Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. Dear Mathologer, I would like to repeat my question I asked in the previous version of the video. Will you please please please ever publish that "insane" (your word! Your promise even) video about the Galois theory. I've really been waiting for it for a very long time now. You promised it after all. You did that in your video about the cubibal polynominals that they didn't teach for over 500 years. Thank you foe keeping your promise in advance. Happy 2025, A.
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  2. General nxn grid is: n^2 + SUM(i = 1 -> n-1) { 2 * (i)^2 }, so it's palindromic
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  3. Burkard, you asked specifically for feedback on how successfully you made these explanations accessible. I am an electronics engineer who has always been fascinated by mathematics, but struggled with the higher, more abstract elements of the subject. My way of learning is like constructing a building: laying foundations and fully understanding foundational material before laying the next layer of bricks. And I always felt uncomfortable just learning calculus identities without understanding their derivation. I think that's one of the big things that held me back. Your videos follow a logical progression, building layers of bricks on top of understanding I already have, and at last I can trust in my gut that these identities are correct because you have shown me, even if I cannot repeat the derivation myself. A few times in this video I had to stop, rewind and check some simplification or rearrangement you did, because it wasn't immediately clear to me, but I got there. Thank you for making this fascinating subject accessible in a way I have never seen before. You have rekindled my love of higher math. By the way, I much prefer the taurean version of Euler's identity: e to the i tau equals one, and I think your beautiful graphical demonstration could easily be adapted to demonstrate this form. I'd love to see your taurean conversion of other identities, which will be similar but I think lead to a far greater intuitive resonance than using pi.
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  4. Wow, damn, I sorta figured out a couple of these things for myself when I was in school (and later on in college) but I was never explicitly taught any of it! It's so cool to see that my thoughts about "wait, is 2^x a kind of discrete version of e^x?" were actually a thing that people had studied and wasn't just a weird quirk!
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  5. Best math channel on youtube by a total landslide, there's simply no competition. Keep up the amazing work Burkard! but I do hope that you don't work too much on youtube to the detriment of your own research (or worse that you burn yourself out); you're doing such a great and important service to math by inspiring so many future mathematicians and showing them ways to think mathematically that almost no degree would ever teach them, but it is crystal clear that you yourself are more than capable of contributing every-bit-as-important new results to mathematics - it is my opinion, and I'm sure also many others', that you are one of the best mathematical minds alive in the world today, and that you can achieve something big (yes, big big , but we know it's not winning a million dollars that you care about :P) Merry christmas! - we wish you the best combination of luck and continued improvement in skill, in choosing your mathematical battles; may you make major breakthroughs in the new year or beyond and push your field forward. I have zero doubt that you'll do us all (and most importantly, yourself) proud!
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  6. I love Zagier's sentence, even without the windmills. It serves as a great exercise in reading proofs. If I ever teach one of those "intro to proofs" class, I would assign the task of deciphering it as some sort of class discussion for the day.
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  7. 27:46 when he paused, I was already like 90% sure he was going to say Euler. If you're ever on a quiz show and one of the questions is "Who first proved X?" if you just guess Euler you're probably good.
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  8. I'm a simple math lover. Mathloger uploads. I click.
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  9. Maths.... is the most beautiful thing ever. So subtle, so abstract and so transcendental... it always existed and will always exist. What we know is a tiny fraction, the most will always be beyond our comprehension.
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  10. In recent years, when I buy shoes they tend to be laced in strange ways so that I always have to relace them (to criss-cross lacing). I wonder why this is. Are there lacings that can be done FASTER in the factory? Have you ever considered speed of lacing as a criterion?
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  11. As soon as you moved the negative fractions below the top line, my first instinct was "Wait...isn't the top part 'outpacing' the bottom part?" Then I lost confidence when you collapsed them, lining up all pos and neg, lol. I was like "but, but, but...." Anyway, I love that stuff!
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  12. I remember when I first noticed the derivative relationship between volume, surface area, and circumference of the sphere. It’s been over a decade since my first calculus course but it’s still so satisfying. When you consider that the derivative is a rate of change, the relationship begins to make perfect sense. I think what really made it click was when we started doing integrals. If you integrate the circumference from r=0 to r=R, you wind up with a sweep of concentric rings that cover the whole area, meaning that the rate of change of the area is given as the circumference as r is varied. Similarly, as you integrate the surface shells across the range of r- values, you wind up with the whole sphere! It’s brilliant once it all clicks and you can see those animations in the mind 😊
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  13. Pilot's computer (E-6B) - which has many other functions as well - including an analog vector adder. When using a slide rule, always estimate the value in your head first to avoid factor of 10 assumptions.
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  14. Typical Fermat. Claiming he has proofs but not delivering. Unlike Mathologer of course 😜
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  15. It also visually shows that log(0) doesn't make any sense, which is cool!
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  16. Hey, I reverse engineered the progression of XP required for each level in final fantasy 5, using this technique (I didn't know about the Gregory-Newton's formula but somehow I figured a formula).. I was 13 at the time or something. I wanted to make a game and for some reason the XP progression was going to be a big part of it. (I ended up making a small dungeon in rpg maker) It turns out that the XP progression is a polynomial, but the last row is a bit random - instead of being the same number (and thus the next one being all zeroes, and the other all zeroes too, etc), eventually it had a random noise of 0, 1 and -1. I figured out that this meant the coefficients of the polynomial weren't perfectly integer, and rounding messed up the differences.
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  17. For the hexagon puzzle: looking at the picture as a 3D stack of blocks, it is obvious that each tiling can be gotten by adding one block at a time. This corresponds to rotating a hexagon with side length 1 by 180 degrees. Thus the number of tiles in each color doesn't change. But the numbers are equal when there are no stacked blocks.
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  18. It always irks me when people teach the subtraction rule and the quotient rule as separate unique rules. The subtraction rule is the addition rule and the constant multiple rule combined. After all, f-g is simply f+c*g where c=-1. Same rules, no need to make a special subtraction rule. The quotient rule can be explained by the product and chain rules combined. f/g = f*h (product) where h = g^-1 (chain). I just dislike having to learn special cases when they're not at all special cases.
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  19. Best one Ramanujan's explanation I have never seen before . Awesome MATOHLOGER .
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  20. I made it to the very end, Burkard. And I don't regret it.
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  21. I can't believe it's actually finally here. I've been waiting for this since the joke at the end of pi is transcendental proof. Thank you, Mathologer. You've somehow outdone yourself yet again!
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  22. First the Mathvengers: Eulergame, then Numberphile hit pi million subs, and now a new Mathologer video?! Is it Christmas? Oh wait, it is.
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  23. "Things are even much hairier in Australia than they appear at first sight." At first sight of this video - it looks really rather un-hairy, I agree... ;)
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  24. A truly beautiful visual and mathematical feast. The complicated symmetry is an extra detail that (for those of us obsessed with symmetry) elevates this to astronomical heights. Thank you, Burkhard for reminding me how much I love Mathematics. A special round of applause to the crystal clear animations.
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  25. Another good one Mathologer. The closing moments were the most important IMO: aspiring mathematicians and those interested should always remember that they're free to redefine expressions and loosen axioms depending on their area of work - much new material can be discovered in this way (e.g. NE Geometry). Similar arguments made for defining what 0**0 should be, and different answers depending on who you ask of course. Perhaps even video worthy :) Thanks again, have a good one.
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  26. I'm surprised there aren't more comments about how your shirt literally says "To infinity and beyond" in math geek. At least, I think it does?
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  27. I cant believe I hadnt heard of Quadratic Reciprocity considering my honours dissertation was on finite fields. Granted, it was efficient computation of matrix opperations under GF2 on GPUs, but still, I cant believe I'd never come across any of this.
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  28. I'd love to see you do a video on dual quaternion skinning. It'd be combining topology, N-D transformations and general matrix/linear algebra. Alot of bad resources on it not really well explaining it's power.
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  29. I taught woodwork for years, in good selective schools and there were always pupils that thought they could build a hollow cube out of six equal squares of plywood. My boss would actually cut them the squares and watch then trying to assemble them. I regret that it did not occur to me to tell them it was impossible because of Fermat's last theorem, that would have been cool. Martin Gardner showed a nice dissection of a 3, 4 and 5 cubes into a small number of pieces that could be assembled into the bigger cube.
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  30. IMO Mathologer perfectly hits the balance between presenting topic in depth and in interesting way. There are other channels more attractive in form, there are channels discussing math deeper, but here I can follow up what's going on, and I want to know where it's going on.
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  31. Awesome
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  32. I remember translating this theorem's Wikipedia page to Hebrew a few years ago, I came up with the coloring proof. but when I saw the swivel proof I just couldn't help but smile. I liked it better than my original proof. this proof made me smile so much. this always happens in your videos, in what you call "aha moment" I experience euphoria. your videos just make me smile from ear to ear, every one of them.
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  33. Am I the only one to notice the infinite sum on the last slide is wrong? It has the product of natural numbers in the denominator instead of just odd numbers. And it includes even powers of x as well.
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  34. 1+2+3 = the amount of seconds between the release of the video and me clicking on it
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  35. Just solved the 4D cube for the first time thanks to your breakdown video! Thanks! It's been a dream of mine since I first learned about it nearly 10 years ago and never thought I'd ever solve one! Great video!
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  36. 7:00 yes it can. The number ends in 81. That's a multiple of 4 + 1.
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  37. "Coins will be gone soon!" Laughs in Zinc Lobbyists
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  38. You triggered my math competition PTSD 😂
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  39. I can hear the ancient Greeks yelling, "Mathematics is just geometry, man!"
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  40. Finally a math topic I’ve never heard about! Thank you Mathologer, you’re great
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  41. My wife called me a nerd when she caught me watching this! 🤓
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  42. Thank you from the heart. You have such kindness, generosity, and humor in your lectures. I trained to be a mathematician but realized that I didn't have any real talent, so I became an Engineer ( all three). And tutored math in my free time.
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  43. Actually, here in Russia we are in fact taught Heron's formula at school, but I'm not really used to it since it was a lockdown year when we were taught this. So if a problem requiring Heron's formula to solve it occurs, I'm always a bit perplexed as it doesn't come to mind at first.
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  44. I have to say that you're one of the few people I know who actually pronounces other mathematicians' names correctly. I appreciate that you take the time to get them correct - it not only shows respect for those mathematicians, but also saves me from having to hear someone say "Weierstrass" with a W or pronounce "Weil" as "Wile" again. Thanks for paying attention to those little details!
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  45. Brilliant
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  46. I rarely ever comment but my appreciation for these master classes merits a comment here. Other channels either prove easy results that look pretty in animation or mention crazy results without giving proofs. What I like about these videos is that they actually take the time to prove those crazy results. I would love to see any videos you may post on Galois Theory or any other weird topic only math insiders get to see and that other channels say "so and so proved this 100 years ago but the proof is really hard"
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  47.  @Mathologer  It does rhyme! and it is deliberately written in the form of seven-word-poem (much like the solution poem to the Chinese remainder theorem). On this same topic, there's another, much more ancient (and famous) "poem" on just the 3x3 magic square: 九宫之义，法以灵龟，二四为肩，六八为足，左三右七，戴九履一，五居中央 Translation: """ The way to fill a 9-square palace, is to imagine a turtle (back): 2 and 4 as the shoulder, 6 and 8 as feet, 3 on the left, 7 on the right, 9 as hat, 1 as shoe, and 5 in the middle """ This text came from an ancient manuscript, that says these numbers/pattern comes on a turtle, and it's a sign of miracle. I guess that's part of what you said in the video: some people think magic square is truly magical.
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  48. Just in case nobody else has pointed it out, may I add that the frictional force between the elastic band and the wheel increases exponentially. This is why cowboys in westerns sometimes do not tie their horses to the rail. A few turns around the rail makes the horse just as secure.
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  49. 28:12 The error is in the final term for the sum of ninth powers, which should be -3/20 nn.
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  50. Don’t…..blow my mind like that
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