Comments by "David Terr" (@dcterr1) on "Mathologer" channel.

  1. Another excellent video! Those medieval Indian mathematicians sure were smart, weren't they? At the same time in Europe, theologians were still debating how many angels could fit on the head of a pin!
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  2. I never cease to be amazed by the Mandelbrot set!
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  3. Another wonderful video! Although I'd seen much of this elsewhere, I never made the connection between Fourier series and epicycles. Very inspiring!
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  4. Very cool video! The only think that's a bit creepy is the animation of the photo and statue, but I suppose this is ultimately an application of calculus as well!
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  5. Another truly amazing video! Ramanujan was amazing, and so are you!
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  6. Hmmmmm, I didn't catch that (HO)³ stood for Ho Ho Ho!
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  7. Wow, this is an amazing result! I'm a 59-year-old mathematician and I've never heard of it before, go figure!
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  8. I wonder if the optimal solutions for 5 or more disks are unprovable even in principle.
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  9. Very nice proof! I was able to follow all the logic, though I'd have to muddle through the details in order to reproduce the proof. I've looked at other proofs of the irrationality of pi, but I was never able to follow them. Good job!
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  10. Great video - I really learned a lot from this one!
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  11. Fanstastic video! It's amazing to see how beautiful and versatile the Pythagorean theorem is and how many generalizations it allows. One could even make the case that the Pythagorean theorem was in a sense the foundation for mathematics! I just think it's unfortunate that Pythagoras got too much credit for it, since he probably wasn't the first to prove it, but I guess that's how history goes.
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  12. Wonderful video highlighting the genius of Ramanujan and the power of continued fractions.
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  13.  @Achill101  Watch from 10:30 to 11:37. This is essentially the same proof Einstein came up with.
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  14. One thing I find fascinating is the connection between the Hanoi graph and the Sierpinski triangle. I wonder if the number 466/855 fits into the Sierpinski triangle in any natural way, without regard to the Tower of Hanoi puzzle.
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  15. Great explanation as always! I love how you manage to derive results without using calculus that seem to require them, and this is a good case in point!
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  16. I find it a bit ironic that Tesla's vortex diagram looks a lot like the Volkswagen logo!
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  17. By the way, Dr. Who was a great show! Great choice for introducing the Tower of Hanoi puzzle!
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  18. Like all your videos, I love this one and I learned lots of useful information as well! In particular, I'm quite intrigued by the ternary tree construction of PPTs. In 2012, I published a paper in The Fibonacci Quarterly entitled "Some Interesting Infinite Families of Primitive Pythagorean Triples", in which I describe four infinite families of PPTs all involving Fibonacci and Lucas numbers, as well as the three you mention in this video, corresponding to the left, middle, and right paths through the tree. Now I suspect the other four families in my paper correspond to other simple paths through this tree, most likely involving a repeating finite sequence of left, middle, and right moves up the tree. This leads to several interesting questions. For starters, what's the correspondence between these repeating patterns and the resulting infinite families of PPTs? Furthermore, as in the cases you illustrate here, do the resulting right triangles each converge on a single one, and what are the proportions [a:b:c] of these limiting right triangles? Now I'd very much like to write another paper, or perhaps a series of papers, in which I try to answer these questions. If you'd like, you can collaborate with me! Let me know if you're interested. If so, we can exchange emails.
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  19.  @benefactor4309  Where were the Kerela schools located? In any case, according to this video, Indian mathematicians knew how to estimate pi way before the 16th century.
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  20.  @benefactor4309  Thanks for the info! I guess this is just another example of Western knowledge stolen from the Eastern world. Very sad IMO!
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  21. Wow, you never stop blowing me away with your videos! I'd offer you a googol dollars for letting me watch it, but I'm afraid I don't have the exact change!
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  22. Great video! I wrote Mathematica code to compute p(666) = 11956824258286445517629485 in 0.0625 seconds. (Never mind that Mathematica already has built-in code for doing so!)
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  23. Geometry was never my specialty unfortunately, so I don't have anything like the intuition Conway had for coming up with his six-point theorem or the windscreen wiper theorem, but I'm still quite awed by both proofs! I'm also awed by the connection with curves of constant width, which I already knew about and strangely I had a feeling would be mentioned later in the video.
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  24.  @Mathologer  Well in all fairness, I used to work at The Exploratorium in San Francisco, where they had an exhibit on them, so I was already pretty familiar with them.
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  25. Perhaps you can point me to some interesting research pertaining to the Buddhabrot set.
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  26. Paradoxes involving infinity are quite intriguing! Some other good examples are Hilbert's Hotel and the Tarski paradox.
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  27. So was anyone able to solve Ramanujan's problem, or did he have to present the solution himself?
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  28. While we're on the subject of formulas for pi, do you happen to know what's the most EFFICIENT method to compute a given number of digits of pi? I recall learning several years ago that by using the FFT for multiplication, N digits of pi can be computed nearly as quickly as the sum of two N-digit numbers! This fact blew my mind! But I'm still curious as to what formula is most efficient known one, so if you know the answer, please tell me.
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  29. By the way, I recall a few years ago solving an Euler problem involving computing the number of ways of making change for 2 (British) pounds using British currency, but I did it the hard way, by brute force! I got the right answer, but my solution involved computer code with about 10 nested loops! I wish I'd known at the time this much simpler method!
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  30. Great video! The animations really help to illuminate what's going on! I have a PhD in math and already knew the Euler-Maclaurin formula, but I still found this video very enlightening!
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  31. Wow, this video really blew my mind! Some very amazing combinatorics! Now I think I know how to solve one of the projecteuler problems that stumped me for several years, namely figuring out the number of domino tilings of a 3 x 2n board.
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  32. Is anyone here familiar with Einstein's proof, which he came up with when he was just 12? It involves adding the areas of the two triangles obtained by dissecting the original right triangle into two smaller similar right triangles.
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  33.  @Achill101  Einstein gave a proof at age 12 involving similar triangles. I think it's referred to in this video. Divide the right triangle into two similar right triangles and consider the areas of each subtraingle as well as the original right triangle. Since these areas add and the hypotenuses of each triangle correspond to the three triangles, and since area is proportional to the square of the hypotenuse, we have a² + b² = c². Pretty neat, huh?
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  34. Your videos are amazing! This is the first one I've watched in a year or so, and I'm just as amazed by this one as by your earlier ones. I learned two important things from this video. First, I learned a very intuitive visual proof that the alternating harmonic series converges to ln(2). Second, I learned another very intuitive proof of Riemann's rearrangement theorem, which I never even knew how to prove before! As always, excellent job!
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  35. Here's an interesting fact, based on the ternary tree of PPTs. By replacing left, middle, and right movements up the tree by the ternary digits 0, 1, and 2, respectively, we obtain an isomorphism between the set of PPTs and the set of 3-adic rational numbers strictly between 0 and 1, i.e., rational numbers in the interval (0, 1) with denominators equal to powers of 3. Furthermore, by considering the set of infinite paths up the tree, we also obtain an isomorphism from [0, 1] to [0, 1], given by x -> lim(u/v) where x is the real number with ternary expansion described above and u and v are the parameters in Euler's characterization of PPTs. Is this isomorphism equal to the identity? If not, what is it?
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  36.  @Mathologer  Thanks for pointing that out - very interesting!
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  37. Great video! I understood everything in the video, but I still need to work with tribonacci rabbits!
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  38. Excellent video, like all your other ones I've watched. Just a few comments: First, I noticed you used 3.18 rather than 3.14 on the circular slide rule. Second, I wanted to comment on the E6-B flight "computer". I took flying lessons a few years ago and actually used one! I was quite disappointed, needless to say, that what was advertised as a "computer" was merely a slide rule, so the technology was about 350 years more primitive than that implied by the name! Nevertheless, I was impressed by its capabilities, which include calculating true and magnetic compass headings taking into account wind velocity, as well as calculating fuel consumption and flight time. I think even Da Vinci would've been impressed!
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  39. I think I liked the visual proof at the end the best, but then again, I already know most of the other stuff. The overhanging parabolic arrangement of blocks was also very cool.
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  40. Wow! I wish I'd learned this stuff before I learned calculus! It's pretty amazing that you don't need calculus to explain why the area under the hyperbola y = 1/x is a logarithm, and you only need to evaluate a simple limit to define e. I was equally amazed by your definitions of the hyperbolic functions in terms of areas invariant to shapeshifting, which also doesn't require calculus. Pretty amazing stuff!
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  41. Is there any connection between curves of constant width and Bezier curves?
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  42. This video left me curious about something. Since it looks like we're able to derive a continued fraction representation of the error term, why can't we use an arbitrary number of terms from this representation to obtain an arbitrarily rapidly converging series? Couldn't we ultimately do better than the best known one? Or is this just a problem of complexity? I suppose we could obtain a series for which each additional term adds another 100 correct digits of pi, but perhaps it's not efficient because these terms have become so complicated! Is this what's going on?
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  43. I stopped the video at 17:57. I first learned the proof of the derivative formulas for the circumference of a circle and the surface area of a sphere when I learned how to solve rate of change problems. Just make the radius of the circle or the sphere, as the case may be, a variable and note that the rate of change of the area of the circle or the volume of the sphere, as the case may be, is just the circumference of the circle or the surface area of the sphere, respectively, times the rate of change of the radius. Voila!
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