Butterfly Curve
Parametric curvesTemple Fay’s 1989 transcendental curve whose sine and exponential terms flutter into a butterfly.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Butterfly Curve?
The butterfly curve was published by Temple H. Fay in 1989 as a showcase of how rich a single polar-style formula can be. The radius e^{cos t} − 2cos 4t − sin⁵(t/12) mixes three rhythms: a slow exponential breathing, a four-fold wing beat from cos 4t, and a very slow sin⁵(t/12) drift with period 24π that keeps each pass slightly different.
Because of that slow drift, the curve does not repeat until t has swept 24π — twelve full turns — layering wing inside wing. It has become a standard test piece for plotting software and a favorite “wow” example in parametric-equation lessons.
Key properties
- Transcendental (not algebraic): the exponential term rules out any polynomial equation.
- Full period is 24π, set by the sin⁵(t/12) term.
- Symmetric about the y-axis (swapping t → −t mirrors x).
- Radius stays within e + 3 of the origin, bounding the wingspan.
- A polar variant r = e^{sin θ} − 2cos 4θ + sin⁵((2θ − π)/24) draws a cousin butterfly.
Where you'll see it
Plotting-software demos, parametric equation showpieces, generative art, and laser projection tests.