Butterfly Curve
Temple Fay’s 1989 transcendental curve whose sine and exponential terms flutter into a butterfly.
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Formulas and key values
- Parametric equations
- Period
- Distance from the center
t is the parameter; the whole butterfly takes t from 0 to 24π.
Explainer video
Make a video of the Butterfly Curve
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Start the video01 · Definition
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.
02 · Properties
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.
03 · Questions
Butterfly Curve: common questions
- What is the equation of the Butterfly Curve?
- Parametric equations: x = sin t(e^(cos t) - 2 cos 4t - sin^5 t/12), y = cos t(e^(cos t) - 2 cos 4t - sin^5 t/12). t is the parameter; the whole butterfly takes t from 0 to 24π.
04 · Keep exploring