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Grant Sanderson
Pi creature caretaker. Math videos: FAQ/contact:
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Grant Sanderson Aug 10
Replying to @hardiks_hah
The slope of f(x) = x^2 at the intersection x=f(x)=1 is 2, so that point should be unstable. And indeed, if set variable x to be 1.0001 or 0.99999 and loop over the operation "x = x^2; print(x)" you'll find that it diverges away from that fixed point.
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Grant Sanderson Aug 7
Replying to @thephysicsgirl
Like minds!
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Grant Sanderson Aug 6
Replying to @thephysicsgirl
Yes, of other collections (not a joke).
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Grant Sanderson Aug 6
Right, exactly. I guess what I'm wondering is if it's obvious that a "typical" group, or a typical simple group anyway, looks more like S_n than C_n in this sense.
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Grant Sanderson Aug 6
Replying to @mordroberon
Yes, but not in a way that would match this pattern. You can try it yourself by factoring a few other numbers around the same size as this one. One thing you'll notice is that it's much more typical to have much larger primes somewhere in the factorization.
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Grant Sanderson Aug 6
Replying to @mattmacauley
I'd love that! You can shoot me a message here:
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Grant Sanderson Aug 6
Replying to @johncarlosbaez
I wonder if there's a way to make rigorous the idea that choosing a "random" finite group will have a size whose prime decomposition "looks like" that of n! for some n.
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Grant Sanderson Aug 6
Replying to @johncarlosbaez
One the one hand this seems strange, e.g. cyclic groups show no such preference. But then again S_n obviously does, so maybe via Cayley's theorem, it's somehow to be expected of groups in general.
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Grant Sanderson Aug 6
Replying to @johncarlosbaez
I had never really noticed this before and would have assumed that it's just because the examples we tend to work with are "simpler" in some sense.
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Grant Sanderson Aug 6
Math Twitter: Is there a known reason behind why the prime factorization of the size of the monster group has a kind of exponential decay pattern to it? That is, lots of 2's, a fraction as many 3's, a fraction as many 5's, ultimately petering out at 71?
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Grant Sanderson Aug 4
It is compulsory, so for the example you mention, it's why it's very important to have a "null" bit that doesn't change what square the configuration points to.
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Grant Sanderson Aug 4
Replying to @dinabogdan03
Maybe a more perfect video would have re-oriented the picture of the hand to match the example, but then again the process of learning to use the right-hand rule involves practicing reorienting an image like that in your head.
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Grant Sanderson Aug 4
Replying to @dinabogdan03
Ah, I see the confusion. The hand is not meant to align perfectly with the axes draw, just to be there as a guiding reminder of how the right-hand rule works.
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Grant Sanderson Aug 4
Replying to @ruanferuere
It's meant as a tongue-in-cheek joke :)
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Grant Sanderson Aug 4
Replying to @nayafia
Woohoo! Congrats. Looking forward to getting my preorder :)
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Grant Sanderson Aug 3
Wordplay + internet replies = 🙄 I wonder how Abbot and Costello would have fared on Twitter. "I think you're misunderstanding that Who is a person, not a pronoun!"
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Grant Sanderson Aug 3
Replying to @3blue1brown
Somehow it's weird to me that "old" has two antonyms: "young" and "new". Do other languages have separate words for old?
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Grant Sanderson Aug 3
Take a computer from 1995 and one from today, which one is "old"? Take the version of you from 1995 and the version from today, which one is "old"? Some things age into the past, others into the future.
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Grant Sanderson Aug 3
I'd actually be genuinely curious to hear if thinks these interactive visualizations are helpful at illustrating the SU(2) -> SO(3) connection.
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Grant Sanderson Aug 3
It's done in the context of a stereographic projection, so much of the lecture is aimed at building intuition for what that is, but hopefully, it can give you some intuition for how each action involves "two perpendicular rotations", and why it's a double cover of SO(3).
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