trigonometry like you've never seen it
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Publicado 2024-05-24
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Todos los comentarios (21)
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I drew a unit piggy and subsequently defined the coswine function
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Fun fact: x²ª + y²ª = 1 for a positive integer has no rational solutions when a > 1 other than (±1, 0) and (0, ±1). For if it did, put both terms over the lowest common denominator, multiply out by that, and you have a counterexample to Fermat's Last Theorem. So all the squircles make their way through the everywhere-dense set of points which have both coordinates rational, without touching a single one of them!
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Hi Michael, The higher dimensional analogues of the 👉interiors👈 of squircles are called "superballs" and have been studied by Rush (myself) and Sloane, "An improvement to the Minkowski-Hiawka bound for packing superballs", Mathematika, June 1987, Volume34, Issue 1, pp 8-18 and by Elkies, Odlyzko and Rush (again myself) "On the packing densities of superballs and other bodies", Invent Math, December 1991, Volume 105, pp 613-639. I am a subscriber and I enjoy your lectures. 👍
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Squircle sounds like an awesome Pokémon name ngl
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Interestingly, it is known that Gamma(1/4) and pi are algebraically independent, so it follows that rho is transcendental.
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mistake at 22:43 in dU. It is Beta(1/4, 1/4), the final value is Gamma^2(1/4)/2Gamma(1/2) = 3.70814
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I was honestly shocked to see that this video had come out just now. Yesterday I spent a few hours, attempting to define the basic trigonometric functions for the squircle. Didn’t know other people had thought about this problem as well, although I’ll admit that isn’t the craziest idea in the world.
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"pi" formula for arbitrary n is rho = Г(1/2n)^2 / (n Г(1/n)) and it does approaches 4 as n->inf 😊
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22:49 I think it should be 1/4-1
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you can parametrize the segment from 0 to pi/2 rad of any squircle xⁿ+yⁿ=1 as (cosª(t),sinª(t)) where a=2/n. Any other value of the angle results in at least one complex value, but you can cover the four quadrands with combinations of absolute values of the parameter functions. So x⁴+y⁴=1 first quadrant can be parametrized as (|cos(t)|½,|sin(t)|½), the second as (-|cos(t)|½,|sin(t)|½), the third as (-|cos(t)|½,-|sin(t)|½) and the fourth as (|cos(t)|½,-|sin(t)|½).
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My first instinct was that x=√cos(t) and y=√sin(t) would solve x⁴+y⁴=1 and would serve as a definition for the squircle, but then was surprised to see that the equivalent of π wouldn't match. I suspect the step of "taking" the definitions x'=-y³ and y'=x³ is what rules out the square root of the original trig functions to match the new functions, which is an interesting development.
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Time to introduce poqueuepine
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Shouldn't the integral around 24:00 be of x^(1/4 - 1) (1-x)^(1/4-1) not 5/4?
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“The squine “ I fucking lost it 😂
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I should be revising fluid dynamics, but 'squine and cosquine' was the nerdiest reason I've ever laughed myself incapacitated.
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Showing that all squircles with n > 1 never pass through points having rational coordinates would prove the Fermat’s theorem for even powers, right?
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It should be 1/4 - 1 instead of 5/4 - 1 to have -3/4
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First time I saw something like that was in a sci fi book about what was riddle an equation for a square (Piers Anthony) back in the 80's. The answer was take the equation for a circle, and instead of 2nd power it would be 2*n power, and let n go to infinity, and the shape will approach a square.
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You forgot to link the book!
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I wonder what the squine and cosquine, as well as the other four functions (tan, csc, sec, cot) but for the squircle, look like?