Color-changing Garments
The Math Behind Transformative Fashion
By: Alannah Sung
Introduction
Math and Fashion are often seen as vastly unrelated fields. After all, one uses equations and analytical skills, while the other involves creativity and artistic expression. In reality, the process of fashion design deeply involves math. From pattern making to innovations in fashion, the math and fashion worlds converge to create exciting and designs that can transform the landscape of modern fashion.
The Project
This project is done for Concordia International School’s AP Precalcalculus class, under the guidance of Dr. Tong. The goal was to create a transformative piece of art utilizing mathematical calculations. My vision was to create a color-changing dress with 3d flower embellishments using thermochromatic dyes. Under the exposure to UV-rays, the flowers on the dress will transform from plain white to a vibrant pink, with pigment concentrated in the inner center of the flowers and branching out to the petals. This creates an image of flowers coming to life, almost as if by magic — when in reality, this can all be explained by math.
Part I: The Design
Firstly, I created a base strapless dress using a pattern found online here. This creates a clean canvas to work with, shifting the focus on the transformative aspect of the design. Although the process of taking measurements and making patterns already involve math, the real calculations come in for the creation of the flowers.
The 3d flowers were made by using the polar equation r = 10 cos (5θ/3) to create a stencil with 5 petals, slightly overlapping each other. This is how the equation looks when graphed:
I created 3 stencils of different sizes by adjusting the maximum radius. The largest flower had petals with a radius of 10, the medium had a radius of 7, and the smallest had a radius of 3. Each flower had 3 layers of petals stacked on top of each other to create depth. After hand-sewing the fabric flowers onto the dress, this was the result:
Part II: The Transformation
Now for the mind-blowing part of the design — it changes colors! Using a type of thermo-chromatic paint called Leuco dyes, the flowers on the dress change from colorless to vibrant within seconds under UV exposure. Well, how does this work? The color on the flowers comes from Leuco Dyes, which are pigments that change color when exposed to Ultraviolet rays (UV) or heat. The dye that I chose changes from white go bright red when exposed to UV. The dyes contain microscopic capsules that contain a colorant, an organic acid, and a solvent. Without exposure to UV, the solvent remains in a solid state, keeping the colorant and acid in close proximity to each other. As a result, they reflect light and create the base color, which is white in this case. As UV rays reach the solvent, the dye undergoes a chemical change, and the colorant and acid separate, uncovering the underlying pigment, red.
Now, how does this connect to math? The answer has to do with different wavelengths of color. Color is simply how our brains percieve light, which is a wave. Every lightwave has a particular wave length — the length of each cycle of the wave.
The sun constantly emits every wavelength of light, and when this is reflected off an object, our brains percieve it as white, just like the Leuco dye before it is affected by UV rays. After the Leuco dye is chemically altered, it absorbs most wavelengths and only reflects one — red. This is what causes the flowers on the dress to change color under UV ray exposure.
