Times-Table Circles: The Mathematics Behind Their Stunning Curves

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).
https://divisbyzero.com/2018/04/02/i-heart...

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).
https://www.scribd.com/document/797275896/...

Abrams, Adam. Modular Multiplication Stimulation.
https://theadamabrams.com/modularmultiplic...

Mathologer (Burkard Polster). Voltex Mathematics (YouTube video).
   • Tesla’s 3-6-9 and Vortex Math: Is this rea...  

University of Waterloo. Generalized Mandelbrot Sets.
https://www.math.uwaterloo.ca/~wgilbert/Fr...