Marsh Fritillary in Polar Equations
Introduction
Well-known throughout our world, butterflies are seen as symbols of transformation and hope. As they flutter through the air people can’t help but be mesmerized by their patterns, however, these patterns serve more uses than just being aesthetically pleasing to the human eye. In fact, these patterns are essential to the butterfly’s survival. It’s ironic that the weakest part of the butterfly — so weak that even the gentlest human touch could tear it — would be the one reason that butterflies are able to survive. These patterns on their wings serve to camouflage the butterfly from predators, allowing it to blend into its surrounding environment.
In my 2022–2023 Honors PrecCalculus class at Concordia International School Shanghai, we were assigned to a project involving recreating something through polar curves. Our teacher, Dr. Peter Tong, suggested that we recreate butterflies through a Desmos graph, so that’s what my class decided to do.
Brainstorming Process
My first step in making the polar graph was to figure out what type of butterfly to graph. I wanted to graph a butterfly that had a complicated pattern to test my knowledge. After looking for a little bit on the website provided in class, I chose to graph the Marsh Fritillary for its iconic coloring and pattern. After that, I chose a picture of the butterfly that followed the criteria provided.
Starting to Graph
When looking at the butterfly, I could see that it would use a lot of linear, parabolic, circular, or ellipse equations to be able to recreate its patterns. To start my initial graph, I imported the reference picture into Desmos, as seen in Figure 2, lowered its opacity, and adjusted its size. After that, I used that photo to determine how I graph different functions.
Starting off, I knew that there would be a lot of equations needed to create this butterfly, so I needed to stay organized. To do this, I created various folders corresponding to the butterfly area. The folders I created correspond to the different areas of the butterfly to be able to be efficient and organized in my process.
Creating the Outline
To start creating my butterfly, I decided to start with the easiest part — the outline of the butterfly. While the outline may seem simple, a butterfly’s wings aren’t perfectly straight, nor do they form a perfect curve, so I decided to use various linear and parabolic equations to match the picture.
I repeated the same process for the main body of the butterfly but used different types of equations that would fit better. For the antennas, I used a linear equation for the left antenna and for the right, I used a linear and parabolic equation. I used circles for the body and rotated an ellipse 45° to the left. An interesting part about this is that the ellipse limit would not work, as one part would either not connect or go too far due to the ellipse being round. To combat this, I duplicated the equation and set separate domains and ranges for each ellipse.
Creating the Patterns of the Butterfly
This step was the most time-consuming step, as it took hundreds of equations to accurately duplicate every single shape seen in the butterfly. Since each separate shape was organic, meaning that it doesn’t fit any of the conics perfectly, I had to combine different conics to match its shape. This provided a challenge, as it meant that I had to set domains and ranges that perfectly met each other to ensure that all the lines intersected each other. To create many of the shapes, I used a variety of ellipses, circles, linear equations, and parabolas to create designs that accurately depicted this process, which is seen in Figure 5.
Changing the Colors
I didn’t like how my butterfly looked with its different colors, as it didn’t look as uniform as I hoped it would. To combat this, I decided to make the lines correspond to the color of the butterfly. To choose the colors, I found the RGB value of the main colors seen on the butterfly: a bright orange, a yellow-orange, and a dark brown. Then, I entered the values into Desmos, to create new color options. Then, I went through all 360 equations on the graph to change the color to match the picture I used.
Converting to Polar
Since this was a polar project, my final step was to convert my equations into their polar form. To do this, I substituted “rcos θ” and “rsin θ” for “x” and “y” values respectively. Then, I used the quadratic formula to input a function in the format that Desmos accepts.
The following link is the link to the Desmos file: https://www.desmos.com/calculator/iqsk4mlmwc
