Understanding QFT - Episode 1: How spin was discovered
HTML-код
- Опубликовано: 3 май 2024
- JJJreact
The history of Quantum Field Theory started with the discovery of spin
Help me pay me student debt (please...): / highlyentropicmind
For one time donations:
Buy me a coffee (best option): www.buymeacoffee.com/highlyentrc
Ko-fi: ko-fi.com/highlyentropicmind
Paypal: paypal.me/HighlyEntropicMind?...
Do you wanna help me with the math animations? Just send me an email with a demo of what you can do and how much you charge: francofelix94@gmail.com
----------------------------------------------------------------------------------------------------------------------
Special thanks to all my patreons, but specially to:
OtiumOtiosum, Valerie Hyde, T Highfill, Ryan Roberts, Carlo Fazioli and Cookie_T and Azure Azalia
----------------------------------------------------------------------------------------------------------------------
Credits:
Script, Host, Filming, Directing, Flaying, Editing, Production: Fernando Franco Félix
Animations and graphics: Oscar Flores and Fernando Franco Félix
Special thanks to Michael Dayah for letting me use his animations from ptable.com
Special thanks to arkiv.dk for letting me use the picture of Bohr, Harald, and their mother Ellen
The song used at 50:15 is "Emotional Epic Choir" by Allen Grey and I paid for the license to use it
----------------------------------------------------------------------------------------------------------------------
Sources:
Most of the research for this video came from:
Mehra, Jagdish, and Helmut Rechenberg. The historical development of quantum theory. Vol. 1. Springer Science & Business Media, 2000.
Details about Bohr's life came from:
Pais, Abraham. Niels Bohr's times: In physics, philosophy, and polity. Oxford University Press, 1991.
Details about Bohr's work came from his essays:
Bohr, Niels. The theory of spectra and atomic constitution. CUP Archive, 1924.
I also used Griffith's book as a guide:
Griffiths, David J., and Darrell F. Schroeter. Introduction to quantum mechanics. Cambridge university press, 2018.
----------------------------------------------------------------------------------------------------------------------
Chapters:
00:00 - Intro
02:05 - Presenting: Niels Bohr
04:51 - Thomson discovers the electrons
06:47 - Why do hot things glow?
09:57 - Planck proposes the photon
13:03 - Einstein discovers the photon
16:18 - Rutherford discovers the nucleus
19:55 - Why don't electrons spiral down towards the nucleus?
22:02 - Patreon stuff
23:37 - Bohr discovers the orbits are quantized
25:41 - Spectrum saga
30:04 - Light is quantized because the orbits are quantized
34:45 - Bohr defends his model from the haters
37:01 - The answer
39:42 - Angular momentum is quantized
41:57 - Why aren't all electrons in the first orbit?
45:32 - Orientations are discovered
48:15 - Atomic structure is discovered
51:03 - Bohr almost discovers spins
52:39 - Why 8 is important in chemistry
54:23 - Bohr explains nearly all spectra
56:18 - Stern-Gerlacht experiment
01:01:24 - Anomalous Zeeman Effect
01:03:31 - Pauli proposes spin
01:04:44 - Ralph Kronig discovers spin
01:06:10 - Spin is discovered, again
01:07:18 - Outro
----------------------------------------------------------------------------------------------------------------------
Attributions for graphical resources not stated in the video itself:
Sspeech bubbles: Image by coolvector on Freepik
www.freepik.com/free-vector/c...
Science center set: Image by pikisuperstar on Freepik
www.freepik.com/free-vector/r...
University classroom: Image by tohamina on Freepik
www.freepik.com/free-photo/em...
Kid's room: Image by macrovector on Freepik
www.freepik.com/free-vector/t...
Graduation cap: Image by macrovector on Freepik
www.freepik.com/free-vector/r...
Water droplet: Image by Petar Milošević, used under cc-sa-4.0
If you are weird enough to have read this far, you may be interested to read about how Rydberg was shafted by corruption at his university:
mathshistory.st-andrews.ac.uk...
mathshistory.st-andrews.ac.uk...
![All That Remains - Divine [Official Music Video]](http://i.ytimg.com/vi/01rW8JxwpO8/mqdefault.jpg)
This channel is the perfect blend of history, science, mathematics, and cringe weebshit. You, my friend, deserve my patronage.
thanks dick cheney
@@mattdriscoll4102 Actually, it was me. QFT, you know.
I was about to close the video, but this comment actually is making me want to give it a chance :)
agreed.
weebshit
jesus man this is criminally underrated, excelent work!!!
this brought science, story, and a little theatrics all into one great educational piece. love it
44 Zero 4z44a4😂z6xx ok ii z.cxxc
I like this longer form of content, you don't just take one buzz-aspect of science but actually cover all the bases of the story while keeping it fresh and entertaining.
Agreed. You must be at least 40 if you still have an hour long attention span 😂. It reminds me of university, I had some great profs, although none with a working anime vocabulary.
Watch a video of Paul Dirac giving a lecture and compare it with this. Dirac was undoubtedly one of the greatest physicists the world has seen but, as a speaker, he had very low energy and kind of droned on. This video covers a vast amount of information yet it maintains a high level of energy for over an hour. I also enjoy the chronological / historical approach to learning scientific theory as I find it makes the material that much more fascinating. These men in the 1920’s changed the world forever with the power of their minds. Thank you Fernando for your strong work and lively presentation. You earned my subscription.
"How electron spins" :
Shows picture of old dead white supremacists 😂😂
To give Dirac credit, he delivered his lectures without the benefit of multiple takes and video editing.
That is not true. Those videos were taken at the end of his life. He was more enthusiastic when he was younger.
@@hebrewisraelitescharleston843
We’d be waiting forever for a black person to discover any of this. I can’t even believe you’d have the nerve to comment here.
I mean to be fair to Dirac he was giving lectures to physists, not a YT audience. Not to take away anything from this video, it truly is amazing.
I knew after 5 minutes of watching this video that I was on a ride I couldn't get off. The narrative style you use is easy to follow and allowed me to visualize the properties of electrons in ways I never have before. Bravo sir! May your channel find it's way to it's highest possible orbit.
00:00 - Overview: Quantum Field Theory (QFT), Spin, electron orbit mystery.
00:41 - Emphasizing observation's role in forming QFT's foundation.
01:56 - Introduction to Niels Bohr's university journey.
02:28 - Bohr's early years and surface tension experiment.
03:33 - Bohr's quest: understanding electron's stable orbits.
04:15 - Late 19th-century knowledge on atoms, light.
05:25 - Unveiling cathode rays, electron discovery.
06:42 - Thomson's "raisin and pudding" atomic model.
07:24 - Max Planck's insights into why objects glow.
08:02 - Planck's solution to the Ultraviolet Catastrophe.
12:27 - Albert Einstein's work on the photoelectric effect.
15:23 - Light: particle-wave duality; atomic existence confirmed.
16:04 - Bohr starts PhD, meets Rutherford.
23:21 - Bohr wrestles with atomic observations.
25:01 - Bohr introduced to hydrogen's spectral lines.
29:57 - Scientists claim series in Rydberg's formula.
32:25 - Principal quantum number in electron orbit energy.
35:22 - Bohr's simplified atomic molecular formation model.
40:55 - Sub-orbits in quantum mechanics introduced.
42:45 - Concept of atom's orbit occupancy limit.
54:59 - Bohr predicts spectral emissions, presents atomic structure.
1:02:46 - Prodigy Wolfgang Pauli enters science scene.
1:06:19 - Uhlenbeck, Goudsmit develop electron self-spin theory.
GIVE A LIKE AND FOLLOW ✅
You are right, I should have added chapters, I don't know how I could have forgotten
@@HighlyEntropicMind its ok .. great video.
@@HighlyEntropicMindThanks for adding them.
Is there a way to pin this chapters list to the top?
If you like to go away from quantum mechanical speculation on speculation, check recent advances by Dr. Vivian Robinson and Dr. Williamson (Williamson has Nobel Prize for Physics), about physical (pure electromagnetic) model explaining both spin and charge.
With a degree in physics already this is like watching your favourite detectives movie for the 40th time 😊
So well told! Thank you! I can't wait for deeper QFT since theory was not my strong suit. 😅
Even with your degree I hope you can learn something from this series. For instance, I bet you were just given the Pauli matrices without explaining why they work, but when we are done, they will feel as obvious as 1+1
@@HighlyEntropicMind Thats right. When they first occured even in theoretical physics lectures they were not motivated very well. Now I know some way to legitimise them but I am still cuious. One more way to look at something can widen the grasp of the entire knowledge one has. :)
Is it like an episode of columbo? Where you know whodunnit but you just enjoy how it's figured out?
@@tricky778 I hope so!
Physicists are basically detectives solving the mysteries of the Universe 😅
I fell asleep at my computer and when I woke up this video was playing. At first I was like, "Did PBS Spacetime get a new host?" then slowly, I realized what was going on. I laughed, I cried, and an hour later, here we are. This is good stuff. I would totally sign up for your Patreon - that pitch hit close to home.. a little too close though cause I'm broker than you but I don't even make videos. I officially like the video and sub to the channel. I'm as engaged as I can get for free. That'll have to do for now. I look forward to the next video with sharpened pencils.
I like that you made Physics funny. Humor always works to get attention. Most Physics professors I have seen are so serious even though they have great knowledge. But learning can also be fun, humorous and joyful. And when it is so, it is literally effortless...
Edit: Salute your editing
This guy is the best teacher a student could get
I like to think I'm just another student, and I'm explaining this because the teachers didn't do their job properly
Agreed. I was thinking about just that.
What a GOOOOOD video. Holy hell, what a ride.
I can not imagine hearing this history for the first time from this video. This guy is such a good story teller that I bet it's a magical ride.
Great explanation of spherical harmonics and I'm very impressed how you could cover the history this consisely and thoroughly while making it thoroughly entertaining.
The key is that I am a VERY slow learner, so when I finally understand something is because I've studied very thoroughly
"Spin" is just a vortex of "fire-ether" i.e. HEAT. Direction of the vortex (from "inside to out", "into physical world" or from out to in) defines if the electric charge is "+" or "-".
@@HighlyEntropicMind Welcome to the club. I say that while some run through the forest and go on happy and well, we learn it through and out, so there's no need to steer clear of it or hurry anymore. All the beasts are known, trees numbered, paths remembered, nights cherished, and you wonder why it was so difficult at the first glance. In other words, you cannot teach something well if you don't understand it deeply, and so you behave freely, and it gives space for jokes and other stuff.
You are absolutely the best presenter of physics information I have seen in a long, long time. Thank you 👍
i'm honored
@@HighlyEntropicMindi love the subtle humor bro keep it up waiting for the next part
Really enjoyed this one. You seemed really comfortable and relaxed. How much you enjoy the topic shows. Love how informal and fun it was. Keep being yourself and I'll keep watching.
You said everything I wanted people to say about this video
I documented all the notes in this video into a big Google Doc and it was almost 20 pages. A great addition to my quantum mechanics studies, thanks a bunch!
I can just give you the script: docs.google.com/document/d/1tKf38PQvWaYrBkpt5F4y7CX1xn844egSCUnBLWaZKj8/edit?usp=sharing
That said, I highly recommend you read Griffith's "Introduction to Quantum Mechanics". You could even download it for free as a pdf. In fact, it would be possible for a person to read so many pirated books about physics they decided to make a youtube series about what they've learned... But don't pirate the books, knowledge has to cost money, only people who pay can learn, that's the law!
Not sure how or why I ended up here, but made it all the way to the end fully focused. You've got a good thing going on here sir, going to explore more!
I not only made it to the end but thoroughly enjoyed the whole video. Thank you so much for your thorough and very hilarious down-to-earth description of both the history and the mechanics behind these groundbreaking discoveries
You're a natural storyteller, and your knowledge is impressive.
Subscribed, these stories help braid together the scope that these physicists had during the early 20th, and I love gaining insight on just what mindset these pioneers had.
This video is absolutely amazing, everything a physics enthusiast could ask for
As someone that was advanced at Math at an early age (Finished Calc BC and took the test at 15), but never clicked with Physics beyond the basics, I would love to see someone like you begin to tie the math to the physics. I was great at solving the Calc equations, but I never even really understood (until recently) that the basics of calculas theory even in that it provides a framework to "calculate" area/volume in a sometimes dynamic context. I didn't know what I was doing, but could do it very well.
"But dont go watch it right now. I need the views, Derek doesn't".
I love the video and this line made me laugh out loud!!!
Thanks for your effort. This form of content is really helpful. I loved that you gave background information on discoveries.
@29:14: "They have two variables, N1 and N2 and they have no idea..." It's amazing how obvious it seems today (given a physics background) what those are. It must have been a complete mystery to them at the time, and very frustrating.
Instantly subscribed after the 'sit down this will be difficult to hear' part 😂😂😭 you made my day lmao. Also cant wait for part 2 (starting my physics masters in september so i really cant wait pls hurry)
Your short monolog was very useful and welcomed.
It gave context... Which I really appreciate.
Mahalo 🤙🏽
"But don't go watch it now, I need the views"!!
Bravo! This is such a relief.
So many videos begin by saying, nasally, "if you haven't seen our other video you should watch that first." You go watch it and it says same thing. You go watch that one and it says same thing. I'm still going back. I believe if I go back far enough, will get to witness Creation and not have to be told about it!
Well done, my friend. Very entertaining, informative!
So glad I found your channel. You finally put the peices together for me when it comes to electrons and how they behave like waves, and in an entertaining way too!
There's still a lot more to say about the wave particle duality, but we'll get to that eventually
Awesome video, hope to see this series continue!
I really hope there is a ep 2. very very much enjoyed your love for topics like this. Thank you.
You sir are a legend.
Coherent, interesting, funny and above all beautifully explained. Cant wait to see more! ❤
I only wish you were around when i was in high school
I couldn't stop watching this video. Can't wait for more!
Thanks!
New sub as this is great educational content all round!!
I also love the clear source atribution of images, and REALLY like the reference to other youtube channels for examples of demo's / other explanations! THIS is akin to the central tenant of Science in the first place of sharing knowledge and sharing other peoples work to aid / explain your own.
This is GREAT content
Thanks for appreciating the attributions and references
You know it's getting good when the presentor starts pacing.
I loved this video. It was informative and I always appreciate a genuine performance 🥰
I’m currently on a journey of crash coursing myself through physics and chemistry, and this video has been the most entertaining and informative so far. I especially love your candor and elder millennial sense of humor. I eagerly await the next video.
Can’t wait to see how this channel matures bro, so much potential here!!! would love to see you hire some animators to visualize these concepts even better!!
Omg I just watched an hour of the history and logic behind the discovery of the spin, mixed with cringy weeb jokes, and i not only understood pretty much everything, i even loved it hearing everything. Good job mate! I hope your channel blows up!
I did my Ph.D. on GR but I do know QFT. This video is a very different way of presenting it, and I think it a helpful way of thinking about it.
First video of yours I've seen. Very excited for the series. Also, your background is the same picture as my phone wallpaper so it looks like you're just standing in my phone, teaching me all these amazing things. 100% subscribed for the future of this series
I would kill to have you as a professor 😄 Sincerely hope this video blows up!
I've taught, I hate it more than the students, this is better
@@HighlyEntropicMind Haha so I guess when you say something that you think is cool, you can pretend we all have a look of wonder on our faces instead of just getting blank stares
Students falling asleep must really piss you off...
@@petevenuti7355 I fell asleep in class constantly
@@HighlyEntropicMind 😂 , standing desk?
Good you have got one Pauli .A sketch with true colours with the main actors.
The cartoon of electrons dance on net of (Space -time) makes the puzzle interesting.
Excellent channel
I love your background. As soon as I saw that picture from JWST when it was released I had to set it as my background and I am glad to see other people like it too!`
I’ve had this explained to me many times. But yourswas the most palatable. Thanks for this!!
This guy is hilarious. And informative. But the humor is what sets him apart. +1 sub.
Damn I love this, I need episode 2 now. Gracias por hacer este contenido
Brilliant video! I’ve watched it multiple times and enjoy it every single time. Can’t wait for the next in the series.
i‘m so glad i found this video. i‘ve been trying to wrap my head around this topic for a while now and your style of explanation was amazing. i really enjoyed the memes and humor. love the TARDIS pen, you earned a sub!
Great presentation 🎉
Another big dub. Excellent work Fer. I now understand why electrons can both behave like waves (when they emit energy) and like individual particles (due to the specific frequencies/energies of orbits). I have more questions than ever, and I cannot wait for your next video. Keep pushing, brother.
I'm so happy you liked it. Please ask any questions you have. I expect people will have questions and I plan to make a video answering them
One question is this: I know you explained it in the video but maybe it went over my head. What exactly is ‘m’ and ‘loopy l’. I understand ‘n’ is the energy level or orbital of the electron, but I’m confused on those two variables specifically. Also, why is M confined to -1, 0, and 1
@@adamcorrigan1939 Curvy l represents angular momentum, basically, the speed at which the electron spins around the nucleus. Just keep in mind that to know the actual momentum you have to use the formula shown in the video (the one Sommerfeld found)
Now, m is just the orientation of this orbit. This is easy to visualize as a 2D loop tilted in some direction... But then when you actually do the math to find the shape of the orbits, you find the spherical harmonics, which are not 2D loops, but 3D shells, and the electron could be anywhere on it
The important part is that regardless of where in the orbit the electron is you know (roughly) at what distance from the nucleus it is (n), how fast is it moving (curvy l), and how much of that momentum is in the direction of the magnetic field (m), or how much momentum would be in the direction of the magnetic field, if there was one
Thank you for the explanation. I would like to see an explanation for the orbital of a magnetic or unstable element to compare it to the stable silver one. Also I guess what still is confusing me is why m can have a value of 2/-2? Is that electron feeling twice the force of the magnetic field? If so, is this double force due to direction of alignment of the harmonic sphere as you described?
@@adamcorrigan1939 Remember that the suborbit has an axis of rotation. Now, imagine two arrows like two hands on a clock, with the same origin but pointing in different directions. One of this arrows is the direction of magnetic field, the other is the axis of rotation. Now imagine we were to shine a light so that we could see the shadow of the axis of rotation projected onto the arrow of the arrow of the magnetic field
The size of this shadow is quantized, this is the quantum number m. Depending on the orientation of the orbit this number will be larger or smaller
Now when m is negative it means that the axis of rotation is pointing down, that means that the electron is spinning in the opposite direction
Love the depth of research and knowledge you disseminate! Definitely will watch again!
First time I've seen some put it all together. Excellent job!
the real spherical harmonics was the atoms we made along the way
you gained a sub! shared you video everywhere lol, amazing content, please keep doing this!
Triple points for the Japanime analogies. Why are there not more physics channels on the cutting edge? 🤔 🧐😢😮😅😂 1:03:26
Thank you for reinvigorating my interest in understanding the universe. Your method of explanation is perfect for my method of understanding!
I love everything about this video. The jokes and anime references get me each time. Plus "Por que no los dos" is my favorite saying, because of that old commercial, lmao. One thing I wanted to add about Bohr, he put off his Honeymoon to write one of his seminal papers; now that's a real scientist.
Double bonus points for a tardis microphone 🎤 51:37
Please ask your questions! I'll see what are the most popular ones and make a video answering them
Is your shirt phyrexian?
@@bishnu13 Yes, it says "reveal the language"
Can you explain why the electron orbits are quantized? Nature doesnt allow it? Why? Or is this what we will already see in a future video mentioned at 39mins?
Also, instead od electrons being dimensionless spinning rotating points on separate quantized orbits could they not be represented as objects (or even waves) moving in higher dimensions ?
I keep on seeing a 3d representation of multidimensional spheres hen i look at the images of the electron orbits…
Ive always wanted to work out the maths for that but im lacking in the maths department,
Looking forward to your next video!
@@Keneo1 The symmetries of nature can only be realized if there are some very specific values of angular momentum, but then you could ask "why does nature have symmetries?"
At some point we can only describe reality and admit that we may never know why it seems to work that way
bro i just finished the whole video and the way you just constantly grab my attention and tell an amazing story is just pure skill. Great stuff!!!!!!!!!!!!!!!!!!
This video helped me finally "grok" several core aspects of quantum mechanics that had eluded me. All in an entertaining package.
I watched the whole thing and I didn't even know about your channel before I found this video. Keep doing what youre doing and no one gets hurt.
Bonus points for cutting in a clip from Dr. Stone with proper context...
I love that show so much
@@HighlyEntropicMind did they continue beyond season 1?
@@dgpreston5593yup!
@@dgpreston5593 season three ended like two weeks ago, and it seems like season 4 is gonna be amazing
@@HighlyEntropicMind wow, I have to find where it is streaming....
I can already tell this is the highest quality video you've ever made
Just wait until you see the next one. Using imaginary numbers in the Pauli matrices will feel obvious
Excellent video. Thank you for giving me a bit more intuition on this topic.
Thank you for making me smile. Your goofy AF and so am I brother. Great job
Quantum spin is even dimensional. That's one reason it has such unfamiliar properties.
This is why I am subscribed: great work!
Just one question: what is an "isekai"?
it's a genre of anime, and for some reason many of these shows have protagonist that look nearly identical
@@HighlyEntropicMind - I could guess that much from the pictures, it was a silly joke question... I'm too old and non-Japanese to bother with anime.
@@HighlyEntropicMind Ive had a steady job for a while and if I manage to lock it up Ill 100% formally study physics
But I swear to god if anime shows up in science one more time im fully commiting my life to building a turbo servorocket on the moon big enough to crash it into earth
Great video! I was engaged the whole time and learned a lot. Can't wait for the next one!
Dude. Started this video and thought 'nah, can't listen to someone i can't understand talk about particle physics'. But then I came back and gave it a second chance... thank you! It finally clicked, and I was highly entertained the whole time. Great stuff. Subbed!
Can’t take a sample of the Sun? The Parker Solar Probe takes that as a challenge.
Good point
Excellent work… you video helped me finally get a grasp on a few fundamentals that have unto now eluded me. Thank you ☺️ Can’t wait for the next one !
Wow, that was just amazing to watch. I learned and laughed at the same time while watching this movie. Thanks a lot and God bless!
Fantastic blend of science, humor and nerd. Well done!
I loved this video!! You really have a knack for being on camera and captivating an audience!! It was truly AWSOME.
This was too good to pause- I looked up and 68 minutes were gone. A very humorous and outstanding delivery of a complex topic with super clear explanations!
Science history is the context in which science happens - it is always import to understand the context, Well done.
Instantly earned my subscription, you're an excellent presenter and I look forward to more!
The same question I thought so for so many years is finally answered. Lengthy but really needed video.
I just recommended (and I am a physicist too) this to many friends! Excellent overview, wthout forgetting the march to knowledge and all the people involved. Kudos to you sir!
Thank you very much for this excellent explanation and historic perspective of QFT.
Wow! You just explained me the basics so so well! What an amazing video 🙏🏼
The history of these discoveries and how one led to another really helps to deepen the understanding. Thanks for the video
Loving your energy man, keep it going!
I love the early days of physics! ❤ They are all my hero’s! And thanks for the awesome physics history!
I am so onboard for this entire journey. Take me, science man.
Loved it! I was supposed to study but wasn't feeling like it. Then I chanced upon your video and it got me hooked. Telling myself right now this can compensate for the studying I should've done earlier. Waiting for episode 2!
I LOVE your style, one of my new favourite RUclipsrs
i love this historical approach to physics!!! it makes it so much more fun.
Along with the other nerds I can agree that this is the perfect blend. Thank you and I look forward to seeing part 2!
Amazing video! It's super entertaining yet very educational. Very well researched and ... well I could only say nice stuff about it. Thank you very much.
this is a wonderful take on this discovery. thanks for making it so easy to understand and for the humor. esp the humor! i can’t wait for more!
Just a quick historic note to this very nice video. One of the ways Bohr convinced others (and Einstein is one of these from his correspondence) was Bohr’s explanation of the Pickering-Fowler lines, which is one of the first examples of high-precision measurements. Pickering, an astronomer, found Hydrogen-like spectra, but with half-odd integer quantum numbers in special stars. Fowler did experiments on earth and found the lines only occurred when helium was present. To tell the difference in the spectra, one uses the reduced mass, and the mass of Helium worked better than hydrogen at the fourth digit of accuracy in the wavelengths. Experiments had just about this precision, and showed the lines came from Helium, not hydrogen. Einstein said to have this agreement means the theory must be right. But, of course, that was incorrect, as Bohr’s theory was later superseded with modern quantum mechanics.
Very informative and very entertaining! Subscribed and looking forward to episode 2!
Interesting video. I like how the original looks like a pirated copy - there's even several seconds where Fernando has wandered off screen. But I watched it all, and it's a great story. Then I watched it again. Subscribed.
I'm working on improving the quality, so it doesn't look like a pirated copy
@@HighlyEntropicMind I always get your videos from torrent. 😆
Absolutely loved this
Loved the Pauli and Kronig conversation 😂
New sub. Aside from the human aspects of this, not much of it was new to me, but I'm very interested to get deeper into the math. Possibly not the consensus view, but the more math the merrier, as long as you build up to each piece of course... Excited to see the next part!
Amazing really enjoyed your video and the historical context of these discoveries needed this.
Fantastic work! Thank you.
Thanks for your excellent explanation! Keep on going …
I love your video man!!! Keep them coming; te felicito