The Peacock Butterfly in Polar Equations
Butterflies are common insects that can be found in nearly every part of the world; they amaze people with their remarkable beauty. Even though most people across the globe know or have seen butterflies, it is difficult for them to pay attention to these charming organisms, especially since their lifespan only lasts a few weeks. Despite this, there are 20,000 butterfly species existing in the world with a variety of colorful patterns. In Dr. Tong’s Honors Precalculus class at Concordia International School Shanghai during the 2022–2023 school year, students were asked to draw these beautiful butterflies with astonishing and detailed patterns using their knowledge of polar equations that they learned throughout the course.
Brainstorming
First, I had to choose a butterfly type that I wanted to graph using the knowledge learned in the class throughout the year. I was looking for different types of butterflies online, knowing that I wanted to look for one that is not only visually appealing but also has unique and complex patterns. I found this Peacock butterfly, which is seen in Figure 1. The patterns that might look like simple gatherings of circles but are fascinatingly complicated in detail with the charming colors that give a sense of mystery of the butterfly were more than enough to catch my attention, leading me to the decision to do my project based on this butterfly.
Starting
At first, I inserted a good-quality photo of the butterfly into the Desmos graph. As I have never tried tracing a photo on a graph using polar equations, I could not easily think of equations I should start with for this project. This made me go through lots of trials and errors throughout the whole process, which I enjoyed as part of learning. Since the butterfly in the photo was tilted itself, I first rotated the photo on the graph to work on graphing it more smoothly, as seen in Figure 2.
Then, I adjusted other factors like the size and opacity of the photo before actually starting to work on the graph when it would be too late for me to change those at that point. Although I wanted to bring the butterfly photo to the origin point (0, 0) of the graph to match the center, as the butterfly was not a complete symmetry, I could not match the center with the origin point of the graph fully.
Since I wanted to stay organized through the process, I created several folders for different body parts, which later helped me to easily make changes to the equations.
I attempted to make the whole butterfly outline with different methods, using one equation that forms the shape of butterflies, several circles that could form a butterfly shape, and others. However, after multiple errors, I decided that working in detail for each part by fragmenting the parts of the body into different equations would be the best way to illustrate the butterfly on the graph, and thus started to do so.
Outline & detailed patterns
Plotting the Antenna
I first started by depicting the antenna of the butterfly. Using inequalities to express the ellipse-shaped tip part of the antenna allowed me to illustrate the antenna of the butterfly better. Then I used linear equations to show the rest part of the antenna that connected to the body.
Plotting the Body
Then, I worked on the main body part of the butterfly. I once again had to fragment the graph into different parts as the body part was not in any clearly defined shape. By performing this process, it allowed me to be more detailed about depicting the butterfly.
Graphing the Left Wing
Things got even more complicated as I started working on the wings. As the outline of the wings was not simple circles, I had to use many equations for circles and set various domains and ranges to follow the outline of the wings. Then, after finishing the outline of the left wing, I worked on the patterns on its wings. I mostly used equations for circles and linear equations for the patterns, requiring me to change the domains and ranges correctly in detail.
Graphing the Right Wing
Since the two wings were not symmetrical, I had to work on the right wing all over again separately from the left wing. Also, similar to this one, the wing was not in a perfect curve, making me graph each curve segment one by one individually using circle equations by matching the domains and ranges for all as well. For the patterns, I followed a similar pattern as described above.
Changing Colors / Finishing the Project
Although the butterfly was made up of all different colors and looked pretty in its way, I wanted the butterfly to look more organized and clearer, similar to Figure 1. Therefore, I first changed the colors of all equations to black. Then, I tried changing different lines of the graph into different colors based on the original photo using the limited color palette provided on Desmos. As a result, even though the colors did not exactly match the original photo due to the limitation in colors of the Desmos graph, the final product of the Peacock butterfly polar graph was done, as seen in Figure 7.
The equations that I used can be checked in detail using the link below: https://www.desmos.com/calculator/lovbproslo
