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Simple trigonometric functions can birth a butterfly. A single parameter t, fed into nested sines and exponentials, traces the outline of a living wing: x = sin(t)(e^cos(t) − 2 cos(4t)) y = cos(t)(e^cos(t) − 2 cos(4t)) The same principle scales hearts from cubic sines and double-angle cosines, then lifts into three dimensions where Lorenz parameters and spherical harmonics weave closed loops on an ellipsoid. Mathematics does not invent beauty; it reveals the shapes already latent in its simplest rules.