBehiel thank you so much
I've watched several different videos about the Hopf vibration in the last few weeks, and this is the clearest and best laid out - just brilliant. Liked and subscribed. (what are you using for the smooth animations? Manim?)
Thanks, glad you enjoyed the video! :)
For the animations I use Python, mostly matplotlib, but if it’s 3D then I’ll often use plotly which takes longer to render but is better at getting the perspective right in 3D. Sometimes matplotlib will get confused with 3D plots. To make the animations smooth, I always render at 60 fps and at a super high resolution, like twice the resolution it ends up being when the final video is rendered. There are other tricks to making things look smooth, like setting the transparency alpha factor so new objects fade into view, and slowly adjusting the camera position in 3D. In general I’ve learned that it’s best to avoid having anything sit still for too long, otherwise it feels like the video is frozen, so giving the perspective a subtle sine wave adjustment helps to give a dynamic feel to objects that aren’t moving anymore, for example those twisted toroidal fiber bundles after they reached the end of their twist.
@@RichBehiel Thanks for the info! Right now I'm just creating things in Geogebra 3D, animating with sliders, and screen recording. I've dabbled with Matplotlib but still experimenting, and Plotly has been on my to-learn list.
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!
Great video! Love me some AG!
Thanks, glad you enjoyed it! :)
This was beautiful and brings joy
Thank you so much for making the video! The visuals are just… BEAUTIFUL. The monologue is accurate and I love it very much! May I ask for the name of the background music at the monologue btw? It’s brilliant ❤
How do you make these amazing, accurate visualizations? Would love to know!
I use Python, for 2D animations the matplotlib module, and plotly for 3D.
Hey look, that's what an electron is!
I'd suggest starting with "Is the electron a photon with toroidal topology?" by Williamson and van der Mark, then finish with "The Hopf Fibration and Encoding Torus Knots in Light Fields" by John Vincent for a deep dive. Wild stuff.
I’ve read the former, and I’ll check out the latter this evening. Thanks for the recommendation! :)
I lean very strongly in the direction that the electron can be brought into the photon field, as some kind of fibration or knot or whatever. Problem is, I haven’t found a model yet that works. They mostly fall apart when it comes to demonstrating stability.
@@RichBehiel the second work deals exactly with the stability of the hopf fibration as a solution of Maxwell's equations, but does not make the connection.
Oh man that’s awesome, I’m looking forward to reading it!
@@RichBehiel so, was it a good read? I'm really interested in these developments because I feel like we've been stuck in the wrong pair of shoes with the standard model, but unfortunately my physics training stops at the undergraduate level.
I enjoyed it, and it’s interesting to think about potential applications with regards to information processing. Only problem is those topologically nontrivial solutions to Maxwell’s equations seem to have a continuous set of allowed energies. But we really want something with a single ground state energy (electron) which can be excited into the unstable muon and tau states. To get there, we would need some kind of topological constraint or something… and this is where these ideas usually fall short of being convincing. The devil is always in the details.
The dream would be to show that a topologically nontrivial solution of Maxwell’s equations (or expansion thereof) has a calculable mass, is stable, has two unstable excited modes, and bonus points if the charge can be calculated as a consequence of its geometry. That’s an *extremely* tall order though, and might even be hopelessly misguided. But it’s still the thing that keeps me up at night.
An interesting analogue in condensed matter are Abrikosov vortices in superconducting films. They have quantized magnetic flux which can be calculated from first principles based on the geometry of the Ginsburg-Landau order parameter (macroscopic wavefunction), and there’s even an emergent coulomb-like interaction which causes the vortices to repel each other, though this force actually becomes weakly attractive in the sparse vortex limit due to nonlocal effects. But anyway, these vortices are totally well understood, I work with them every day, and they are suspiciously similar to what I imagine an elementary particle ought to look like. Especially in light of the various analogies between the Higgs field and superconductivity. But superfluid/superconductor models of the vacuum have a long history, and not a whole lot of interesting progress.
as a crocheter who is merely a math enthusiast
im very amused to learn about fiber bundles
math fibers,,,
they look like really cursed balls of yarn XD
PS: what the heck did you use to make the animated graphics? theyre so cool
Cursed balls of yarn 😂 So true.
I used Python, with plotly.
@@RichBehiel ah python i see. thanks for answering !
@@nyuh I noticed you made a visualization with blender for AoC. You are probably aware that you can script practically everything in blender using python, but I wanted to mention it in case you aren't. So you could basically use blender to "plot"/render/animate very cool mathematical shapes.
@@michaeldamolsen ooohh yeah youre right
i didnt realize
maybe ill try it out sometime
Hi, I was wondering what software did you use to crate the animations? They look really good!
Thanks! :) Python, plotly module.
This question I ask out of pure ignorance:
What would happen if have a function z=f(x,y) such that we have a domain (call it \D). I am guessing that you can define a fiber bundle such that if you have a disc of radius \epsilon this fiber will extend from f = 0 to f(x, y). What if you have a singularity in this domain? I have no idea what that would be called, how would you handle this topology? Are there similar extensions to higher dimensions?
I really don't know about this, so any reference would be appreciated.
amazing... thanks man for the introduction... my new THINK ig
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.
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🎉
Fanttastic explanation and vizualization - now I "understood" zthe Fibration - and awesom monolgue! Thanks You!
Thanks Willy, glad you enjoyed the video! :)
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