Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. For me is impressive how almost everything can be explained geometrically but commonly isn't thaught like that, even when I find it more beautiful and comprehensible
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  2. When dealing with actual shoes what goes commonly overlooked is the order in which the laces overlap at the intersection. It matters immensely. For experiment tie your shoe with standard zigzag but make sure every overlap lays right over left. Walk around for a day and then try again with overlaps left over right now. You'll notice they pull vastly differently on the foot. The lace overlapping the other lace is free to adjust slightly while pinning down other down restricting its movement. I have a bony knuckle top cent of my feet so I make sure the overlaps at this spot are laced a certain way. It's so much more comfortable than overlapped the other way. All this while never changing the actual layout of the lacing.
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  3. "And mathematicians wonder why people think they're weird." My mother was a singer, actress, secretary, homemaker, and social butterfly. NOT a math person (that was my dad). One day, I was trying to explain a math problem to her and I needed to pick a small number to use in a simple example. So I said "How about 1?", and she starts laughing. "Why 1? It's so small! Why don't you pick a REAL number?" 😄
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  4. The last remarks hinting to asymptotic series had me on the verge of my chair while watching the video. I am a physics PhD student and your "masterclasses" are gold to me even if they have nothing to do with my research. The amount of care, precision, passion and entertainment that is present in your videos is outstanding; making such complex and rich subjects accessible to the amateurs and casuals as well as interesting and non trivial to the experts is a miracle made possible by your talent and commitment. I am not saying this to appear sappy, it is just that I believe that it is priceless the fact that I can have access to high-quality content like this for free. I just want to show appreciation for the creators that spend time and energy to put together a video like this one. Thank you very much and keep up the good work!
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  5. Donald E. Knuth calls this "finite calculus", as opposed to the usual "infinite calculus" that is commonly taught. His book, "Concrete Mathematics: A Foundation for Computer Science", uses this "finite calc" to tackle subjects such as hypergeometric functions, generating functions and asymptotics, and derives lots of analogies between the two variants. For example, establishing a power rule, an analogue to exponentional and logarithmic functions and even a "summation by parts" technique. Finite calc is a powerful tool that lets us work wonders, and makes lots of sums we usually cant tackle, easily reducible. I'd say check the book out!
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  6. I found calculus to be really easy when I first learned it, but it was always the algebra that held me back. Just as they say, people take calculus to finally fail algebra.
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  7. Wow… this whole finding differences of a set of points of a polynomial is something I figured out in 11th grade. I made this program on my calculator that you input a table of points and it will calculate the area under the curve, and I found as you say, for ‘well behaved nice functions’ it was super accurate. Kinda proud I was able to figure this all out with only knowing very basically calculus at that point in my life.
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  8. I'm here to brag about my bragging rights for choosing yellow arbitrarly
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  9. the rubber band explanation for logs clicks so naturally, in a way that few other explanations do!
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  10. As a German with a rather casual interest in mathematics, I just want to point out how nicely Moessner presented his idea in the original text. No overstatement, no attempt at making it seem more complex or grandiose than it is, and a straight to the point, easy to understand text. It's literally just "here is a generalisation that does not seem to have been noticed so far, have a look". Great text and an awesome video!
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  11. Unlike mathematicians, real-world painters know that the area they can paint with a given volume of paint depends on how thick they slather the paint on the surface. Volume = Area x Height. They also know that they cannot spread the paint infinitesimally thin—otherwise just one drop of paint would be more than sufficient to paint everything. So no paradox for them.
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  12. I was not that surprised by the square pattern, nor the linear one. But I was truly amazed when he pointed out that in the linear pattern the natural numbers appear exactly one by one 😇
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  13. 9:39 I love this thing about 2^n being equivalent to e^x. It's as if someone came along and very simplistically assumed that "we're dealing with integers, so we should round e down to 2". It's crazy! :D
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  14. A year ago I left a comment on one of these video's saying I was so inspired I was going to make my own math education you tube video's. I have something very special for everyone coming very soon, it's a free software project that I created while working on a tool to make animations for my video's and is almost ready to be released. I just published the first video on my channel, check it out!
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  15. There must be millions of people who have heard of the Tower of Hanoi puzzle and the simple algorithm that generates the simplest solution. But what happens when you are playing the game not with three pegs, as in the original puzzle, but with 4, 5, 6 etc. pegs? Hardly anybody seems to know that there are also really really beautiful solutions which are believed to be optimal but whose optimality has only been proved for four pegs. Even less people know that you can boil down all these optimal solutions into simple no-brainer recipes that allow you to effortless execute these solutions from scratch. Clearly a job for the Mathologer. Enjoy :)
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  16. Feedback: I've just finished a master's degree in mathematics, now starting with a PhD. Nevertheless, there are many things I'm learning from these videos. It's cool seeing recreational math videos reaching this level.
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  17. I think your visual derivations of the fundamental mathematical concepts is amazing. It helps the people who struggle with visualising the algebraic operations with something visually tangible and making connections which were forgotten by the maths teachers.
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  18. Well, after that one, I (probably irrationally) suppose mathematicians tend to avoid too much sun exposure cos tan is a sin.
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  19. "I made it to the very end."
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  20. Most memorable part: In university Mathologer apparently came up with an original finiteness proof for Kempner's series, and the grader failed the homework because they couldn't be bothered to check a solution that was different from the one on the answer sheet.
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  21. 40 minutes of Professor Polster + links to 400 hours of reading = 1 helluva education Every single time, thank you for all you're doing, Mathologer!
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  22. "Are there any other special numbers?" Well obviously 2 works :p
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  23. Truly the man who knew infinity.
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  24. I will never tire of 99999999 in a strong German accent
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  25. I have indeed been taught this at school. Not in the math class, but during Latin class. Our Latin teacher also knew ancient Greek and had found the Greek text about this interesting enough to teach us about it. I found it very enlightening.
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  26. Greetings from Melbourne. For a change I am posting this video at a reasonable time, 8:51 a.m. on a Sunday morning. We are still in lockdown around here, but things appear to be improving: 63 new cases. There is a very interesting footnote to what I am talking about today contained in the description of this video. Check it out :)
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  27. always something uncanny about that ai generated yassified sophie germain
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  28. This video makes a lot of cents.
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  29. The good, the bad and the even
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  30. 42:40 "To be continued" I see what you did there.
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  31. I have made a career of mathematics, but these videos make me feel that childhood joy of mathematics all over. Thank you so much for making these.
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  32. 17:40 "This recipe also solves another Hanoish puzzle." I think the correct adjective is "Hannoying".
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  33. For the record, this is also a path to discovering derivatives and calculus. For many reasons it's a shame this wasn't stared at harder a while back. I wonder what we're just a few steps from discovering
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  34. One neat fact that was left out is that when the four numbers in the box are viewed as fractions, the two fractions are equal to the tangents of half of each of the two acute angles of the triangle.
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  35. RIP conway, he will be missed
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  36. It seems a miracle indeed that anyone would do a half-hour video on the Moessner Miracle, until you read the name Mathologer.
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  37. I'm almost 40 and back in school to become an engineer, not having touched math since my teens. This type of visualization is so helpful getting my old brain back on track! I have a feeling I'm gonna be replaying this video a lot 🤓
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  38. as easy as proving fermats last theorem, walk in the park.
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  39. Mathologer often uses the expression “mathematical spidey sense”. He is right. Math is not invented or discovered,; it is sensed.
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  40. First challenge: no, you can cut out the angle of the board
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  41. This was a foundation subject taught in actuarial studies long before the advent of PCs and spreadsheets. Much of the early work of actuaries relied upon such techniques to analyse empirical mortality and morbidity data in life insurance. It’s actually a cornerstone of a broader field called numerical analysis. Another long forgotten subject that might merit your attention is spherical geometry, the basis for navigation and astronomical work. It’s parallels and extensions to Euclidean geometry are fascinating.
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  42. I wasn't even taught the divisibility test, let alone the remainder function! This is really cool!
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  43. Super interesting! Slight error: @6:11 you go to 3.18 on the diagram (each sub-sub-tick is .02).
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  44. Conway’s circle has the larger diameter. The “diameter” of the shape of constant width is exactly one of the chords we were discussing. But in Conway’s circle, those same chords are not diameters, so are less than a diameter, so the circle has a greater diameter.
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  45. When I saw Burkard with hair, my head turned into a black hole.
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  46. This channel has brought me intellectual ecstasy for years
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  47. <3
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  48. Cube time! I stopped the video around 5:00 where you said those slices don't fit nicely around the cube. Well I say they fit pretty good if you just grab a couple of little unit helpers. So we get two 1^3 helper cubes to add to 5^3 + 6^3, then a 5 cube plus six 6x6 slabs plus two unit helpers in the corners we get the 7 cube putting everything together! So 2*1^3 + 5^3 + 6^3 = 7^3. So lets go further. Following from our two previous families, the central pattern for cubes looks like its gonna be 6=6*1, 18=6*(1+2), 36=6*(1+2+3), etc. So lets look at the second one. 16^3 + 17^3 + 18^3 ~= 19^3 + 20^3. We're gonna need more helper cubes, this time two sets - one for each big cube on the right. Putting stuff together we get 2*(1^3 + 2^3) + 16^3 + 17^3 + 18^3 = 19^3 + 20^3 makes perfect fit! Now it looks like our helper cubes have a pattern very similar to the central number: 6*1 <-> 2*1^3 and 6*(1+2) <-> 2*(1^3+2^3). So now I'm gonna go right into the third cube equation with central number 36=6*(1+2+3) and say that we need 2*(1^3+2^3+3^3) helper cubes. So just jump right to the equation 2*(1^3+2^3+3^3) + 33^3 + 34^3 + 35^3 + 36^3 = 37^3 + 38^3 + 39^3, and it works! If you want to keep the Christmas tree aesthetic, we can call our helper cubes the ornaments :)
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  49. I'd be really interested in that "infinite soldiers and moves" solution
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  50. You know it's going to be a great video when you're only one minute in and you've already seen three excellent results.
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