Hearted Youtube comments on Mathologer (@Mathologer) channel.

  1. Whenever you feel marooned in a parallel universe and nothing seems to remind you of Kansas, five words will snap you back into familiar territory, the land of reason and kindness - "Welcome to another Mathologer video!" Or three notes for that matter.
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  2.  @friedrichschumann740  Selbstverstaendlich! 🙃
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  3. "Mathologerization" The act of explaining Math "Mathologerers" People that attempt to follow along, and never give up. "Matholgerized" The Math, explained
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  4. 2:15 Let (a, b), where a≥1 and b≥0, be the lowest side of a square written in vector notation. Precisely, it is the side containing the bottommost (and leftmost in case of a tie) vertex as its left endpoint. This enumerates all possible squares uniquely. There is room for (n - (a+b))^2 such squares in the grid, so the total number of squares is (A2415 on OEIS): sum[k = 1 to n-1] sum[a+b=k | a≥1, b≥0] (n - k)^2 = sum[k = 1 to n-1] k (n - k)^2 = n^2 (n^2 - 1) / 12
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  5. our monthly dose of underrated mathematics
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  6. I'm not a math historian but I believe that it is because the Greeks studied sections of the cone. If you cut the cone parallel to the base, you get a circle. If you cut the cone oblique to the base, you get an ellipse as long as the cut is not parallel with the sides. If you cut the cone parallel to the side, you get a parabola. If you cut it more oblique than the sides, you get a hyperbola. https://en.wikipedia.org/wiki/Apollonius_of_Perga
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  7. For the curious, r and s are solutions to cubic equations, so WolframAlpha present them with formulas involving complex numbers, even when the final result is real.But when asked to simplify, it found a closed form using trig functions: (1 + Sqrt[7] Cos[ArcTan[3 Sqrt[3]]/3] + Sqrt[21] Sin[ArcTan[3 Sqrt[3]]/3])/3
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  8. The pigeon that disliked this video got stuck in pigeonhole
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  9. maybe once explain us the Poincaré Theorem proof :D
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  10. This method likely originated due to lack of proof of claims which would have been very common in the past as documentation was not robust in the past, it assumes that all claims are not completely true and are likely with equal probability and we end up with this type of distribution.
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  11. I'm just simply amazed that it never occurred to me that solving "next element of sequence" puzzles on IQ tests is actually just applying differential calculus. I guess I just never gave it deeper thought... Amazing video, great explanation :)
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  12. 37:25 Proof: Let n be the number of cards, Let k be the number of cards bigger than the last one. By transferring the last card to the other side i eliminated k instances in which sometimes bigger than it is before it but also added (n-1)-k instances in which its bigger the other numbers (because the rest of the numbers are smaller then it). So eventually i just added n-1-2k . Since its a power of two i can ignore the -2k (its even). Now, i am left with just n-1 added to the power. If n is even then n-1 is odd and therefore the sine changes. If n is odd then n-1 is even and therefore the sine doesn't change. [|||]
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  13. Due to covid, I finally managed to have enough time to watch a whole mathologer video :)
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  14. The simple part: any odd number n that can be written as the sum of two squares must be the sum of an even square a^2 and an odd square b^2. Now a^2=0 (mod 4) and b^2=1 (mod 4), so that n must be 1 (mod 4).
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  15. The most memorable part was Tristan's fractal. Fractals are beautiful and they always show up when you expect them the least.
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  16. Bonus problem: Show that the sequence shown at the start of the video (x_k = 1, 2, 4, 8, 16, 31, ...) is the maximum number of pieces that can be formed by slicing any convex four dimensional polyhedroid using k–1 hyperplanes.
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  17. I just spent twenty or so minutes trying to make a number-wall for the partition function by hand. I may have given myself a mathematical concussion in the process😵Great stuff! Maybe next time I'll start with a simpler sequence though 😂😭
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  18. Actually you don't even need to know anything about binary numbers. The proof is equal even if we read the number as decimal.
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  19. I love watching your videos even though this is way out of my field. You make math very interesting and dare I say it? Fun. Thanks for the hard work.
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  20. ​ @Mathologer  Yeah but I heard his disciples never got the angles right...
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  21. i feel like 4k+1 and 4k-1 is much prettier
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  22. Definitely love the short form video option. I have about 50 of your videos in a list to watch later, but I rarely have time for a 45 minute video. (I'm slowly getting through the "mystery of the sums" one... lovely stuff).
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  23. Always nice when you upload a new video!
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  24. Merry Christmas! 6:11 - You could pick the two blacks in the top left corner. It would isolate the corner green square, so not every combination of 4 squares removed is tileable. 10:13 - say m = 2p-1 and n=2q-1. The denominators in the cosines will be 2p and 2q. Carrying out the product, when j=p and m=q, we will have a term (4cos²(π/2)+4cos²(π/2)), which is 0, cancelling out everything else 13:50 - Lets say T(n) is the number of ways to tile a 2xn rectangle. First two are obviously T(1)=1 and T(2)=2. For the nth one, lets look at it from left to right. We can start by placing a tile vertically, which will isolate a 2x(n-1) rect. - so T(n-1) ways of doing it in this case. If we instead place a tile horizontally on the top, we will be forced to place another one directly below, so we don't isolate the bottom left square, this then isolates a 2x(n-2) - so T(n-2) ways of doing it in this case. ---- We have T(n) = T(n-1) + T(n-2). Since 1 and 2 are fibonacci numbers, the sequence will keep spitting out fibonacci numbers 14:33 - It's 666. I did it by considering all possible ways the center square can be filled and carrying out the possibilities. It was also helpful to see that the 2x3 rectangles at the edges are always tiled independently. I was determined to do all the homework in this video, but hell no I wont calculate that determinant, sorry 30:12 - I'll leave this one in the back of my mind, but for now I'm not a real math master. I'm also not a programmer, but this feels like somehting fun to program 37:28 - Just look at the cube stack straight from one of the sides, all you'll see is an nxn wall, either blue, yellow, or gray
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  25. I have a simple solution to who found this proof: By the elimination of variables. As a variable, I eliminate myself. Good luck with the other 7 billion variables.
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  26. solution to there bricks with overhang of 2 units: place one brick with overhang of 1. then place another bring on top of this one all the way to the right with its own overhang of 1 unit. clearly this will fall. now place the third brick to the left of the second, making the top layer 4 units (2 bricks) long, and the bottom layer centered around it. Done.
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  27. Just remember to use multiplied triangle instead of measuring just up to five what ever units you're using. For example measure to 30, 40, and 50. The larger the triangle, the less chance there is for the inevitable measurement error when doing it haphazardly by hand.
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  28. Gotta say, as someone who knows a fair number of artists first-hand, seeing AI "art" being used is always a bit gut-wrenching knowing what they are going through right now because of AI.
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  29. im already a big fan of long-form content in general, but this channel has such a distinct and enjoyable format that it easily sits in the top of the list. instead of continually building to a large and complex problem/explanation, its like you meander through a selection of related small problems that can be appreciated on their own or as a collective. it makes them very nice to watch and rewatch
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  30. And turns out, there is a 3D equivalent for magic squares called a magic cube. Going beyond 3D, we have magic hypercube. How can I never hear of this before? I love that the well-known classic problem has a lot of different variations to tinker about. Especially when each has a unique approach to the problem. The wonder of math always amazes me.
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  31. was this the first STRAND type puzzle?
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  32. HELLO OUT THERE...OUt there...There...ere...
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  33. The level of video and animation is amazing. This is a huge and talented work! СПАСИБО БОЛЬШОЕ 👌
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  34. I don't understand why so many people are afraid of calculus. Calculus, geometry, analytical geometry and linear algebra were the easiest math disciplines I had. Anyway amazing video, I did maths years ago, I abandoned it for software development, but I love how I'm always learning something new with your videos.
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  35. Whenever I feel smart, I watch Mathologer to knock my ego down a few pegs.
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  36. Nice one! I have also done the versions of AM-GM and Cauchy Schwarz like you did here - love this diagram for those visual proofs! Used it to animate Priebe’s and Ramos’ sine of a sum too. Was gonna create a mashup of them to show they are all the same but here you’ve done it better. Thanks!
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  37. 19:32 Can't believe you missed using the color sequence Red, Black, Blue, Yellow, Pink. This would be a delightful easter egg to the Power Rangers fans.
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  38. If the tower has an odd number of discs, first move the smallest to the place you want the tower to end up. If even, move to the space you don't want the tower to end. Then just consistently follow the direction you moved the smaller disc to (clockwise/counterclockwise if arranged as a triangle, to the left/right with wraparound if arranged in a line).
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  39. Thank you the video was fantastic!! I never heard of number walls before, and the graphical presentation was very informative. I look forward to learning more from the links.
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  40. Simple solution for the sum of 7-star angles: 1. Without loss of generality, assume the 7 points lies on a circle 2. Those 7 angles are produced by 7 unique arcs that form a whole circle 3. For a certain arc, the angle at centre produce by that arc is always twice of the angle at circumference (Theorem "Angle at centre is twice angle at circumference") 4. Therefore, twice of the sum of all angles is 360 degrees. 5. Sum of required angles = 180 degrees
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  41. 50 minutes that didn't feel like 50 minutes. I was ready for you to get into the zeta function and gamma (both the function and the Euler-Mascheroni constant). Re: your request at the end: I'm coming up on 30 years out of high school and I haven't taken a formal mathematics course other than introductory statistics since 1992. I got into recreational mathematics thanks to the late Martin Gardner and a few other writers, and then I found channels like yours and 3blue1brown on YouTube a few years ago. Lately I've been pondering the Riemann Hypothesis and the Collatz Conjecture, and I'm currently re-reading Julian Havil's book Gamma, about the Euler-Mascheroni constant. My instinct is that it's transcendental, but I'd love for somebody to prove it. I'm really looking forward to seeing what you have to say about gamma and the zeta function... and take as long as you like on those topics! Edit: Yes, I've seen your previous videos on Riemann and related topics. Also, something I remember discovering while playing with numbers on my own in high school algebra class came to mind. I remember finding that the perfect squares had differences of consecutive odd numbers, and then when I took this to higher powers I found the factorial of the exponent at the bottom of the formula, but I was never able to complete the formula for the full general case. I still have my original spreadsheet file from 1990 with my calculations.
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  42. this is not the reason why this system was introduced actually, remember that those times written records of things were perishable or even not present at all, so the money that was claimed had no solid proof on paper but only testimonies from witnesses, there was a probability of claim being exaggerated so it tries to punish high claims without solid proof, imagine that someone claimed 90% of the money without solid proof, it would be unfair to just equally divide on that basis because the others would only get like 10% of the money without solid proof that this distribution is fair, the talmudic system assumes that claims are not 100% accurate like that in the case of the inheritance problem in the video, its a silly system anyways because claims without proof truly have not real fair way to be resolved.
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  43. That was a wonderfully powerful intro. So beautiful and fully deserving of the absolutely epic music.
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  44. In information theory there are two related problems where the solutions are similar to the topic of this video. One problem is like, you have three information channels with different noise levels, how should you split data to transmit in each channel to maximize the total data transfer speed. The solution is called “water filling algorithm”. There’s a dual problem with the solution as “reverse water filling problem“. Such problems don’t have any game theory flavor and the objective is very clear. The solutions are also analogous to communicating vessels.
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  45. Your concerns are well founded. I was a math and science teacher at two private schools in the United States (each time for two years). The textbook selections never aligned well with my ideals of making math and science relevant, useful and entertaining. For 3 of the four years, I was able to accomplish that in spite of the books. During the 4th year the academic police state finally caught up with me, insisting that the reasons we teach anything are foremost to increase test scores and grow our market share through test-based reputation. That authority banned all non-standard curricula and forced me out of the profession I loved. All the texts from which I taught were ostensibly aligned with their goals. They were also filled with endless drills, BS examples, incomplete history, and frankly serpentine reasoning far more likely to confuse that to convey any valuable understanding. I’ve lived 4 years in Germany (which was a bit better) and 4 more in South Korea (which seemed much better). Unfortunately, the Korean kids were under immense pressure to perform, making almost all of them profoundly depressed and/or stressed from perhaps 10 years of age. Most of the great contributors to the betterment of our world recognize the importance of conquering fun challenges. Both mindless, meaningless repetition and idiotic complication turn beautifully curious and malleable children into miserable adults. When teaching became primarily the indoctrination of future workers, it necessarily ceased teaching young people to think critically and creatively in favor of teaching them to do what those in authority tell them. This bodes terribly for the future and causes me to be deeply concerned for our posterity. Thank you for making learning the entertaining challenge it’s meant to be!
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  46. This is amazing. Everything truly is connected to everything. I could hardly have been more surprised if Pascal's triangle had also made an appearance. 🙂
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  47. It makes sense that the area is the derivative of the volume, if you think that the volume is created by adding all the surfaces of the spheres with smaller radiuses. Basically like blowing up a perfectly spherical balloon. That trick should work for any shape, not only spheres. For example a cube defined by the 8 points (+/-r,+/-r,+/-r) has a volume of 8r^3. If you drive that by r, you get 24r^2 and that is exactly the surface of such a cube with side length 2r.
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  48. It would be nice to see a follow up filling in the details that were left on the table at the end of the video. A video on the connection between continued fractions, cutting sequences, and trajectories of billiard tables could also be a fun "spiritual successor". ;)
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  49. Tesla's 3-6-9 and Vortex Math is the key to Youtube views.
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  50. It's weird to see you mentioning falling powers, and their derivative... a long time ago I was bored, and started writing down the math for such functions to pass the time. I came to it because of the combination formula! It's very cool to see it actually has a functional purpose, more than I thought! I definitely noticed it could be helpful, but I never thought it could be THIS. This is so basic, and yet so important for the rest of calculus we learn in school, it's really amazing. Teaching this first, at least a little, could really help with people's intuition of Calculus.
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