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