Post AoL5XyDNLltvmKAALI by catselbow@fosstodon.org
(DIR) More posts by catselbow@fosstodon.org
(DIR) Post #AoKx6J4FQYHCi7Bw48 by futurebird@sauropods.win
2024-11-23T14:24:13Z
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Math people. Do you have an intuitive *geometric* reason why interlacing the power series of sin x and cos x gives the series for eˣ?Do you have a way that you *visualize* growth whose rate is exactly its present value coming out of the unit circle as sine and cosine do?I know the equations, and the proofs and they aren't convoluted and probably "enough" but I hope someone can share their mental images. Even if they aren't fully formed.
(DIR) Post #AoKxnPcKx1CTTbgtAO by sci_photos@troet.cafe
2024-11-23T14:31:59Z
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@futurebird Aren't you missing the imaginary factor? This … somehow “bends” the exponential growth onto the unit circle … ?
(DIR) Post #AoKy30z8Ij0diKnFS4 by mcc@mastodon.social
2024-11-23T14:34:46Z
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@futurebird *thinks* i think that's a really good question (the first one) but i'd worry it's actually not possible because when you start screwing with power series expansions in that way you're kinda like… doing things that are'nt "real"? anyway i'd try maybe nesting boxes within boxes within boxes
(DIR) Post #AoKyDoJpjY4lcGcj68 by FlockOfCats@famichiki.jp
2024-11-23T14:36:41Z
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@futurebird Not sure if this helps and you’ve probably seen it, but this graphic on Wikipedia is cool 🤓 https://en.wikipedia.org/wiki/Euler's_formula?wprov=sfti1#History
(DIR) Post #AoKyPBhH9ej5J80u12 by futurebird@sauropods.win
2024-11-23T14:38:50Z
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@FlockOfCats That explains sine and cosine but how does it explain the exponential function?
(DIR) Post #AoKySTNpqyOB1uUG2K by futurebird@sauropods.win
2024-11-23T14:39:26Z
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@mcc "not real" just means along another axis.
(DIR) Post #AoKyrvpjDMJwVNfoau by mcc@mastodon.social
2024-11-23T14:43:54Z
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@futurebird sorry i didn't mean "real" as in "not imaginary" i meant like… i dunno you can do Shenanigans sometimes. rearranging terms for different results etcBy the way, here is a video I dug up when I saw your question because I remembered it having a really good geometric visualization but I couldn't remember what it was a visualization of. Watching it, I'm not sure it's relevant. But it does visualize a power series at one point…https://www.youtube.com/watch?v=G0Fa5Zl-Z3c
(DIR) Post #AoL03UjNuEJRsUmeQq by llewelly@sauropods.win
2024-11-23T14:57:19Z
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@futurebird I don't know, but maybe comparing these two gifs may help, especially if you can do it frame by frame:https://en.wikipedia.org/wiki/File:Sine.gifhttps://en.wikipedia.org/wiki/Exponential_function#/media/File:Exp_series.gif
(DIR) Post #AoL0G9pCwgP6nsXszg by futurebird@sauropods.win
2024-11-23T14:59:37Z
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@llewelly kinda... The Taylor series is a better and better approximation of the sine and the cosine ... and it can also be used for the exponential. But ... how geometrically do these two things come together to do that?The algebra might be the best way I suppose.
(DIR) Post #AoL0l3epvQYseYQeps by llewelly@sauropods.win
2024-11-23T15:05:11Z
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@futurebird my apologies; I forgot the distinction between the taylor series and the power series.
(DIR) Post #AoL0qQOio0XAd0RzRw by futurebird@sauropods.win
2024-11-23T15:06:01Z
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@llewelly no this is great it's another good place to look for the "visualization" that I don't even know if it exists!
(DIR) Post #AoL1p76l8f4gSzzHO4 by FlockOfCats@famichiki.jp
2024-11-23T15:16:37Z
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@futurebird Hmmm good question….i don’t know! If you imagine the unit circle that is getting traced out, does the exponential shake out of it if you look at the coordinates on the unit circle vs the velocity at that point?Like at (1,0) the rate of change in x is 0 (changing dissection) and all the change is toward positive Y. Hmmm …sorry just think aloud, not helping lol
(DIR) Post #AoL5XyDNLltvmKAALI by catselbow@fosstodon.org
2024-11-23T15:58:51Z
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@futurebird Just thinking out loud, but maybe it would be helpful to start out by thinking about the hyperbolic cosine and sine, instead. For example, see some of the wikipedia illustrations:https://en.wikipedia.org/wiki/Hyperbolic_functionsAlso, do you pronounce sinh as "sinch" or as "shine"? I've always heard and said "sinch", but recently I've heard a couple of mathy people say "shine", and it sounds weird.
(DIR) Post #AoL5mOELQBsKWDUYyG by futurebird@sauropods.win
2024-11-23T16:01:26Z
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@catselbow That *is* weird IMO.
(DIR) Post #AoL78UN9MkzDtPiDa4 by geonz@mathstodon.xyz
2024-11-23T16:16:37Z
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@futurebird @mcc as soon as it's abstract it's not real, but we can still visualize & express -- that's what's so cool about math! Except I don't get as far as trig.... if I were at work I might peek at that "visual trigonometry" book on the shelf out there...
(DIR) Post #AoL7wavFHtn4gbYIwi by dangrsmind@sfba.social
2024-11-23T16:25:41Z
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@futurebird RIP Ralph Abraham who could draw a single picture to explain complex ideas in analysis and algebra that would have certainly explained this. He was an amazing teacher.
(DIR) Post #AoLAe0HR5DYph0l504 by grtyvr@infosec.exchange
2024-11-23T16:55:58Z
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@futurebird not sure this goes anywhere yet but I suspect the fact they are orthogonal will play into it....
(DIR) Post #AoLCadgsoWo6fjBVVg by Zamfr@mstdn.social
2024-11-23T17:17:41Z
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@futurebird There is a nice attempt in Visual conplex analysis, by Tristan Needham.He first encourages you to see exp(x) as (1 + x/n)**n for n-> infinity. Interest paid out continuously, you know.That has a nice geometric equivalent for complex exp(x + iy). Namely, a stack of triangles with height y. As n goes to infinity, that stack is a spiral, or a circle if the real part is zero
(DIR) Post #AoLJuTpHtmdZY5WX8y by Zamfr@mstdn.social
2024-11-23T17:53:37Z
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@futurebird Later he has introduced the complex derivativeas 'amplitwist', with the imaginary part the twist.Then you can define exp(z) as the complex function whose complex derivative equals itself, and that becomes a circle for exp(iy)
(DIR) Post #AoLRLMzE8ULvpkwSLA by adredish@neuromatch.social
2024-11-23T20:03:04Z
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@futurebird @albertcardona Not sure, but does this help?https://tauday.com/tau-manifesto
(DIR) Post #AoLTADeEiS3barvzY8 by australopithecus@mastodon.social
2024-11-23T20:23:28Z
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@futurebird I never got deep into power series so I can't offer any insights on that side of things, but intuitively, "growth" is just another name for the slope, plus "slope is rise over run" is practically a mantra, plus sin is rise and cos is run from polar coordinates, so ... /shrug, this just sounds like "name the shape with constant change of slope through θ" to me
(DIR) Post #AoLZLu4HwFJG4H4gl6 by robert_cassidy@urbanists.social
2024-11-23T21:32:48Z
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@futurebird Oddly, because I do rely on visualization in a lot of situations in the cases you ask about I don’t, but I can intuit the answer - in the sense I can give a decent guess and look at an answer and tell if it’s clearly wrong. I think in the second question I’m tapping in more often to the cause of the growth than the math directly and intuit off of that. I’m also a physics and data science guy so my intuition jumps back and forth a lot.
(DIR) Post #AoLcyCQmPk0DsyY0S8 by mattmcirvin@mathstodon.xyz
2024-11-23T22:13:20Z
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@futurebird I've just internalized that e^ix is a helix, sin and cos are its shadows on coordinate planes
(DIR) Post #AoLf7WShVb3GWd1G3U by darabos@mastodon.online
2024-11-23T22:37:26Z
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@futurebird I've watched a bunch of YouTube videos on the subject now. My conclusion is that https://youtu.be/B1J6Ou4q8vE is by far the most action packed discussion of this. Also if I understand you correctly you are interlacing and also taking the absolute value of the elements? I think all geometrical intuition is out the window at that point.
(DIR) Post #AoLlkQh5UFx9efgTR2 by mattmcirvin@mathstodon.xyz
2024-11-23T23:51:41Z
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@futurebird @FlockOfCats Exponential growth comes from geometric growth, that is, you add an increment that's proportional in size to the thing itself. Take the continuous limit, and you get the exponential function.Now, instead, add a little *perpendicular* vector proportional in size to the original quantity. You get similar skinny right triangles that make a geometric spiral in the plane. But now, when you take the continuum limit of skinny triangles, the spiral tightens up into a circle. That's the exponential of ix.The connection to the exponential function is this "add a little proportional bit, and repeat" operation. All the Taylor series is doing is separating out the terms with different powers of i. So the odd and even ones give the projections of circular motion, which are trig functions.
(DIR) Post #AoLr8U3t7JIyowdcsi by johncarlosbaez@mathstodon.xyz
2024-11-24T00:52:03Z
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@futurebird - I visualize this stuff as part of the approach to trig described here:https://math.ucr.edu/home/baez/trig.html
(DIR) Post #AoMhtx0SVwC1olzP2O by mrdk@mathstodon.xyz
2024-11-24T10:43:17Z
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@futurebird Well, the first point is that the function exp(ix) has an “orthogonal” growth rate, since it satisfies the equation exp(ix)' = i exp(ix)The equation says that a particle that moves along the function x ↦ exp(ix) has a speed vector that is orthogonal to its position vector. A movement along a circle around the origin of the complex plane has this property.So the idea behind is:1. The function x ↦ exp(ix) has “orthogonal growth“.2. Orthogonal growth produces circle movement.3. Circle movement produces sine and cosine.