Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. That “hexagon exists in a cubic lattice” is why two sorts of crystal lattices in chemistry are identical. I can’t remember which ones, but I think it’s hexagonal and either face-centred-cubic or body-centred-cubic. Also there’s both tetrahedra and octohedra within a cubic lattice, which tesselate with each other in 3D space. The way I’d look at that initial problem, finding equilateral triangles in a cubic lattice, is that all points in a cubic lattice are either 1 or sqrt(2) from their neighbours, and an equilateral triangle needs a sqrt(3) in there. Plus or minus an inverting scale-factor. But on a cubic lattice, the distance between diagonally opposite points is sqrt(3). Not exactly rigorous, but intuitive to me.
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  2. 0:06 Just started and I already pressed like for the t-shirt
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  3. Woah, I was able to guess the shortcut rule! Do I get bragging rights? 🙃
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  4. I the 1950's I was taught to check math problems by something the teacher called "casting out nines." I didnt know why it worked but was intrigued by it. 60 years later I stubble across the answer.
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  5. "Don't worry. Be happy. And let's leave the demons for later." - Life advice from Mathologer, 2019
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  6. So excited when the notification popped out!
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  7. 3:55 omg I just realized that’s a pumpkin pi.
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  8. I learned of Heron's formula when I was about 8 years old from a table of mathematical formulae in the back of a dictionary, but I was never taught it in school, and I spent most of my life being curious about how to derive it in an elegant and geometrical manner. Your video is fantastic, it has unlocked some remaining pieces of the puzzle I had not figured out.
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  9. Wow that last part of the proof was really nice :)
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  10. Some years ago, a company made a perfect mathematical paint as described. Unfortunately they couldn't get it to market because the paint leaked past the molecules of every container they put it in.
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  11. Worth mentioning quadrilateral as a special case - we have 90 and 180 ears to add (180 are essentially midpoints). When we add 90 degrees first we get Van Aubel's theorem as a special case. When we build 180 ears first (i.e. taking midpoints, giving us a parallelogram) we get Thebault's theorem (which also follows from Van Aubel).
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  12. Adding 1 = 1 at the top would be very satisfying. Length of 1. Justified with the sum if a single number = product of a single number.
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  13. Most memorable: that the harmonic series narrowly misses all integers by ever shrinking margins
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  14. I'm doing my part! 😄
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  15. Everyone: maths is boring :( Mathsloger : let me take care of it. ;) Btw your videos are very interesting and full of knowledge...... Love from india 🇮🇳❤❤
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  16. First challenge: The chessboard cannot always be tiled after removing 2 black & 2 green. Suppose we remove the two black squares adjacent to the upper-left corner and the two green squares adjacent to the lower-left corner. Then the left corner squares are no longer adjacent to any other squares, so the board cannot be tiled. EDIT: Second challenge: if m and n are odd, then ⌈m/2⌉ = (m+1)/2 and ⌈n/2⌉ = (n+1)/2. Now the j=(m+1)/2, k=(n+1)/2 term of the product is 4cos²(π/2) + 4cos²(π/2) = 0, so the product is 0. Moreover, if m is not odd, then 0 ≤ j < (m+1)/2 in all terms of the product, hence 0 ≤ jπ/(m+1) < π/2 in all terms, so cos²(jπ/(m+1)) > 0 in all terms, so each term of the product is nonzero. This means the formula gives a nonzero answer whenever m is even -- symmetrically, the answer is nonzero whenever n is even. Thus, the formula returns 0 if and only if m and n are both odd.
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  17. I thought this vid was going to be more than 30 minutes long, but it's still nice to have an occasional short video from time to time! :) As for the puzzles: 5³ + 6³ ≈ 7³ (just off by two which just so happens to be the cube root of the next number 8) The answer to the second puzzle is 2 since 10² + 11² + 12² = 13² + 14² = 365 (I knew it was 365 earlier since when mentioning this equation some people like to point out that there are about 365 days in a year!) Funnily enough the sequence in the third puzzle goes 5, 6, 6.893..., 7.80557... which seem to round up to 5, 6, 7, and 8. Now I'm curious to know if this rounding pattern continues or if it's just a coincidence . . .
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  18. I prefer Flammable Math's avoid positivity shirt -|x|
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  19. That's it, I'm stealing that. I'm no longer a Software Engineer, I'm an Algorithmologist.
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  20. quick answer: a lot
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  21. Merry Christmas! I think the easiest way to show pi = lim N(r) / r^2 is to say that the squares centered in grid points <= r units away from (0; 0) must cover a circle with radius r - 1 and center (0; 0) but are contained in a circle with radius r + 1 and center (0; 0). Which gives us pi * (r + 1)^2 / r^2 >= N(r) / r^2 >= pi * (r - 1)^2 / r^2 and since both leftmost and rightmost side of this inequality approach pi as r->inf, we get what we wanted
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  22. There’s a reason that the all-time mathematical greats like Euler, Ramanujan, Laplace, and Fermat were fascinated by magic squares and other patterns! It’s the knack and habit of recognizing those underlying structures which led them to some of the greatest insights and advances in mathematical history. THIS is why it’s so important to teach our kids more than rote memorization of numbers, facts, tables, and theories; we need to teach them how to see them as patterns which lead to other patterns within even further patterns. People with the gift of innate intuition about patterns are the people who change the world.
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  23. Mathologer: explains rules about two colours added -> the third one me: ok, addition mod 3, easy Mathologer: two same colours -> that colour me: hmm me: something something group theory??? Mathologer: -(a+b) mod 3 me: takes maths degree certificate off wall
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  24. I had my worst two weeks lately, and this 30ish minutes was the first I was feeling joyful. Thank you!
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  25. On the rational main-angles in a goniometer. "At some point I'll do a whole Mathologer Video in German. Promised."
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  26. Glasses: +50 iq Nerdy math t-shirt: +80 iq German accent: +6000 iq
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  27. One of the coolest parts of the Gleich paper is that it leads very naturally to the question of how to express normal powers in terms of falling powers; e.g., n^3 = 1 + 7n + 6n(n-1) + n(n-1)(n-2). Equivalently, is there a pattern in the first numbers of the rows of the difference scheme for f(n) = n^k? Go read the paper in the description for the answer!
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  28. A day with a new Mathologer video is a better day
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  29. The disaster has been averted. Glad you’re back!
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  30. What amazes me is just how accessible the Mathologer videos are. I would rate myself as 'clever' up through High School mathematics (pre-calculus), but also a bit lazy and underperforming. I am not a 'math guy.' I remember the general idea of trig but only the most basic identities, a handful of geometric proofs, some of the definitions in calculus but none of the methods or shortcuts. And yet, excepting a very few, I almost always manage to watch these explanations and follow along - delighted and inspired the whole way through. The man is truly a very gifted teacher and a master of this "you-tube" way of presentation. Thank you, Dr. Polster, for leading us on this wonderful journey.
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  31. I really appreciate the "insane" videos. A lot of math content on YouTube is watered down or repeated, so it's nice to see this stuff that I've never heard of.
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  32. What a weekend! 3b1b + mythologer in the same weekend!
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  33. I have never seen any of these proofs - I graduated HS in '56 - got a BSEE in '60 - worked in Aerospace 38 years - retired and became a HS math teacher in 2001 - retired in 2014. Never offered any of these proofs to my students. I think they (and I) were shortchanged.  Love this stuff.
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  34. lmfao I tried to google "Niche method". Thinking this was some mathematician Then after seeing results for Nike, realized the joke. "Just Do It" facepalm wow I'm slower than I thought
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  35. Decades upon decades of maths: agonal spasm over sheets of paper Mathologer: so basically we use Pascal's triangle here and that's it
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  36. This looks highly related to the discrete derivative and discrete integral which is cool stuff..
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  37. I love your Ramanujan inspired TAXI 1729 T-shirt
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  38. Hell ... at around 23:50 I clearly understood (several times) the term 'the center of mess' until I realized that this has absolutely no mathematical meaning. But in that moment, it made perfect sense to me ... the point clearly was in the center of all that mess.
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  39. I find really wholesome this man's dedication to speak and explain so passionly for 50+ minutes straight. As a phisicist that I am, I love how matematicians like this one continiously inspire us all everytime they can. Keep on the good work, stay amazed and happy holidays!
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  40. German mathematician: "Here's another kitten, in a cube. Very cute. Feeling revived?" Quantum mechanics students: "NO ERWIN, PLEASE, NOT AGAIN!"
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  41. I've been tying the "Ian's knot" ever since 5th grade (roughly 11 years ago) and only today I just found out where the name came from.
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  42. "You're here for the mysterious iris in the thumbnail, aren't you?" Sorry, but no. I'm here because it's another Mathologer video :)
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  43. For the bug problem: An equivalent question would be to ask how long it takes for any one ant to reach the ant it is chasing. The chased ant always moves orthogonally to the chaser, so it is neither going towards it nor away from it. Thus the distance between the two ants simply decreases by the speed of the chaser. It thus takes 1 unit of time for the ants to meet.
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  44. I already know Heron's formula and Brahmagupta's formulas but I haven't seen a proof for them until now, also that general formula for the area of a quadrilateral is beautiful, thanks
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  45. Why isn't this in my subscription feed... I had to search up the channel name to find it
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  46. "If you're a 5 dimensional creature, that's a no brainer... Just start by visualizing 5D shells of 5D cubes" 🤯👽🤓
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  47. Wow. Such an amazing and carefully constructed video. Also, the important warning about ‘changing the rules’ shows (to me in my opinion) how seriously and solemnly the ethic of education is handled on this channel. You have a very satisfied and excited new subscriber!
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  48. As an American born-and-raised who was in the public system as both student and teacher… our math education is disgustingly deficient in number theory. High school graduates (even some going into stem fields) do not even know the Euclidean algorithm. They have almost no experience working with modular arithmetic. Too many decades of parents complaining about this “useless” math subject has led to them and their children being mystified by the simplest of number theory diagrams. Thank you mathologer for making so much explanatory content paced for victims of the US public school number theory book banning. (Inb4 some other American tells a story about their one teacher that taught them number theory)
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  49. I searched for Mathologer the other day and noticed he had missed some months of uploading, watched some videos, only to see another upload today. Lovely!
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  50. This seems very similar to second order automata - each cell's value is determined by the three cells over it, and the cell 2 over it.
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