[math-fun] I wonder how old this is.
The arclength of one period of sin x = the circumference of an ellipse with semiaxes 1 and √2. In[112]:= #1 == #2 == FunctionExpand@#1== N@# &[ArcLength[Circle[{0, 0}, {√2, 1}]], ArcLength[Sin@x, {x, 0, 2 π]}]] Out[112]= 4 EllipticE[-1] == 4 √2 EllipticE[1/2] == 4 √2 π^(3/2)/Gamma[1/4]^2 + Gamma[1/4]^2/√(2π) == 7.64039557805542 𝚪(¼) is the rightful value of the symbol 𝛕. And someone should write Beckmann II: A History of 𝛕. —rwg
Whoa, it's stronger than that. The *in*definite integrals are identical: In[183]:= Assuming[Abs@x <π,Simplify[ArcLength[Sin@t, {t, 0, x}] - ArcLength[Circle[{0, 0}, {1,√2}, {0, x}]]]] Out[183]= 0 This invites the animation of a pair of ellipses pinching the sine wave while bumpily rolling along it, with the point of contact and centers moving at a constant horizontal speed. I expect the ellipses to be in the "tall" phase at the extrema of the sinusoid. —rwg On Wed, Dec 16, 2020 at 2:22 AM Bill Gosper <billgosper@gmail.com> wrote:
The arclength of one period of sin x = the circumference of an ellipse with semiaxes 1 and √2.
In[112]:= #1 == #2 == FunctionExpand@#1== N@# &[ArcLength[Circle[{0, 0}, {√2, 1}]], ArcLength[Sin@x, {x, 0, 2 π]}]]
Out[112]= 4 EllipticE[-1] == 4 √2 EllipticE[1/2] == 4 √2 π^(3/2)/Gamma[1/4]^2 + Gamma[1/4]^2/√(2π) == 7.64039557805542
𝚪(¼) is the rightful value of the symbol 𝛕. And someone should write Beckmann II: A History of 𝛕. —rwg
I believe this is about as old as making sine wave-fringed paper by wrapping a cylindrical candle in a sheet of paper, cutting it at an angle, observing that the surface of the cut is an ellipse, unrolling the paper and observing that the edge is a sine wave. E.g. https://www.cutoutfoldup.com/405-cut-a-sine-wave-with-one-straight-cut.php Leo On Wed, Dec 16, 2020 at 2:22 AM Bill Gosper <billgosper@gmail.com> wrote:
The arclength of one period of sin x = the circumference of an ellipse with semiaxes 1 and √2.
In[112]:= #1 == #2 == FunctionExpand@#1== N@# &[ArcLength[Circle[{0, 0}, {√2, 1}]], ArcLength[Sin@x, {x, 0, 2 π]}]]
Out[112]= 4 EllipticE[-1] == 4 √2 EllipticE[1/2] == 4 √2 π^(3/2)/Gamma[1/4]^2 + Gamma[1/4]^2/√(2π) == 7.64039557805542
𝚪(¼) is the rightful value of the symbol 𝛕. And someone should write Beckmann II: A History of 𝛕. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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