N-Units Away Curves

              I have been studying this mathematical problem in one form or another since childhood. As a small child playing with crayons, I discovered that when successive curves are drawn alongside one another, the smooth curves eventually begin to “crumple” and “become pointy,” as I used to describe it. Little did I know at the time that I had stumbled upon a fascinating phenomenon involving the loss of differentiability in a particular class of parametric equations.

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              Even since then, I have pursued this problem across multiple stages of my life: from first discovering it with crayons in my youth, to successfully describing its underlying parametric equations in high school, to finally solving the phenomenon as a college independent research study. What started as a child's curiousity in his doodles eventually evolved into a grand and challenging mathematics research project. Herein is a sample of my work as a mathematician: