Today is the birthday of Gauss (born 4/30/1777), who defined 1796 the double periodic sinus and cosinus lemniscaticus (lemniscatic elliptic function). Here is a way to visualize them.
w = 2 EllipticK[-1]; sinlem1 =
ParametricPlot[{EllipticF[ArcSin[r], -1], r}, {r, 0, 1},
AspectRatio -> 1/GoldenRatio, PlotStyle -> Red];
sinlem2 =
ParametricPlot[{w - EllipticF[ArcSin[r], -1], r}, {r, -1, 1},
AspectRatio -> 1/GoldenRatio, PlotStyle -> Red];
sinlem3 =
ParametricPlot[{2 w + EllipticF[ArcSin[r], -1], r}, {r, -1, 1},
AspectRatio -> 1/GoldenRatio, PlotStyle -> Red];
coslem1 =
ParametricPlot[{w/2 - EllipticF[ArcSin[r], -1], r}, {r, -1, 1},
AspectRatio -> 1/GoldenRatio, PlotStyle -> Blue];
coslem2 =
ParametricPlot[{3/2 w + EllipticF[ArcSin[r], -1], r}, {r, -1, 1},
AspectRatio -> 1/GoldenRatio, PlotStyle -> Blue];
coslem3 =
ParametricPlot[{5/2 w - EllipticF[ArcSin[r], -1], r}, {r, 0, 1},
AspectRatio -> 1/GoldenRatio, PlotStyle -> Blue];
scl = Show[sinlem1, sinlem2, sinlem3, coslem1, coslem2, coslem3,
PlotRange -> All]