A cycloid is obtained by tracing a point on a circle that is rolling along another circle. Us ordinary people know these from our childhood toy called Spirograph. Miseducated people with a homocentric world view know them as planetary orbits.

    Cycloids enjoy having singular points. These are being resolved when lifted (luck!), and the lifts are also knotted (more luck!).

      One can again obtain a knotted Möbius strip this way:

      There are many different cycloids, and I could fill many pages with more images. Let’s finish with a 5-pointed star to ward against evil.

        Again, the spiky singularities disappear, as is best seen in the side views.

          Below is an animation of a rotating knotted minimal Möbus strip based on a the top cycloid, with normal vector also rotating. Enjoy.

          Resources

          Mathematica Notebook

          PoVRay Sources