A Quick Intro to Fiber Bundles (Hopf Fibration)

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Fiber bundles are useful and interesting mathematical structures, with applications in quantum mechanics and other areas of math and physics, but sadly they are tragically underappreciated. In this video, I'd like to spread awareness about the beauty of fiber bundles.
Read more about fiber bundles: en.wikipedia.org/wiki/Fiber_b...
Read more about the Hopf fibration: en.wikipedia.org/wiki/Hopf_fi...
A nice, accessible guide to the Hopf fibration in some more depth: nilesjohnson.net/hopf-article...
#math #physics #quantum #topology #philosophy #trippy #science #geometry #Hopf

Опубликовано:

17 май 2024

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Комментарии : 317

@logankennedy7082 Год назад

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!

@RichBehiel Год назад

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?

@logankennedy7082 Год назад

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.

@daltanionwaves Год назад

@@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.

@RichBehiel Год назад

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.

@kodfkdleepd2876 Год назад

A lot? Define a lot.

@WindmillEntertainmentGames Год назад

that ending monologue was really beautiful. great video

@RichBehiel Год назад

Thanks! :)

@Coolguydudeness1234 11 месяцев назад

I thought the segway at the end was about to be a sponsor but it turned out to just be some beautiful philosophy

@RichBehiel 11 месяцев назад

Money is temporary, math is eternal :)

@zray2937 Год назад

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 11 месяцев назад

what physics do you do / where do you do it out of curiosity i also do physics

@Classical741 11 месяцев назад

@@bobjones5869 I can understand how you might think that, but there is nothing preventing anyone from following their own curiosity and interests.

@bobjones5869 11 месяцев назад

@@Classical741 i was just wondering where you study physics that’s all

@schmud68 11 месяцев назад

@@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.

@twelfthhausjones6753 9 месяцев назад

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.

@OpenSourceAnarchist 7 дней назад

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

@iamnotalive9920 4 месяца назад

the "monologoue" at the end gave me goosebumps

@ELECTR0HERMIT 5 месяцев назад

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.

@Smart_Economy_Architecture-Lab 8 месяцев назад

Wonderful introduction

@RichBehiel 8 месяцев назад

Thanks! :)

@Anandharajan.a 11 месяцев назад

RU-vid is going to change the way of learning forever

@DanielJacksonsNotFromSG1 11 месяцев назад

What a beautiful closing articulation of platonic ideals!

@a.l.4162 Год назад

10:43 Your end monologue is fantastic. It reminds me of Heidegger's approach to philosophy. Thank you.

@RichBehiel Год назад

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.

@GilesMcRiker Год назад

Heidegger is my favourite Nazi philosopher!

@Achrononmaster Год назад

@@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.

@RichBehiel Год назад

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.

@bongfellatiophdgand2316 11 месяцев назад

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.

@matthewocadiz7333 8 месяцев назад

Monologue was fire

@jeanklein2198 11 месяцев назад

The monologue in the end was very pretty

@WingsOfADream1 Месяц назад

yerp.

@konstantinrebrov675 8 месяцев назад

I feel like a wizard every time I watch your videos, and understanding what you're talking about.

@notnilc2107 11 месяцев назад

the last section accurately describes my relationship with maths.

@devfromthefuture506 9 месяцев назад

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.

@uniqueglow9541 11 месяцев назад

You're projecting, a lot... And philosophizing. Nice video.

@steveboel12 5 месяцев назад

This is so cool. Can't wait for the follow ul videos on this topic

@AA-gl1dr Год назад

I love this and I am so thankful that you exist and decided to teach the world.

@RichBehiel Год назад

Wow, that’s a very kind comment - thank you! That made my day :)

@AdrianBoyko Год назад

“Rich Behiel is a way for The Universe to teach itself” - Carl Sagan

@InZiDes Год назад

Fiber bundles (and the generalization) are everywhere in Algebraic Geometry. Is vastly appreciated.

@eduardoeller183 Год назад

You're animation and exposition are flawless.

@RichBehiel Год назад

Thanks Eduardo! :)

@ozzymandius666 Год назад

@@RichBehiel The invisible structures that permeate the universe, that get called into time (and sometimes space) when illuminated by consciousness. A somewhat more modern analog of Cartesian Dualism. Math is discovered, not invented, if you will. Anyway, you have earned an instant channel subscription. I truly enjoy videos that require at least a modicum of mathematical sophistication.

@mayabenowitz6729 11 месяцев назад

Most underappreciated piece of mathematics and arguably one of the *most* important pieces of mathematical physics!

@liamfeatherly458 Год назад

Hey man what a cool final monologue!

@RichBehiel Год назад

Thanks! :)

@leonstenutz6003 Год назад

❤ "... 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 ... 😊

@rxphi5382 8 месяцев назад

The animations at 2:52 are just crazy😍

@johnbrooks6243 Год назад

"Disc is the area of the circle" "Circle is just the edge of the circle " -Richard behiel 2023

@juansoto3469 11 месяцев назад

Gorgeous! Thank you

@V.squared 5 месяцев назад

I love the ending monologue. thank you.

@RichBehiel 5 месяцев назад

Thanks for watching! :)

@V.squared 5 месяцев назад

It always nice to know others think the same things...take care, my friend...

@Achrononmaster Год назад

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.

@alifeleparanj3688 7 месяцев назад

I love the monologue part! I printed the script for that section and it is now on the wall of my office :)

@RichBehiel 7 месяцев назад

Wow, that’s awesome! I’m glad you enjoyed it :)

@TheWyrdSmythe Год назад

“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.

@RichBehiel Год назад

The intelligibility of Nature is the greatest darn mystery there has ever been.

@Pygmygerbil88 Год назад

Breath takingly beautiful

@RichBehiel Год назад

Thanks, glad you enjoyed it! :)

@arnavshukla2408 2 месяца назад

That monologue, just wow. I'm out of words to describe how elegantly this video was produced. Thanks Richard

@RichBehiel 2 месяца назад

Thanks for the kind comment, and I’m glad you enjoyed the video! :)

@swamihuman9395 Месяц назад

- 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.]

@suibrad Год назад

this is phenomenal and underrated

@RichBehiel Год назад

Thanks! :)

@yaotianji2681 Год назад

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.

@mark11tz 11 месяцев назад

The magnetic lines in the tokamak fusion reactor seems to be a kind of fiber bundle at 2:46.

@anthonybrakus5280 3 месяца назад

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 🙏🏾

@user-vt6em5fl9v 3 месяца назад

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.

@RichBehiel 3 месяца назад

Thanks for the kind comment! :) Reality sure is beautiful!

@BleachWizz 11 месяцев назад

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.

@angelamusiemangela 5 месяцев назад

Questa si è la nostra ricerca ,la Fiandra delle Fiandre , Finalmente! Grazie mille per la presentazione. Un bacione!

@MrSilverSerf Год назад

Wow! This is absolutely gorgeous and so inspiring!!! Eagerly wait for next videos!

@pectenmaximus231 11 месяцев назад

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.

@SPVLaboratories Год назад

really good narration of this topic, looking forward to what other things you have to say about mathematics in the future

@JBMJaworski Год назад

Beautiful Hypnotic of Mathematics! :) 😍

@nahometesfay1112 11 месяцев назад

This us the first I visualized a rotating tesseract as a rigid rotating object. It's amazing.

@darrellhaufler2750 8 месяцев назад

Top notch videos and animations! I wish I could have picked your brain while preparing my thesis defense.

@RichBehiel 8 месяцев назад

Thanks! :) What was your thesis on?

@VHenrik007 Год назад

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!

@RichBehiel Год назад

Thanks for the kind comment, glad you enjoyed the video! :)

@ColeCoug Год назад

absolutely love this channel so happy I found it

@leewilson3839 11 месяцев назад

It's hard to simplify and visualise this math but it's very cool

@btg837 11 месяцев назад

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!

@RichBehiel 11 месяцев назад

Thanks for the very kind comment! :)

@mastershooter64 Год назад

YESSS!! MAN!! Fiber bundles are so frickinnn cool!!! finallly someone appreciates it!!!

@EduNauram Год назад

Spectacular video

@jeyko666 Год назад

This is so great! Looking forward to more videos

@RichBehiel Год назад

Thanks! :)

@alis.2 Месяц назад

You and fiber bundles have something in common, both of you are underappreciated.

@willywalter6366 Год назад

Fanttastic explanation and vizualization - now I "understood" zthe Fibration - and awesom monolgue! Thanks You!

@RichBehiel Год назад

Thanks Willy, glad you enjoyed the video! :)

@sudoboat 7 месяцев назад

Whatever you smoked before making this video I want some of that, too! Awesome work!

@2pizen 8 месяцев назад

Great monologue well! thank you mathematical stars!

@jamesbentonticer4706 10 месяцев назад

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.

@RichBehiel 10 месяцев назад

Thanks, I’m glad you enjoyed the videos! :)

@kono152 Год назад

these animations are fantastic, can't wait to learn about this

@RichBehiel Год назад

Thanks, glad you enjoyed the video! :)

@KALE18 Год назад

I think this concept is used to describe some topological defects in spintronics

@Quazgaa Год назад

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

@brendawilliams8062 Год назад

What a way to go🎉

@brockgrace7470 Год назад

I almost understood some of that.Thank you for a cool video.

@michaelcharlesthearchangel Год назад

Fibrations are so cool & hot at the same time when creating a Quantum W OR M hole!

@electroflame6188 11 месяцев назад

I personally prefer a 4D visualization where you take a 3D slice of the space and move it along the shape.

@hysenndregjoni853 Год назад

This was ethereal!

@nyuh Год назад

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

@RichBehiel Год назад

Cursed balls of yarn 😂 So true. I used Python, with plotly.

@nyuh Год назад

@@RichBehiel ah python i see. thanks for answering !

@michaeldamolsen Год назад

@@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.

@nyuh Год назад

@@michaeldamolsen ooohh yeah youre right i didnt realize maybe ill try it out sometime

@rao_v Год назад

These videos are great! Would you consider doing a video on how you make the animations?

@alexanderlovellette318 Год назад

This was a really great video with awesome visuals that really me want to go study topology holy cow

@RichBehiel Год назад

Thanks, I’m glad you enjoyed the video! :)

@Nernkal Год назад

I was totally on board until the ending monologue. Platonism is heresy! Hilbert will be avenged!

@RichBehiel Год назад

Hilbert was a great man, but Platonism is eternal! 😉

@baldrbraa 9 месяцев назад

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.

@usernameisamyth 11 месяцев назад

amazing... thanks man for the introduction... my new THINK ig

@mrtertg2603 11 месяцев назад

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 😊😉👍

@DerrickJLive Год назад

Great video, thanks

@eonasjohn 10 месяцев назад

Thank you for the video.

@RichBehiel 10 месяцев назад

Thanks for watching! :)

@tedsheridan8725 Год назад

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?)

@RichBehiel Год назад

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.

@tedsheridan8725 Год назад

@@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.

@whatelseison8970 Год назад

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

@topquark22 8 месяцев назад

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

@swamihuman9395 Месяц назад

- Thx. I get it :) - Now, to delve into the applications...

@paulward2333 Год назад

Thank you.

@NonTwinBrothers Год назад

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.

@Swordfish393 11 месяцев назад

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.

@davidwilkie9551 11 месяцев назад

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.

@RichBehiel 11 месяцев назад

Thanks! :)

@plancksepoch3040 Год назад

What a world indeed.

@ntesla66 5 месяцев назад

I like you, you make the world a better place.

@RichBehiel 5 месяцев назад

Thanks for the kind comment! :)

@JessyHeizenberg Год назад

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

@RichBehiel Год назад

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 :)

@JessyHeizenberg Год назад

@@RichBehiel this is great. thanks for the explanation!

@evilotis01 Год назад

oh amazing! i have just been reading about these in Roger Penrose's book and struggling to visualise them

@ogleabout230 Год назад

Great video! Love me some AG!

@RichBehiel Год назад

Thanks, glad you enjoyed it! :)

@ShipOfFreaks 10 месяцев назад

lmao the ending monologue was unexpected but i get it 💯

@prod.winterxphool6227 Год назад

Wow what a fantastic video!

@RichBehiel Год назад

Thanks! :)

@Bea_remembrance Год назад

Great video

@palfers1 9 дней назад

This is strongly reminiscent of a twistor! (as drawn by Sir Roger)

@KipIngram Месяц назад

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.

@RichBehiel Месяц назад

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.

@pra. Год назад

amazing animations!

@RichBehiel Год назад

Thanks! :)

@pra. Год назад

@@RichBehiel by the way, what programs did you use to visualize this?

@RichBehiel Год назад

Python, for 3D animations I use plotly, 2D I use matplotlib.

@reinerwilhelms-tricarico344 Месяц назад

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

@bobasawrus Год назад

Thankyou.

@onlinXman Год назад

this needs more views

@RichBehiel Год назад

I agree! 😅

@drdca8263 11 месяцев назад

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?

@yash1152 2 месяца назад

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.

@aanchaallllllll 8 месяцев назад

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