A circle. A set of evenly spaced points. A simple rule: connect each point n to b · n mod p. From this, something unexpected appears — smooth curves, spirals, and delicate inner shapes that seem far too elegant for such elementary arithmetic. This video investigates the surprising geometry behind times-table circles and explores why their inner patterns align with epicycloids, a family of curves produced by rolling circles. Starting from visual observations, I moved through failed methods and finally uncovered a structural explanation. If you’re curious about the hidden connections between number theory and geometry — or just want to see how simple modular multiplication can generate beautiful, continuous curves — this is for you. References & Sources: Richeson, Dave. I Heart Cardioids (2018). Rozikov, Utkir A. An Introduction to Mathematical Billiards. World Scientific, 2018. (Background on dynamical systems and geometric trajectories.) Pluffe, Simon. The Shape of b·n mod p (2020). Abrams, Adam. Modular Multiplication Stimulation. Mathologer (Burkard Polster). Voltex Mathematics (YouTube video). University of Waterloo. Generalized Mandelbrot Sets. ~wgilbert/FractalGallery/Mandel/