Comments by "David Terr" (@dcterr1) on "3Blue1Brown" channel.

  1. I'm glad my salary isn't a fractal!
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  2. Wow, this is some really amazing stuff! Generating functions are remarkably useful, no doubt, but I've never seen them used in this way before. Another excellent text with lots of information on generating functions as well as their utility is Concrete Mathematics by Graham, Knuth, and Patashnik. Thanks for making this video and all your other ones, you rock!
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  3. Not necessarily. The boundary of the Mandelbrot set, which is one of the most well-known fractals, has a dimension of 2.
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  4. Wow, you always blow me away with your videos! I would've never imagined that Newton's method involves an inherently fractal structure such as that shown here. Moreover, this fractal seems USEFUL in a way that the Mandelbrot doesn't seem to be, or at least, not in a way in which it's usually presented. Keep up the good work!
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  5. Wow, you never cease to blow my mind with your videos! This is a seemingly quite difficult problem with a very nice, simple solution once you gain the necessary insights into how to solve it, which is what makes it a wonderful problem in my opinion. I also love how you contrast Alice's and Bob's approach to this problem. By the way, some of you may think I'm sexist for making this claim, but I think Alice's approach is the one more likely taken by a female mathematician and Bob's is the one more likely taken by a male mathematician, since women tend to be more right-brained and men more left-brained. In any case, one of the lessons of this video is that both approaches are equally important in mathematics and thus, if you accept my claim about men and women, although math has been traditionally dominated my men (as is unfortunately the case for just about all disciplines), it now seems to me that women may be potentially just as good mathematicians as men, but with a different approach.
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  6. It's quite clear to me that Alice and Bob have a much healthier approach to romance than Jack and Jill. Rather than climbing a hill and falling down together, they're involved in collaborating on solving math problems!
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  7. Fractals to the rescue!
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  8. Wow, very mind-bogglingly elegant solution! I'll have to work on the challenge now!
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  9. Incidentally, did you know that it's possible to draw a world map on a Mobius strip in such a way that opposite points on the earth correspond to opposite points on the Mobius strip, i.e., points for which if you stuck a pin through them, the point on the opposite side represents the point on the opposite side of the globe? Pretty cool, don't you think? I wonder if this fact could help us solve the inscribed square problem!
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  10. Excellent explanation! This is even better than the first draft, since this one goes into more detail and was easier for me to follow.
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  11. Very educational and relevant video! Thanks for posting this!
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  12. Great video! I've never seen this construction before, but I'm amazed at how it illustrates Dirichlet's theorem as well as properties of the continued fraction of pi. This makes me wonder what other important facts about primes are illustrated by Ulam spirals.
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  13. Wow, this was a very enlightening derivation of Euler's formula! (pardon the pun!) I never imagined this formula was so closely related to circular geometry. I wonder if it's possible to devise similar geometric proofs to the formulas for zeta(2n) in general.
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  14. Here's something I still don't understand. It seems to me that in order for the light wave induced by the incoming wave to have a large amplitude, the frequency of the incoming light needs to be close to the natural vibrating frequency of the atoms in the refracting medium, in which case the induced wave will have a phase delay of 90 degrees, resulting in a refractive wave that gets kicked back and hence an index of refraction greater than 1. So in order to obtain an index of refraction less than 1, wouldn't the phase of the induced wave need to be ahead of the phase of the incoming wave, and how is this possible?
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  15. Great visual explanation of winding numbers and how to compute complex zeros of analytic functions!
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  16. Here's a question I've been thinking about for a while. Although it seems that AI could pass the Turing test based on classical algorithms such as these, and perhaps it even already has, what about consciousness? Can classical computers be conscious, or does consciousness lie strictly within the domain of quantum mechanics? I'm inclined to think the latter these days. Perhaps someday we'll have conscious as well as intelligent quantum computers, but I don't think that will ever be possible with classical computers. I'd like to know your opinions about this.
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  17.  @victorlevoso8984  The reason I'm inclined to think that quantum computing is necessary for consciousness is that according to Godel's incompleteness theorems, absolute knowledge is impossible, whence there are unprovable mathematical results and hence we require being able to transcend logic in order to arrive at the truth, which I believe is the role of consciousness, which can only arise from quantum mechanics, since this is not a deterministic theory, consciousness ultimately being the source of quantum uncertainty. On the other hand, any computer that follows deterministic rules to arrive at its results, no matter how sophisticated these algorithms may be, necessarily lacks consciousness since it lacks this quantum uncertainty. Perhaps you disagree, and that's fine, since I know you're in good company. For instance, Hofstadter seems to take your point of view, arguing in his book I Am a Strange Loop that consciousness is an emergent phenomenon, arising from sufficiently complex systems, such as the transformers described in this video, which according to him I suppose would be conscious in some sense, though I tend to disagree.
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  18. Toward the end of this video, I thought of what I think is a rather clever application of the final result. What this means is that if you could be illuminated by a light source of a given intensity simultaneously from all directions, you would tan exactly four times as fast as if you were illuminated from a faraway source with the same intensity and rotated uniformly in all directions.
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  19. When I saw the answer to the problem, I was immediately reminded of Burnside's lemma. It seems the result is equal to the number of necklaces with 5 beads, each with a number going from 0 to 2^400 - 1, in which rotations of the necklace are not counted as distinct. Can you think of a nice isomorphism between this problem and yours? I suspect binary expansions will come into play, but I can't quite come up with an isomorphism, though I suspect there is one.
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  20. I really enjoy all your videos and this is one of the best I've seen! Can't wait for your follow up involving how to compute 2D convolutions efficiently!
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  21. This was a very helpful video for me to watch. I can't say I fully understand everything covered in the video, but at least now I have a better feeling for some of the terminology associated with cryptocurrency, such as hash functions and the block chain.
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  22. Wow, I just watched this again at the suggestion of a friend, and I must say that this time it blew my mind! I never cared much for topology, but now I can see how useful it is, and I want to learn more. Great job!!!!
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  23. I have a three-part question you may or may not be able to answer. If we start with n-1 fixed roots of a degree-n polynomial used to generate one of our fractals, what can we say about the "black" points, i.e., the locus of the nth root resulting in stable dynamics? Is this locus ever empty? If not, does it always look like a collection of Mandelbrot sets? Is it in general possible to estimate its total area?
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  24. Very enlightening video! Topology was never my strong area, but now I can see how useful it can be!
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  25. Wow, I never appreciated phase space so much until I watched this video! Now I see why it's a useful concept.
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  26. I never actually saw a proof of the Wallis product formula before or realized why it was true, so I found this video very enlightening!
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  27. Very enlightening notation!
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  28. I've seen this proof before, but this is the demonstration I've seen of it.
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  29. Fascinating video! I can't say I understand everything you try to explain here, but I can see how a lot of math goes into ChatGPT in order to make it seem intelligent - math that could perhaps even lead to computers passing the Turing test.
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  30. Yet another excellent 3Blue1Brown video! I got a bit lost toward the end, unfortunately, but you peaked my interest enough to want to learn more! Great job, once again!
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  31. Wonderful demonstration! I always assumed you needed calculus to determine the surface area of a sphere and never saw a connection to the area of a circle with the same radius.
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  32. Go ask Alice when she's 10 feet tall and her shadow has an area of 15 square feet.
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