trigonometry like you've never seen it
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Published 2024-05-24
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All Comments (21)
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I drew a unit piggy and subsequently defined the coswine function
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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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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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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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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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22:49 I think it should be 1/4-1
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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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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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“The squine “ I fucking lost it 😂
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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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-3/4 is not equal to 5/4-1
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Squigonometry is my new favorite math unit
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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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I should be revising fluid dynamics, but 'squine and cosquine' was the nerdiest reason I've ever laughed myself incapacitated.
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Time to introduce poqueuepine
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Crazy, that's exactly what I was thinking about a few days ago, was thinking of doing a line integral on a squircle and find the limit as it approaches a square, so I needed to parameterize it
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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!
