Butterfly Curve

By the end of this lab you'll be able to:

• Plot the Temple H. Fay butterfly curve using its parametric polar equation
• Understand how exp(cos(t)) and cos(4t) combine to create the wing shapes
• See how a single parametric equation can produce organic, complex shapes

The butterfly curve is a famous parametric curve discovered by Temple H. Fay in 1989. Its polar equation r = exp(cos t) - 2cos(4t) + sin⁵(t/12) generates a shape that uncannily resembles butterfly wings.

Welcome to the Butterfly Curve!

This curve was discovered in 1989 by Temple H. Fay. One equation makes a butterfly — math and art in perfect harmony! Let's code it together!

How It Works

For t from 0 to 12π, compute r = exp(cos(t)) - 2*cos(4*t) + sin(t/12)^5. Then convert polar to Cartesian: x = scale * r * cos(t), y = scale * r * sin(t). The many-lobed result needs 12π to complete.

Sample Code

import turtle
import math
monty = turtle.Turtle()
monty.speed(0)
monty.hideturtle()

scale = 50
steps = 2000
monty.pencolor('royalblue')
monty.pensize(1)
monty.penup()

for i in range(steps + 1):
    t = 12 * math.pi * i / steps
    r = (math.exp(math.cos(t)) - 2 * math.cos(4 * t)
         + math.sin(t / 12) ** 5)
    x = scale * r * math.cos(t)
    y = scale * r * math.sin(t)
    if i == 0:
        monty.goto(x, y)
        monty.pendown()
    else:
        monty.goto(x, y)

What Do You Think Will Happen?

The formula mixes exp(cos(t)) with cos(4t). Will the result look like wings, a star, or something completely different? Make your guess — then click Run to find out!

Try It Now

▶ Run ↻ Reset

Butterfly wings — the exp(cos t) term creates the large lobes while cos(4t) adds smaller oscillations that form the wing details. Were you right?

How It Works

exp(cos(t)) oscillates between exp(-1) ≈ 0.37 and exp(1) ≈ 2.72. The subtracted 2*cos(4*t) can make r negative, adding interior details. The sin(t/12)^5 term creates the wing-tip elongation by varying slowly over the 12π range.

Explanation Table

Line	What it does
math.exp(math.cos(t))	Main lobe — oscillates between 0.37 and 2.72
2 * math.cos(4 * t)	Subtracts 4 oscillations — creates wing detail
math.sin(t / 12) ** 5	Slowly varying term — elongates wing tips
12 * math.pi	Full sweep for the curve to close

Learning Check

Your Turn — Remove One Term

Remove math.sin(t / 12) ** 5 from the formula (set it to 0). How does the curve change without that term? Predict, then run it!

▶ Run ↻ Reset

Without sin(t/12)^5, the curve is more symmetric — two equal lobes instead of the elongated wing-tip shape.

Experiments

1. Change scale. Try scale = 70. You'll know it worked when the curve is larger.

2. Increase steps. Try steps = 5000. You'll know it worked when the curve is smoother.

3. Use cos(4t) coefficient 3 instead of 2. The inner lobes will change. You'll know it worked when the wing detail looks different.

4. Plot the curve in multiple colors. Break the loop into segments of steps//6 points each and change the color for each segment. You'll know it worked when different parts of the curve have different colors.

Beautiful Math!

You coded one of the most elegant parametric curves in mathematics! The butterfly curve shows how exp, cos, and sin can conspire to make art. Up next: Epicycloid Star — circles rolling outside circles.