I recently saw someone using the Amaziograph app for the iPad, it allows you to create drawings with rotational symmetry and people were making really fancy doily shapes by swiping their finger across the screen. I knew it would lend itself well to line art but as I investigated these apps I discovered they were all raster based and wouldn't easily give vector artwork that could be cut so I had to find another way.
It's fairly straight forward to replicate in Inkscape. Inkscape allows you to clone an object and the clones all update when the original changes. I drew a simple line and then cloned it dozens of times, arranging them all with rotational symmetry. As I modified the base line, all the other lines updated and these interesting patterns grew in complexity. The drawing was already vectored so it was easy to get it across to the laser. The whole pattern was cut in a single laser line so I filmed the process too. (svg here)