The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The following three formulas in terms of polar coordinates produce butterfly like curves.

The following MATLAB script produces these three beautiful curves.

figure(1)
t=0:.01:24*pi;
r=exp(cos(t))-2*cos(4*t)+sin(t/12).^5;
polar(t,r),gtext('THE BUTTERFLY')
figure(2)
t=0:.01:8*pi;
r=exp(cos(t))-2*cos(4*t)+sin(t/4).^3;
polar(t,r),gtext('THE BUTTERFLY')
figure(3)
t=0:.01:2*pi;
r=exp(cos(t))-2*cos(4*t);
polar(t,r),gtext('THE BUTTERFLY')

Same graphs can be drawn with these Mathematica commands.

PolarPlot[Exp[Cos[t]] - 2 Cos[4 t] + Sin[t/12]^5, {t, 0, 24 Pi}, Axes -> False, PlotStyle -> Red]
PolarPlot[Exp[Cos[t]] - 2 Cos[4 t] + Sin[t/4]^3, {t, 0, 8 Pi}, Axes -> False, PlotStyle -> Red]
PolarPlot[Exp[Cos[t]] - 2 Cos[4 t], {t, 0, 2 Pi}, Axes -> False, PlotStyle -> Red]

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