The Duke of Burgundy Butterfly- A Polar Equation Portrayal

Butterflies are some of the most beautiful and fascinating creatures in the natural world. Their delicate wings are adorned with vibrant colors and intricate patterns, and they are known for their graceful and effortless flight. As a high school student at Concordia International School Shanghai, our class was challenged this year by our teacher, Dr. Tong, to use mathematical functions to describe the shape and movement of butterfly wings, providing insight into the complex and elegant patterns found in nature. Through the use of polar equations, we gained a deeper understanding of the beauty and wonder of these remarkable creatures.

The butterfly I chose to portray through polar comics equations was the Duke of Burgundy. I found this butterfly on a website of butterfly collages as this image immediately caught my attention. This butterfly has orange chequered patterns crossed with lively vivid veins on its soft brown wings. The Duke of Burgundy butterfly, as its name reveals, is a very majestic and eloquent representation of nature and beauty.

Figure 1: The Duke of Burgundy Butterfly

Step 1: The first step I worked on was tracing the middle body part of the butterfly. This was the easiest part of the whole project, as most of the patterns in this area were not very complicated to outline. I used a combination of ellipses, parabolas, and linear equations to draw it. I played around with the dotted and dashed lines to highlight the unique texture of this butterfly. For the body, I started by using ellipse conics to outline. I used the standard ellipse conics equation: (x-h)²/a² + (y-k)/b²=1.

Figure 2: First draft equations of the butterfly body

Figure 3: Superimposed body equations on butterfly

Step 2: As I worked, I realized that the standard ellipse did not fit some of the patterns, such as the butterfly’s eyes. To solve this, I attempted to substitute x=x’cost-y’sint and y=x’sint+y’cost into the regular ellipse to rotate the conic. However, I ran out of time to calculate the rotated polar formula for every single ellipse. Instead, I used the imposed formula (shown in Figure 4) and the sliders to match the shape of the ellipses I needed to use (such as the eyes in Figure 5).

Figure 4: Slider formula for rotated conics

Figure 5: Rotated conics used on the butterfly eyes

Step 3: I then worked on the overall outline of the butterfly wings. I color-coded this area using orange to contrast with the color of the butterfly. Although the outlining of this part may seem easy, because the wings could not be portrayed by one consistent line, I used multiple parabolas and linear equations in every section to attempt to match the curves of the butterfly wings.

Figure 6: The outlined butterfly wings & stem

Step 4: While graphing the veins of the butterfly, I decided to keep the more visible patterns of the butterfly, so the visuals wouldn’t look too messy. This section wasn’t very hard, as most of the outlines were linear and parabolic conic equations. Although the work was tedious, I eventually grew more efficient and picked up the speed of outlining the butterfly. Figure 7 shows the finished outline of the vines of the butterfly.

Figure 7: Superimposed outline of butterfly wings

Step 5: After getting lost several times on my Desmos webpage, I realized the importance of using folders and graphic organization. I stopped my work and spent time using colors and folders to separate the different areas I was working on, as demonstrated in the picture in Figure 8.

Figure 8: Organized folders of equations by parts of the butterfly

Step 6: The next part, the most time-costly part, was the drawing of the patterns of the butterfly. I chose to use black to color-code the patterns that I outlined. To match the shape of the patterns, I used rotated ellipses and parabolas. The outer patterns were drawn using a combination of linear and parabolic lines, while the inner patterns were mostly circles and ellipses. The hardest part of this section was matching every single shape of the butterfly. Most patterns used at least 4 equations each to outline. I used nearly 100 equations to finish this section.

Figure 9: Superimposed outline of butterfly patterns

Step 7: I added some details and changed some of the colors using the RGB value and slider (of h,s, and v) to fit the brown tone of the butterfly. With Dr. Tong’s recommendation, I also tuned down the opacity (to 2.6) of the background to highlight my work.

Figure 9: Slider that controls hue, saturation, and value parameters.

Figure 10: The lowered opacity of the background

Step 8: I converted some of the equations to polar by substituting x and y as r cos(θ) and r sin(θ). Then, using trigonometric identities and algebraic manipulation, I simplified the equation and express it in terms of r and θ only. Finally, I solved the equations for r in terms of θ to obtain the polar equations.

Step 9: I eventually went over to my butterfly and added some inequalities to fill a few of the shapes in, and changed the background into a polar graph.

Figure 11: The final product

Figure 12: The final superimposed graph

My unique mathematical butterfly had finally emerged; the result was definitely worth all the hours of hard work. Thank you for reading this log!

The Desmos link is available here: https://www.desmos.com/calculator/aim4p21ltm