Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. 8:07 If you can make a 12 pointed star with this, you can design a CLOCK with this fancy movement!
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  2. To take it one (very small step further), this shape of constant width has the smallest diameter of all shapes of constant width containing all 6 points!
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  3. Me every time a Mathologer video comes out: visible excitement
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  4. Your explanation did work on me prof. Burkard. My background is engineering, and I played a lot with Optimization, Partial Differential Equation and Numerical Methods. The 50 mins long does not bother me. I believe 50 mins is just right to sums up millenia of math discovery. The moment you said 10 chapters really sparks my joy, and finishing the whole video is really satisfying. This is indeed a Masterclass professor. Can't wait for the next. Thank you very much!
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  5. Your videos never disappoint, great work. This proof reminded me of that for the Pythagorean theorem that we take 3 copies, scale and bring together.
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  6. 3,4,7 is actually really useful in music. Intervals of 3 and 4 are periodic complements in mod 12, because they also function as the smallest interval in basic trichords, intervals of 7 are the most common. I often visualize the structure of chords in a way similar to the graphic representation used in this video. The triangles and squares form a continuum that you can rotate along to find relatively valent harmonic structures in a set of closely related keys.
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  7. Жаль, что я не нашёл этот канал раньше. Это просто потрясающе! Жаль, что у нас в универе нет таких классных учителей по матеше))
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  8.  @Achill101  That is correct, but since taxes come out first the government doesn't have to make any allegation, investigate anything, or prove anything. They just take the money. Non-governmental creditors can avoid fleeing by this scheme only if they take actions and make investigations.
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  9. I enjoy partitions as one of the many studies in mathematics that can get mind-numbingly complicated, but starts from a place an elementary school student can understand. Amazing.
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  10. 15:50 The missing chunks look approx triangular, so I calculated the size of the missing triangles, and used that to find a correction factor for the (1+1/n)^n approximation of e. If you multiply that approximation by (2n+2)/(2n+1), you get a much more accurate approximation. For n=100, (1+1/n)^n has an error of 1.35%, but ((1+1/n)^n)*(2n+2)/(2n+1) has an error of just 0.001%
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  11. Thanks for crafting such a lovely video. The hours spent in photoshop yielded a beautiful result, and the delightful subject matter made the video a joy to watch.
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  12. ​@muskyoxes That established the existence of photons and was instrumental in the foundation of quantum mechanics. You know, that field of physics which says that God plays dice with the Universe.
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  13.  @funkybibimbap6972  Really? Even with the irony contained in my final sentence?
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  14. I don’t understand why, when you are explaining how to grow a random tiling at 22:41, you split up the empty 2x4 region into two 2x2 regions. Wouldn’t there be more possible tilings if you were allowed to also split it into 2x1 + 2x2 + 2x1? It seems to me that there would be. And while the tiling you get would be identical in tile positions to one you could have gotten another way, the arrow directions would be different and they would grow into different tilings at the next step.
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  15. Another nice trick: If you make any move, you can just repeat the exact same move and you will always get to the starting position. That way you can develop your own algorithms too although they ore often quite long and tedious
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  16. I used my maternal uncle's slide rule at school: 15" boxwood with engraved ivory scales. I believe it goes back to the early 20th or late 19th century. The case was made in Harland & Wollf's shipyard in Belfast, in "repurposed" steel, probably in the early 1920's. I still have it, but I haven't used it for ~60 years.
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  17. Some energy absorbing crushable elements in cars, called 'crush cans', are designed to buckle like the tin can shown, even to the point of having the first band of triangles pre buckled to make sure that the buckling progresses in the correct manner.
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  18. As you started explaining the "decomposition", I was reminded of fourier transforms. However, when you then mentioned that this in fact is a fourier transform, I was quite surprised. Not at all obvious that these two things really are the same (to me anyway). Fascinating!
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  19. Very interesting ! Let's try this : Assuming there's really a Planck length in the Universe and you work with real world coordinates, you couldn't keep shrinking the polygons forever without them converging to a single point. Would that mean there's no such thing as a square grid in the Universe or that triangles are not a thing?
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  20. It’s amazing how algebra and geometry can be connected by such a pretty formula. And the derivation using recurrence is simple and… simply stunning.
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  21. pretty sure everyone who ever played around with a casio in their school life will know what the 36th triangular number is. (since when you add all numbers from 1 onward, exactly 36 will fit on most casios, except for newer ones. They allow for more)
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  22. 2nd problem is 3^n because there are n discs, each of which has 3 'choices' of pegs - same as the number of vertices on the sierpinski triangle 3rd problem can be done by comparing coefficients of 12/59 and 18sqrt(17)/1003, upon which irrational parts drop out - not sure I want to do this.. -btw, thanks Mathologer!
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  23. Thank you for the fantastic presentation!! 🙏
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  24. That was a wonderful video. I understood most of the things except 41:01 I'm in class 12
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  25. It worked brilliantly. By now you know how much this means to all of us, but thank you, yet again.
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  26. You are the greatest math teacher on the internet. I am grateful and happy whenever I watch any of your videos. Thank you.
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  27. 00:10 Yes, the overlords will abolish all cash and your digital money will only be in one central bank governed by the oligarchs.
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  28. When I was taught about scientific notation as a schoolkid the slide rule never entered into the picture as to why it was useful, but one of the great things about it is that it removes a lot of the potential for error from the "moving the decimal point" part. By forcing the mantissa (the part that's not ten to the something) to be between 10 and -10 you get a problem that's easy to work out with a slide rule (or if you're fancy a log lookup table) and the significand (the part that's ten to the something) becomes simple addition of the exponents. This is also how floating point works in computer logic, that's just a clever binary form of scientific notation. Multiplying in a digital computer relies on repeated addition plus some bit shifts to double and half the numbers and the size of values has a defined limit in terms of number of bits/significant figures.
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  29. My high school math teacher (this was back in the 60's; I'm 75 now) made and marketed a very nice circular slide rule. His name was Stanley Arlton. I wish I still had mine.
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  30. Cool paradox this also shows why the curve area length formula has to be secant lines instead of just horizontal and vertical. Approaching something doesn’t always mean that they have the same properties
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  31. Now find the ways of making change for a googol euros. (coins denominations 1, 2, 5, 10, 20, 50 cents, 1, 2 euro)
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  32. Nice debunking as usual. Nodded my head the whole way through. Base10 isn't special beyond its human convenience. It's shocking to think so many people can be wowed by lack of connecting some dots in their own basic mathematical knowledge. But I didn't know much about these interesting diagrams, nor had I thought about decimal places of divisions by two as powers of 5. As always beautiful video.
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  33. Yes - I like to think of it as adding a new layer of paint, and adding up (integrating) all the layers. For some curves/surfaces the tricky part is making sure the new “layer of paint” is the same thickness at all points. Measured normal to the surface, that is. It can be a challenge for some curves (surfaces) where the offset curve (…) is a different type of curve to the original, such as for a parabola.
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  34. The 153, 104, 185 triple at 12:00 is the box 9, 4, 13, 17 and you get there by navigating right right right and left :)
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  35. Finally. Amazing.
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  36. Extremely cool! Made me realize there’s a very practical strategy for handling these problems that, in my opinion, beats the pure programming AND pure math approach: programming this sort of thing correctly is nontrivial, and even with a smart implementation isn’t that fast (my quick and dirty python version takes about 20 seconds for $2500 - I just remembered I only bothered with 1, 5, 10, and 25 cent coins). The pure math approach to obtain a closed form expression through generating functions is mechanical and straightforward (and beautiful) but incredibly unwieldy for generating functions of this size. Even if you use Mathematica to do the algebra, you still face the problem of convincing yourself you didn’t make a mistake in writing the code! My approach, inspired by your video, is to simply observe that since the generating function is a rational function, its coefficients will be exponentials times polynomials in the index (and I think a bit more reasoning gets you to the fact that if you only extract coefficients with the same residue mod 100, they’ll just be polynomials (as you show for residue 0, of course)). For N coins, this polynomial will be of degree N-1. So pages and pages of computer algebra, in the end, are just to get N rational numbers! A much more direct way is to write a piece of code that is just barely fast enough to deliver N values of the function, and then solve the resulting linear system for the coefficients of the polynomial! No need for super slick iterative implementations, which will top out way before a million of dollars anyway; no need for huge amounts of algebra. Fast, easy, and delivers an essentially O(1) algorithm for a huge number of problems (ie. anything with a rational generating function, so anything produced by a finite state machine)!
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  37. Just one line is crossing my mind watching this "It's a kind of magic, MAJIK!!!" ✨
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  38. 28:11 "Euler" ...
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  39. Mathologer, I have an argument at my end. I'm saying infinity IS used in mathematics and my friend says it ISN'T, that it has no place in maths. Of course we both understand that infinity isn't a number and all the rest, and I know about square root of -1 (is that imaginary number i? Which is really useful in maths and just as silly as infinity? Isn't zero itself in a similar category?) too. Riemann's paradox seems to be just one of many examples in which infinity is used (in the practical sense?) in mathematics. Please do a video explaining how the layman's concept of infinity differs from or is the same as the mathematician's!
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  40. Wow that blew my mind that Pascal's triangle mod 2 gives the Sierpinski triangle, and mod 3 makes a similar pattern. I wrote a script to play with this more and it seems that mod any prime number it gives a nice Sierpinski-like pattern, and mod composite numbers give more messy overlapping patterns that relate to the factors. It's always amazing when seemingly disparate areas of math connect together in unexpected ways!
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  41. At 12:00 I naturally thought of that formula on my own while I was taking calculus 2.
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  42. 39:16 tile an (ab)x(ab) grid with (a)x(b) rectangles and draw the top-left to bottom-right diagonal. If it intersects the bottom-right corner of a rectangle then a square is formed by (n)x(m) lots of (a)x(b) so na=bm, n!=b, m!=a so a,b cannot be prime.
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  43. This is a beautiful example of SSS congruence / AA similarity
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  44. I have a question regarding 32:19, the challenge at the end. You claim that the existence of integers x,y with x^2 - y^2 = n (> 0, for simplicity) leads to n being odd. As i found the counter example x = 4, y=2 and therefore n=16 - 4 = 12 being not odd , I probably misunderstood you. Any help is kindly taken. Greetings from Germany.
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  45. Your videos are greatly awaited and they are always worth waiting for. Your videos always generate love for Mathematics. I wish I had a teacher like you in the school days. Lots of love and respect to you. Always ❤
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  46. Your work is simply amazing. I like how the difficulty rises. I always have something to pause and ponder. As usual I will re-watch it a couple of times to try to understand the last part.
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  47. Challenge answer: no two collections from {1, 2, 4, 8, 16, 32, 64} have the same sum.
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  48. 5 minutes gang
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  49. Those crazy polynomials are going to hunt me in my dreams
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  50. This is the reason why I'm getting a PHD in mathematics: the infinite beauty of the numbers.
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