Making Mathematical Art by Combining Circles

The Mathenaeum puzzle activity Circles on Circles and it's associated app Epicycloids illustrate some aspects of this. Spirographs (as in the examples on the right) are generated in this way, and, indeed, something as simple as two coupled pendulums (see left) can display chaotic motion (something we explore in the Quadratic Chaos activity).

To generate patterns like these mathematically (for analysis, or for computer code), we use the x-y plane, and the following geometrical facts:

• For some constant r, and a parameter φ that varies from 0 to 360°, a plot of the points
      (x,y) = (r cos φ, r sin φ)
  forms a circle of radius r,
• We can rotate a point in the plane by the angle θ about the origin by transforming the coordinates according to:
      x → x  cos θ + y sin θ
      y → y cos θ −  x sin θ.

The Wiggly Creature Constructor below uses a particular mathematical model based on circles and rotations. See what "creatures" you can create by setting different values for the model parameters.