Thanks to those who caught my speako describing Kolmogorov as a "19th-century" mathematician. Of course, I meant the 1900's. His work is quite recent and remains relevant to a number of active research efforts in a surprising breadth of fields. It's crazy to me that his name is as relevant to fluid dynamicists as to those studying machine learning.

@augustuscaeser1358

5 жыл бұрын

Sir can I ask you to do on something like how .... Long list sir

@maclean119

5 жыл бұрын

How can we apply Willmore flow and morse theoretic application to Navier-Stokes Equations and is it even possible ?

@maclean119

5 жыл бұрын

@Seb Not look into Euler-Lagrange equation great place to start.

@rickflare9077

5 жыл бұрын

Looks Just like a magnet under a Ferocell....#KenWheeler

@artlenski8115

5 жыл бұрын

Kolmogorov is the father of modern probability theory ie axiomatic probability theory, he also contributed to the field of functions approximation ie function analysis, these two fields are basically the foundation of machine learning.

I got to see a lot of this footage in person, and it somehow looks better with Grant's tranquil voiceover. A+

@schregen

5 жыл бұрын

Hey it's the girl from the video! Pretty cool 🍄

@3blue1brown

5 жыл бұрын

Thanks for making this happen! So much fun :)

@JohnSmith-td7hd

5 жыл бұрын

At some point, could you make really hot smoke rings? I want to know if the rings shrink as they lose heat. Also, please film this in infrared just to make sure you're not missing anything. Also it would look cool :)

@hasiumcreeper5384

5 жыл бұрын

Physics Girl, you truly are the Queen of Collabs.

@joshmyer9

5 жыл бұрын

OMG, 11:27 of video, and, unless I missed it, he didn't even find a second to apologize for your vortex cannon. SMH, Grant, SMH. (Such a great collab, I really enjoyed both videos!)

Really nice summary. I'm a beneficiary of all that theory in my career as a civil engineer working on dam spillways and hydraulic structures. For complex projects, we still build scale physical models because even the CFD software doesn't do a great job predicting turbulent flow conditions.

@domenicobianchi8

5 жыл бұрын

I studied turbulence. The only cfd that makes sense is DNS which cannot be used now and probably ever with this kind of computers. So, dont worry, everybody need to stick to real physical models

@bene20080

5 жыл бұрын

@@domenicobianchi8 Certainly not. Lots of CFD have their use cases. There are lots of instances where you don't need the details of a DNS simulation. Also there are other, where you can't do a DNS, because you do not have the computation power, but can also not do any physical model, because you can not get the right equipment / it is too expensive, or not even possible to put any sensor at the phenomenon. So, give up, or what?

@jameswright4732

5 жыл бұрын

​@@domenicobianchi8 I mean, to a certain extent yes, DNS is the only "correct" way to perform CFD calculations. However, LES is certainly a valid option in a lot of circumstances and hybrid RANS-LES models are quickly bridging the equally large valley between RANS and LES turbulence models. Fluid's needs physical experiments like any other form of computational modeling. But I will 100% agree that where an FEA model is often perfectly fine without any experimental validation, CFD is just a number until it gets correlated with an actual physical result.

@zes7215

5 жыл бұрын

wrr

@ASJC27

5 жыл бұрын

LES and DES if done properly are super accurate in any case I've come across (and can be more accurate than experiments in difficult cases). But these are relatively new techniques and super computer clusters powerful enough are only now starting to be common enough to use it. Even then, these simulations take forever. The typical RANS simulations used today are inherently "wrong" and require a lot of validation to make right. But for a lot of real world Reynolds ranges and flow types they are good enough and require much less computational resources.

Another thing about green lasers: Do not buy cheap green lasers!! They do not radiate pure green light - in the worst case the biggest part of their radiation is in the infrared spectrum. You will not even know how strong this light is, and it can blind you in no time. More expensive lasers filter this infrared part out, or use more efficient ways to create green light - ways that don't produce the "byproduct" infrared light. Just to make sure this was said and everyone who wants to try, knows.

@jasondoe2596

5 жыл бұрын

Yeap, Brainiac75's channel has a great video on the topic. Seriously, the (very nice) experiment shown here is quite dangerous, even with a high-quality, properly filtered laser. Only attempt with safety glasses, without other people in the room (or near any windows), and capture it on video.

@jasondoe2596

5 жыл бұрын

PS. I mean safety glasses *matched* to the wavelength of the laser, *and* the wavelengths of possible IR radiation for this particular laser type. If done right, you'll probably only be able to see the results on video.

@Dudleymiddleton

5 жыл бұрын

I have a laser pen(2 of them ,actually) that emits blacklight UV, it makes bleached cartridge paper and anything "Hi-viz" glow very brightly. Would that work better than green ?

@ZweiZombies

5 жыл бұрын

@@Dudleymiddleton I personally wouldn't know, so maybe you can Google away for the safety specifications :)

@thefunkybside

5 жыл бұрын

The IR emissions are not a byproduct being produced, it's the raw output of the diode before it's doubled by a crystal to the good old 532nm lime green we all know. Good lasers aren't avoiding production of byproduct, they're simply doing a better job converting the raw output to the desired wavelength + using filters to prevent any leakage of of the IR.

Thanks for the shoutout. I did a double take at the opening of Diana's video wondering if I was looking at my own footage.

"high swirly-swirly factor"

@T3sl4

5 жыл бұрын

"Speedything goes in, speedything goes out"

@ASJC27

5 жыл бұрын

I've never heard it referred to as such. I know it as the "swirl factor", but I do like swirly-swirly better.

@mduckernz

5 жыл бұрын

Which is, of course, distinct from the swirly-whirly factor

@matthewellisor5835

5 жыл бұрын

Cried from laughing at this.

@tornes

5 жыл бұрын

@Todd Starbuck that sounds familiar to me hahaha

I wrote my doctoral dissertation on optical probes into fluid turbulence. Using proper beam paths and correlation functions, it is possible to directly measure the Kolmogorov scale in real time. I'm really happy to see this video because there is so much fascinating and important physics and mathematics to be found in fluid turbulence.

@joonasmakinen4807

5 жыл бұрын

Joseph D. Harris, thank you for sharing! I found your PhD thesis: drum.lib.umd.edu/bitstream/handle/1903/11461/Harris_umd_0117E_11928.pdf?sequence=1

@josephd.harris6954

3 жыл бұрын

@@joonasmakinen4807 Thank you for looking me up!

@FrequencyDomainLife

2 жыл бұрын

Small world!!! I'm working with your same adviser 10 years later doing something similar!

@josephd.harris6954

2 жыл бұрын

@@FrequencyDomainLife Say hi to Chris for me.

@edgarfeller

2 жыл бұрын

So electrical engineer can do research on turbulence or i'm wrong ?

Hi, I am 16 years old and am trying to pursue a degree in Physics... I never really comment on videos or anything but the depth and intensity of the subjects you talk and explain in an animated manner gives a whole new level of perspective to the world of Mathematics and Science... I can't express my happiness when i discovered your channel almost a year back and have completed most of your videos.... i just wanted to say that please don't stop making such informative videos... you r really helping me to be open to the world and be more diverse with my studies and i truly really feel grateful... thanks a lot

@citizenmachine

Жыл бұрын

Hello. I don't know who you are and where you are from, but your words "...to be open to the world..." moved me deeply as I myself have had literally the same thought in the same sense running through my head while trying to study. By the way, neither do I ever comment on any videos or respond to others. So how is your study going?

@keziahshaji5916

Жыл бұрын

@@citizenmachine It's been 2 years, like meeting up with yourself from the past? Well, studies are at the point where I had completed highschool, and am going to major in Theoretical Physics, if you want to know. Again, I'm pleasantly surprised by the unexpected comment.

@HxTurtle

Жыл бұрын

that section here is kinda outta world. how could someone even dig out a two year old comment that (sadly) produced zero responses. it must have gotten buried very deep, meanwhile, imma thinking/assuming. commenting can be quite interesting as some reactions might even include "world opening" moments as well. it's surprisingly easy to swift through lower quality ones, which happens but thankfully not that often as it is sometimes made look like.

I just gotta say, im a phd student studying computational fluid dynamics and turbulence and I showed this to several of my lab mates and we all absolutely loved it!! ty for making this somehow both accessible to a broad audience and still true to the complexity of the subject matter, its not an easy feat and this video does it very well!!!

@alanmakoso1115

5 жыл бұрын

Same. I did something relating to boundary layer separation for the Science and Engineering Fair of Houston and analyzing turbulence with CFD is one of the most important parts. This video truly helped me with the CFD parts of it.

@emiliofermi9994

Жыл бұрын

Are you majoring mechanical engineering?

"When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first." -Werner Heisenberg (maybe)

@kevinburns8473

3 жыл бұрын

I recall reading this in James Gleick's book Chaos, had quite the chuckle when I hit that.

@Jackisaboss1208

3 жыл бұрын

@@kevinburns8473 Good book! I’m reading it now

Please make Navier Stockes in depth

@HoshinoMirai

5 жыл бұрын

Agreed!

@scose

5 жыл бұрын

Yes

@18WiddaBullit

5 жыл бұрын

NO! We're waiting for measure theory and lebesgue integral since forever. First things first.

@scose

5 жыл бұрын

@@18WiddaBullit fair enough, I like measure theory

@professorhasinabanu2199

5 жыл бұрын

Totally agree.

Hey! This technique is used in actual fluid dynamic experiments (or something very close to it anyways). It's called Particle Image Velocimetry (PIV). Basically, you take a bunch of really expensive dust particles (expensive because they're specially designed to reflect an even amount of light regardless of orientation and also because they need to be very small/light), and disperse them in a flow field. Then you take a planar laser sheet (much like what you have) and have a camera take two pictures of the sheet within a very small time frame. By comparing the two pictures (via software generally), you have an instantaneous look at the exact velocity field of the fluid! It's a super nice way of getting CFD-level fidelity and results from an actual experiment (though it is quite difficult to setup in certain circumstances).

8:34: "I know, yikes!" Inner Grant: "I know, cool, right!"

This is by far my favourite KZread chanel. I really appreciate how you make videos talking about mathematics on such a high level, with so much enjoy and really well done animations. Math is something that I love for basicly my whole life, but I never loved as much as I do now thanks to your chanell. Thank you for making every single video here on youtube. :) Ps: I'm from Bazil, and english is not my first languege, so sorry about any grammar mistake or stuff.

SO happy that you made a fluid dynamics video!!! I have been patiently waiting for someone to come around and do a video on it.

Physics Girl sent me here and now I'm guided back there. Alone the idea to create a planar laser "blade" is so fascinating I could loop through these two videos for some more iterations.

Absolutely love your work! My PhD supervisor Arkady Tsinober was one of Kolmogorov's students and a unique person who not only understands a math, but also can make experiments. I built a multy-hotwire and Coldwater probe and measured all 9 velocity derivatives for checking Taylor hypothesis of frozen turbulence.

my Probability Theory professor was Kolmogorov's PhD student, pretty sure Kolmogorov was not 19th century :)

@danielsteel5251

5 жыл бұрын

Who was your prof? Gelfand? Arnold? Martin-Löf? Uspensky? Dynkin? ... (Kolmogorov apparently had no time for advising non-genius students.)

@zairaner1489

5 жыл бұрын

Well you are right, he was 20st century. Showing once more how late stochastics was made senseful

@Stayhigh71

5 жыл бұрын

My Econometrics professor was Kolmogorov's student in MSU, he is always proud to mention he was taught by such a great mathematician.

@smaugly

5 жыл бұрын

@@Stayhigh71 I'm just responding to this because I am an MSU student as well, and I think this is cool. I believe this is my first comment on KZread, you should feel special ;)

I love this video so much! I've spent the last 5 years studying and applying Kolmogorov turbulence to the propagation of laser light through the atmosphere. It's great to see the visualizations of turbulent flow!

Thank you so much for your amazing videos!!!! I use them for wonderful visualizations to assist in teaching my children some of my deepest passions. Not only are your visualizations beautiful, they do indeed appear to assist in that 'deeper understanding' you express. I recommend your channel to many other parents and want you to know how much I appreciate the hard work you clearly put into making these videos.

Love the visualizations done by Physics Girl; in the fluids world this is almost identical to an experimental technique called stereo particle image velocimetry (PIV). With PIV more distinguishable particles are suspended in the flow and imaging used to find a vector field at each time step recorded.

You’re the only reason I didn’t give up maths after those boring hours of doing elementary algebra over and over again. And now, you’re making me fall in love with physics.

omg i'm an atmospheric science grad student and i'm literally studying this stuff right now started to follow 3b1b when it only had like 20k subs and i did not see this ever coming lol. thanks so much for making this (series) possible. can't wait to see your future works about the energy cascade!

I'd love to have an explanation on the intuition behind this 5/3 mystical formula. Maybe a future video?

I am a CFD Engineer and would love to watch videos on this topic. This topic can also help us learn many mathematical concepts in a fun manner. Please make more videos. Thanking you for the effort :)

Fluid dynamics ftw! You absolutely need to make a video on the Lorenz system and strange attractors!!

@xgalarion8659

5 жыл бұрын

Maybe you can give him some piano background music...

@connorcriss

5 жыл бұрын

Yeeeesssss

@NoNTr1v1aL

5 жыл бұрын

Everybody like this so he sees it!

I’d love a video about differential forms, orientation, and the generalized stoke’s theorem! Thanks for the totally epic video Grant!

This is a great demonstration of certain types of flow visualization in engineering and fluid dynamics research, such as particle image velocimetry. The same technique is used, and images are taken and the particles are tracked to calculate the flow field. Another great and simple technique to visualize flow is schlerian imagery

I'm taking Fluid Mechanics right now, and this video really helped with my understanding of what constitutes/creates turbulent flow!

Congratulations for another incredible video. Your work is helping me to teach my students, I shiw them some of your art because it's what they are. That's why I pay to help to be able to continue. Your work is really top, is a gift to humanity.

You are a living "Richard Feynmann" for us.... The way u simplify ur explanations is amazing

You just keep amazing me! Finally the concept of energy cascading from larger to smaller spectrums clicked in! Your teaching abilities are simply out of this world.

I ADORE YOU FOR REAL! You, physics girl, Dan Walsh and the whole educational community of creators, are shifting the trend towards loving science and making it Cool! You All inspire me and I hope someday to follow your lead, and communicate my passions to the whole world.

I've just heard a lecture on turbulence modelling today at uni. What a coincident you upload a great video about this interesting topic!

Wow! Finally got the explanation for the Kaimal power spectrum used in Eurocode 1 for wind loading. Thank you so much 🙏

6:22 There's a difference between incompressible fluids and incompressible flows. All fluids are compressible, even water (though it's compressibility is on the order of steel). Air, while highly compressible, can be treated as incompressible in a lot of fluid flow problems. An incompressible flow simply means that the difference in density at the highest and lowest pressures of the fluid is negligible (as in if the change in density is around 1%, it doesn't have a significant effect on the flow). The generally agreed barrier between incompressible and compressible flows is around 0.4 Mach. This is true because A) Mach and fluid compressibility are integrally related to each other and B) the flow speed of a fluid system represents the lowest pressure (free stream) and highest pressure (stagnation pressure, ie. the pressure when you turn all the fluid's kinetic energy into pressure). 0.4 Mach generally means it's around 5% maximum possible change in density, though it could be significantly less depending on the fluid system. Edit: Great video! Please keep doing higher level fluid videos! Conformal mapping and all. You do such a great job at breaking down these complex systems.

@ArthurCammers

5 жыл бұрын

LoL I guess the universe (space between stuff and all the stuff) is compressible or else the Big Bang and all inflationary theories about the young universe are false.

@jeffbeasley8235

5 жыл бұрын

I hoped that someone mentioned this in the comments. Even air is considered incompressible at lower speed flows. A ton of people aren't aware of that! The "compressibility" of a fluid depends on its particular flow... An idea that isn't intuitive at all.

@zebratangozebra

4 жыл бұрын

Thinking of the pushback I would get telling people water is compressible.

Your videos are very very great. As I am studying aviation engineering I have to learn a lot about turbulent flow so thank you for making this video!

You propagate your fascination for science so effectively through your videos! Also you have such clarity in speech and expression! Great teacher!

Each and every video of his is a work of art. Absolutely loved it!

loved the video. finally can give my friends an idea of what i‘m doing. just a minor correction: you said „when something compressible like air“. compressibility is not a property of the fluid but the fluid flow.

Fantastic combo! A couple of my favourite KZread personalities. You and Diana never fail to impress!

Great video explaining such an intricate subject! Note at 8:40 : Kolmogorov only really worked in real space (as opposed to Fourier space) so the energy distribution as a function of D actually scales with D^2/3 - Fourier transform that and you get the more well known k^-5/3.

I can proudly say that watching a new 3B1B video is the highlight of my day. Thank you for inspiring so many people by sharing with them the beauty of math and physics.

I've been watching your videos for a long time now, and I'm inspired by your dedication to explain things with such creativity and clarity. Keep up the good work :). (Btw could you do a video on the navier stokes equation by any chance?)

Thanks for the technical note about the Fourier transform -- I've always wondered how on Earth you would quantify/define an "eddy", but a Fourier transform is a perfect way to do so.

I love this video. It's so beautiful and simplistic about an otherwise messy subject

How can this channel get any cooler?! I sent this link to my daughter who is studying high school physics..she loved it

i studied at kolmogorov's school in moscow and i am exited seeing him in your video!

Cool stuff! I also did this experiment last january and have a video on my channel, albeit in lower quality. If you can shoot two consecutive rings, you can make the last ring go through the first one, which is called leap frogging. Laser planes are a very cool way to visualize flows!

Awesome visualization, as always! You may be interested in the SINDy algorithm of Steven Brunton et al., which is explicitly able to identify a nonlinear dynamical system such as vortex shedding based solely on data.

The best video so far, keep it up! Great work guys

A+ collaboration. Physics Girl’s and your styles are really good compliments to each other.

This is absolutely fascinating. And very complicated mathematical physics. Gives a clue of how difficult the Navier-Stokes problem is.

8:50 Hey, the inertial subrange boundaries aren't dependent only on the fluid, it depends on your geometry, and the velocity of the flow being considered. For example, for turbulent flow over a flat plate, you would consider the characteristic length L as being the length of the plate. Then the upper bound of the inertial subrange is about 1/6 L, and the lower bound is about 60 L (ρUL/µ)^(-3/4), where ρ is the density of the fluid, U is the mean velocity over the plate, and µ is the flow viscosity. Altogether Re = ρUL/µ is the Reynolds number. What defines this range, is that the kinetic energy only cascades down, but it isn't produced (like at bigger scales) neither dissipated (like at smaller scales).

A whole series on fluid dynamics would be so cool :)

8:04 - I've had that poem stuck in my head since Chaos Theory and Systems Analysis.

LOVED this video, keep up the quality!!

Man, your shows are always awesome to watch and learn a lot eventhough one might think that one knows "everything". 👍🏽👍🏽👍🏽

Though this had very little math content, I am happy to see you doing collab videos

Detailed and comprehensive, nice job

great video as usual!! please can you make a video where you'll talk about the concept of stability and you'll also talk about lyapunov's theory?

I fly freestyle drones, they weigh around 600g and have about 6kg of thrust. I would love to be able to visualise all the turbulence they create and leave in their wake. A huge help in learning to fly them well is realising that you are 'swimming' around not flying. Thanks for another great video!

6:15 I think one important observation regarding incompressible flow, is that yes, indeed air is compressible, but so is liquid water, and all fluids for that matter (in fact a little bit of compressibility is what allows the concept of pressure). But when we say incompressible flow, it's important to consider the speed of the flow, and the pressure gradients associated with it. For many applications, with no pressure gradients (like in external flow, like the one shown in the footage) and with speeds well below the speed of sound (definitely like shown in the footage), compressibility effects are quite negligible. So although air is compressible, for external flow, and low Mach numbers, the incompressibility hypothesis is really reasonable.

@DriverMate

Жыл бұрын

True, but then he says he wanted to talk about exact solutions without approximations

As always Fantiasic work!!! Thanks a lot !!!

Wow! The efforts put in to visualize these concepts...

I think it'll be hella cool if you can use the planar laser with a large pressure chamber with clear walls just to see what pressure does to the turbulence features, maybe even their temperature dependences. Comparing these might be really good visualization and helpful to those learning thermodynamics/stat-physics for the first time. Great job!

The beauty of discussion throughout is amazing.

I see an upload from 3Blue1Brown, I try to run as fast as I can. Really love these videos

Superb. You keep surprising with the beauty of your videos.

Just for clarity, 3:34 depicts either inviscid flow or is missing the Kutta-Condition at the trailing edge of the airfoil. The flow leaving the trailing edge of the airfoil should be parallel to the slope of the airfoil at that location. EDIT: The stagnation point seems to be just on the upper surface of the airfoil (just barely in front of the trailing edge) In order for the stagnation point to be at this location, the flow must accelerate infinitely around the cusp of the airfoil. The Kutta Conditions ensures that the stagnation point is on the trailing edge.

I just love the way you teach us new things

Just in time for my fluid dynamics exam! Awesome video

OMG you explain it so clearly and so simply!

"Eddies in the time space continuum" "Ah, is he?"

@qwertyTRiG

4 жыл бұрын

Yes. That's his sofa!

@stapler942

3 жыл бұрын

Came here to say this.

@novameowww

3 жыл бұрын

Life has many doors, ed-boy

2:29 aw great dude, you've un-eyed the pi's Now they're just P

This is, by far, the best KZread channel ever. Thank you! :)

I am truly amazed! Really love you guys!! I learned a lot!

This is a basic version of Particle Imaging Velocimetry. So cool!

The laminar flow part reminds me of the continuation of that video on divergence, rotation, complex analysis an fluid dynamics :p

Nice to see the descriptions in the equations. When I see an equation I need know what the letters stand for,

dude your editings are out of this world

Spent the summer working with a technique called Planar Laser-Induced Fluorescence, which utilizes passive flow of a fluorescent substance to visualize scalar transport phenomena (i.e. concentration). I'm excited to see you drawing attention to this useful method. Coupled with velocimetry measurements, planar lasers are unstoppable!

I've taken a graduate level turbulence class, and let me say that this is probably a better introduction to the topic than what I got there. Also, if you use the right particles (something a little bigger than smoke at this scale), you can use the planar laser to do what's called Particle Image Velocimetry. This is just a fancy way of saying that you take two pictures very close in time, match the particles up (usually with a computer), and do a delta()/delta(t) for each particle. You can map the whole 2-D velocity field at a single point in time using this technique, and even use it to track transient flows if you use a video camera. I worked in a lab that did these sorts of experiments in grad school, and it's basically just a fancy, more controlled version of your setup here.

This particular demonstration is relevant to convection currents and 0-1-2-ness logarithmic sync-duration condensation. The hexagonal polar vortex on Neptune is an exercise in time-timing sync-duration vector-values analysis like the odds and evens between Primes and Cofactors applies above.

Apart from educative, this video is incredibly beautiful. Thanks for bringing this wonderful content to us.

My PhD research was using a multi-fractal description of the energy cascade to model subgrid eddies in turbulent flow. Lots of maths and FORTRAN. But the NS equations are probably simpler to describe than most people think, especially when you consider the fluid moving through a small box.

Well explained sir ;-) As an interesting side-note: The turbulent cascade charachteristics described by Kolmogorov & co for fluid dynamics also apply to some other chaotic systems, as for example the fluctuations in financial markets. (Ghashghaie, Shoaleh, et al. "Turbulent cascades in foreign exchange markets." Nature 381.6585 (1996): 767.)

Looks great Grant! I really like the car-breaking introduction to dissipation. I'll probably steal that. 😀

@bernatfont6510

5 жыл бұрын

Your animation looks great too! 😁

Fantastic video,,thankyou, very well explained

i had always assumed vortex rings turned inward. great video.

beautiful explanation

Since you did smoke flow visualization, a video on PIV (Particle Image Velocimetry) would be quite cool to show people how scientists actually measure real flows using the same setup you had!

Both yours and Dianas' videos are my favourite part of KZread!

I'd love to see you make a video about critical Reynolds numbers!

The new opening animation looks great.

What a pleasure to see my dearest curly branch of maths here! Kolmogorov 6:41 has definitely lived in 20th century, though.

@unflexian

5 жыл бұрын

I vow to call fluid dynamics curly math in any and all enviorments.

Mmmm... I love this! Ever since I was able to recognize that I had thoughts I saw this 'chaos' as beautiful patterns. Nothing is ever the same as something else. You don't see the same sunset as I do; I am sitting to your left and I have you in my peripheral vision and on my mind and I feel a shoulder on the right side of my chest while I have my arm over your other shoulder. You do not. My molecules share a history with yours, but ever so slightly different. The path for every particle to where it is now is different and all combinations are unique, despite obeying the same natural laws. Those natural laws are what bind us, the paths are unique. Another example: The branch of a tree doesn't 'happen' to be there and have that shape and color and composition. It's the result of genes, wind, sunlight, temperature of the ground and surrounding air, nutrients, water, how much of which was where at what time in what combination and the damage and help that other flora and fauna gave it. All of those are unique, not just for every tree or branch but for every cell and cell division, giving each tree its own unique 'character', showing it's entire history, all the interactions with all life around it and all particles, the climate, the weather. Take a step back and look at the species as a whole and how it developed through time with changing climate and surrounding flora and fauna. Take another step back and see how the planet formed. Step back and see the universe too shape. All the way back to just after the beginning when hydrogen was all there was. - That picture of the background radiation is the earliest baby picture of all of us together that we have; think about that for a second - Every leaf and every branch is a logical result of all of our histories and every particle has its own history that's interconnected with other particles. Every human has their own history that is interconnected with everything else. We're not random, we're a logical result. We're not chaotic either, we are a logical result. Hard and often impossible to predict, yes. There are too many variables for accurate predictions. We simply don't have and probably will never have perfect knowledge of the initial conditions. That's also why we still have the relative freedom to make decisions and the responsibility to gather information to make the best possible decisions, while most likely being in an absolutely deterministic universe, where true chaos simply cannot exist. All patterns are simply the logical result of our history and I don't know if the laws of nature are a part of that history or only a cause of it. Maybe there are other universe where at the beginning something was different that resulted in different laws of nature and possibly some of those laws couldn't work and the universe collapsed and some worked well and went on existing. Who knows? What we do know is that the laws of nature in _this_ universe made everything to be the way it is now, including your thoughts at this very second. The history of the entire (visible) universe (and to a lesser extent that of what is not visible anymore but once was connected to the part of the universe we are causally connected to) is in everything you see and think. The entire history of the universe and the history of the person, including everything they ate and the love they received is in the colors and the seemingly chaotic, but definitely unique patterns of the iris of the person you love most. What that history looks like is a mystery, but it is definitely there. You now know the color of the eyes of the person you love and where the specks are in the iris of the person looking back at you. Now take a step back in your mind and look at how you're interacting with each other and at what you want to do next. Your future is as unique as your history. Choose something. Choose a general direction you want to pursue and choose an action as a step on the path in that direction and do it.

Really well done! Thumbs up and now I'm a subscribed!

The cool thing to me about this, is that this is only in two dimensions. The ring is on another plain but we just see a slice of it drifting past. It’s a great way of showing 3D passing 2D like how 4D would pass through 3D. For a 3D representation of turbulence, you could use water and dyes. Or even colored smoke. We see it every day.

A few years back I thought to actually do some proper revision of the math I used to know about N-S, (because it's fun to imagine mostly), but the more I think about it the more I remember why I'm not academic material of that nature, (a bit "agricultural" a friend once described it, he was talking about a Harley Davidson, but you get the picture). So taking the long way around to redefine the conception from first principles is Actuality-> inevitably, if you want to know why the N-S Equation is a thing of real elemental meaning. The videos are very good demos of the transverse/cross-sectioning layout of the wave-package, at this particular scale/relative spin-resonance imaging rates for which the little whirly poem applies. Having just watched a video on the likely origins of living organisms on Earth, and seeing how this equivalent, functional transverse snapshot of bio-logical chemistry operates, the idea of math, resonances in harmonics and reciprocals for all prime numberness of multi-phase-synched moving flow of the turbulence is daunting. At best, maybe some chunking of emulated/synthetic chemistry will be a solution, (abstracted pictures within pictures), using atoms as vertices in the Quantum Fields Modulation Mechanism of QM-Time Principle In-form-ation formulae in the equivalent time-duration conception, (case by case?) That's why Science is a process, not a thing, and all practitioners and practices are transferable. ----- The 137 numberness dominance node, Shaw's Algorithm and the wave-particle amplitude and frequency density-intensity real-numberness condensation modulation cause-effect of/by Superspin.., might be a Stroboscopic Specrtoscope Spectrum Analysing methodology of the e-Pi-i sync-duration Hologram?. It is all talk from me, and the only Programmer I know is in Accounting..