Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. In 1967, I took an Engineering Measurements course at U.C. Davis. One of the measuring tools was a polar planimeter which allows you to find the area of any figure by tracing its perimeter. If you have a plot of land drawn on a map. you can easily find the enclosed area. This is a very practical application of Heron's Formula.
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  2. We were meditating over these principles in 1988, when I was in 7th grade and we demonstrated tons of problems around that. How was this discovered only 10 years ago?
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  3. If there is any mathematician here that wishes he or she could go back in time so they could discover proofs for fame consider this: All of those past mathematicians would trade places with you in a heartbeat to reach the level of understanding we have now of maths and the world. We have these proofs, because of their frustration of the ignorance of the population around them.
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  4. Look up "Kitab ul Hind". It's a book written by an Arabic scholar when he was visiting India (India under the muslim turkic occupation). He translated knowledge into Arabic and took it back to the Islamic world in the 11th century. In fact, the so called "Islamic golgen age" wouldn't have happened if it weren't for their findings in India. Europe took that knowledge from the Arabs in the coming centuries.
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  5.  @Mathologer  yes I do. Because I love to make experts correct when they do something wrong. But man this video just blew my head. But my wit and will to make experts correct is damn huge, so I will continue to work on these summation formulas even though It will take a Quadrillion year! BTW Congratulations to Mathologer for hitting 500K. You know I really don't get sick of watching your video till end. I enjoy it. And trust me I mean it from my heart. !!! LOVE YOU MATHOLOGER!!!
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  6. The movie "Fermat's Room" is indeed excellent, I'm glad it's getting a bit of publicity!
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  7. A very weird rabbit hole I went down related to the four bugs problem: The function that gives the path length in terms of the step size is f(x) = x/(1 - sqrt(x^2 + (1-x)^2)), and we saw that the limit of f(x) as x -> 0 is 1. What I did was look at the Taylor series of f(x) around x = 0 and what I noticed was the terms up to x^15 all had positive coefficients, but from x^16 on the signs of the coefficients are periodic with period 8 (starting from x^19 the pattern is 4 +'s then 4 -'s). I think I understand why it's periodic with period 8; it has to do with the roots of x^2 + (1-x)^2 being (1+i)/2 and its conjugate, where the angle corresponding to (1+i)/2 in the complex plane is 1/8 of a turn. And I also sort of have a formula for the coefficients: if the Taylor coefficients of 1/sqrt(x^2 + (1-x)^2) are a[n] and the Taylor coefficients of f(x) are b[n], then b[n+1] - b[n] = a[n-1] - a[n] + a[n+1]/2. But the a[n] sequence itself is probably not expressible in closed form so I'm not sure how to get more info on b[n].
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  8. The vid has been up for 15 minutes. 1000+ views. 100+ likes. Just an ordinary fantastic Mathologer vid.
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  9. I like your calendar idea :)
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  10. Same energy: why does a sidereal year last an extra day compared to a solar year? You have to count the extra rotation of earth that results from its revolution around the sun. Or, the earth doesn't have to spin quite one full turn around its axis per day, since it will move around its orbit and make up the rest of the rotation (about 4 minutes per day)
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  11. I think the proof that the rational numbers (or fractions, actually, and thus rational numbers) is countably infinite is one of the most beautiful. But it's not included in the paper! What a travesty.
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  12. this channel is absolutely fantastic!
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  13. Thompson's book is superb. It's like being led through the steps of calculus by a friendly Victorian uncle
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  14. I'll call it a Parker Anagram.
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  15. The toy maker himself wouldn't expect this video to come out one day during that time
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  16. 0:35: "We're always on the lookout for the mathematical sole of things"
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  17. The -1/24 doesn’t really do much of anything anyway so let’s throw it out! Ha ha Ha ha ha.
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  18. Once more a greatly enjoyable introduction to a topic that is tangentially relevant to a lot of my mathematical interests but I knew little about. I can't wait to watch what you Mathologerize next (maybe some simplification of Mihăilescu's theorem? A big ask, I know)
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  19. If you don't already know that 10² + 11² + 12² = 13² + 14² then you can still calculate it quickly by noticing that it's approximately 5 x 12² = 720, plus an "error term" of (13-12)(13+12) - (12-11)(12+11) + (14-12)(14+12) - (12-10)(12+10) = (13+12-12-11) + 2*(14+12-12-10) = 2+8 = 10. In fact, in general, (a-2)² + (a-1)² + a² + (a+1)² + (a+2)² = 5a² + 10.
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  20. I have had difficulty with arithmetic since I learned to count. As a result I looked for patterns in numbers to make life easier. I learned the adding digits to find out if a number was a multiple of 3 and that a multiple of 9 always has the digits add up to 9. In fact I used a similar trick to add up a series of numbers from 1 to n by noticing that 1+n is the same sum as 2+(n-1) and so on...... I learned later in life that Gauss figured out this trick at age 9. I figured it out at age 11.
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  21. This day just got a lot better 🥰
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  22. The variations on the Harmonic series were definitely my favourite - who even thought to ask such a strange question as "What's the Harmonic Series, but if you remove all the terms with a nine in them?" It would never have occurred to me to ask a question like that.
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  23. 9:40 I don't know if I was taught it, but it always seemed trivial that this could be done from the moment I first heard about all this.
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  24. 29:49 The width of the white rectangle is sqrt(2) - 1 . Its height is 1 - (sqrt(2) - 1) = 2 - sqrt(2) = sqrt(2) * (sqrt(2) - 1) . This height divided by the width is: sqrt(2) * (sqrt(2) - 1)/(sqrt(2) - 1) = sqrt(2) .
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  25. Hi, I learned it in school in the GDR, in an extended math class in the begin of the 1970ths. It was rather interesting and fascinating.
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  26. just starting to really enjoy maths and discovered it as one of my passions, (doing hours everyday), your videos are very fun and useful! much appreciated!
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  27. 15:59 What is sed zeven? :)
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  28. As year 10 students were explicitly mentioned:- First the horizontal strech by a factor of 2, means that the input x was changed to x/2, hence plugging that in gives 2/x. The vertical squish divides that by 2, leaving us with 1/x.
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  29. 2 minutes into the video amd here's what's on my mind: the rule is the same as the average modulo 3, meaning (x+y)/2. If you assign a number to each color, and use this average, the answer will be the number for the corresponding color. For example: (1+0)/2 =1/2 = 2 (inverse of 2 modulo 3 is 2). Average of x and x will be x. This also reminds me of tricolorability of knots. (Or whatever it's called. I forget)
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  30. The amount of effort you put into making these animations are truly astonishing.
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  31. ich liebe dich
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  32. Sir please make a video on collatz conjecture and staircase paradox
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  33. This is the most mind boggling video ever made on this equation. and definitely a masterclass. How Ramanujan thought this is simply impossible to fathom. It appears that when it comes to Maths first comes god and then comes Ramanujan.
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  34. 4k-1 would have been more elegant than 4k+3, IMO
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  35. Something I was waiting for but wasn't mentioned: the horizontal symmetry, which can be explained by simply numbering the points with negative numbers going anticlockwise.
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  36. If mathematics is one side of the video, the music is the other. Thank you for both astonishing mathematic topic and making me discover so great music. <3
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  37. Her hair isn't even curly 😭
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  38. Lol I stumbled into it because I on and off for a few years looked at varying degrees of triangular sums and their closed forms. Of course I didn't arrive anywhere from that other than just saying "oh well isn't this neat". I eventually, looking at matrix algebra, blundered into the idea of using systems of equations to determine coefficients for the general polynomial solution which disappointingly contented my former self. Faulhaber's formula is what you want to look at if you want all the closed form solutions for polynomial sums/series. It's interesting.
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  39. Proud to b Indian❤️
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  40. It's uncountable because for any arbitrarily large integer n, there are 2ⁿ ways to pick the first n terms (either positive or negative) and then the remaining terms are fully determined by the over/under method.
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  41. no one expects the spanish inquisition😂
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  42. Thanks for the video! One more bump you might want to mention is that we're assuming continuous functions and sometimes assuming continuous derivatives. Works great for the Mathologermobile, which I assume can't teleport or go from 0 to 60 in zero seconds.
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  43. Exactly! I am almost 50 years old, and I can distinctly remember playing with these 'folds' in the EXACT same way as shown in the video when I was around 10 years old with my own arts and craft projects at home. In fact, I might still have it laying around somewhere in some boxes at the attic. I suspect when he says 'discovered' he actually means either A) officially described in some math paper, and/or B) a proof was found. Which are VERY different things than 'known/discovered' .
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  44. Really enjoyed it the first time around and looking forward to a second. Is junior Mathologer a grad student or your son or maybe even both?
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  45. I am a simple man. More Mathloger is better than less Mathloger.
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  46. Hey, I won gold because of this channel (long story lol), and have a suggestion for a small addition to the part two. The correspondence between the map of Lambert and Kepert is done by taking circle segments between two points, and varying the "angle" of the circle segment. 0° is a line segment, and 90° is a halfcircle, like used in the video. One of my own proofs, of the cylic quadrilateral angle theorem (that has undoubtedly been found by someone else as well) is that given a quadrilateral, you can look at the line segments like they are circle segments. The "circle segment quadrilaterals" have invariant α–β+γ–δ. Since you can merge two pairs of circle segments, you basically directly get the theorem. This even works for hyperbolic geometry! It is a nice proof, using unorthodox techniques, so it probably has to show up on this channel eventually, although it likely won't fit. It also has a dual theorem, where the opposing sides of a quadrilateral sum to the same length if and only if the quadrilateral has an inscribed circle.
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  47. Never thought I'd see such a detailed video on this topic. I've heard a little bit about all of these concepts (Aztec squares, Kesteleyn's formula, rhombic tilings, etc.) while watching Federico Ardila's great lecture series on combinatorics on YouTube, and I really think the accessibility of this subject benefits from visual-oriented, thorough, and intuition-driven videos like these. As always, great video.
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  48.  @Mathologer  Sure, just checking us. There goes my hypothesis that you started writing 120 degrees, then realized you'd basically already written it since cos 120 = -cos 180 - 120 = -cos 60, but left a half-written entry which got merged with the entry for 180 degrees.
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  49. Once you mentioned the higher dimensions I expected the Pascal triangle will pop up somehow in this videos.
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  50. When I was a kid, I created my own lacing, where each eyelet pair had one string going across, out the first eyelet and in the corresponding one, and that was all that was visible - no diagonal lacings. Everything else was under the eyelets and did not cross, and were thus invisible. I was very proud of it. I still lace my shoes with this style.
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