A spirograph is a geometric drawing device that produces mathematical roulette curves of the variety known as hypotrochoids and epitrochoids. In this talk, we will derive a parametric equation to describe hypotrochoids. Then, we will look at hypocycloids with $n$ cusps, their connection to $SU(n)$, and how they can roll snugly inside of each other. This talk should be very accessible to undergraduates. The only prerequisites are some familiarity with matrices and complex numbers.