Butterfly Curve
Lesson notes
Overview
An intricate polar curve with wing-like lobes from exponentials of cosines. This topic is rated intermediate and is number 199 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.
What you are seeing
The panel above shows a path traced by a parametric curve as its parameters change. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
The butterfly curve, discovered by Temple Fay, is the polar curve r = e^(sin θ) − 2cos 4θ + sin⁵((2θ − π)/24). Over many turns it draws a pattern with wings and antennae. It is a good example of how combining a few simple functions, an exponential, a cosine and a slow high power of sine, creates a rich and surprising shape that no single simple equation would predict.
Reading the relationship
The expression "r = e^(sin θ) − 2cos 4θ + sin⁵(…)" is a compact summary of the idea. It uses the symbols r, e, θ, c, o, s; name what each one stands for before you start. Notice that it contains an exponential, which changes by the same factor for every equal step, and a trigonometric function, so the result repeats as the angle goes round. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Run the animation once with the default settings and describe in words what happens. 2. Change one slider at a time and note which quantity responds most. 3. Push each slider to its extreme and decide whether the result still makes sense.
Try this
Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.
Common pitfalls
A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.
How it connects
It builds on “Trefoil Projection”, the topic before it in Mathematics. Once you are comfortable, continue with “Heart Curve”.
Why it matters
Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.
Where you might use it
- Revising “Butterfly Curve” before a class, exam or certification
- Checking working on homework and exam questions by watching the graph respond
- Modelling growth, change and uncertainty in finance, data science and statistics
- Building the reasoning behind the algorithms and engineering tools people use every day
Frequently asked questions
What is Butterfly Curve?
An intricate polar curve with wing-like lobes from exponentials of cosines. It is an intermediate topic in Mathematics.
How do I use the Butterfly Curve simulation?
The panel at the top shows a path traced by a parametric curve as its parameters change. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression r = e^(sin θ) − 2cos 4θ + sin⁵(…) packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Builds on the basics with relationships, methods and trade-offs. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Trefoil Projection”, the previous topic in Mathematics. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.