A harmonograph is a mechanical device consisting of two or more pendulums attached to a pen. The swinging pendulums control the motion of the pen, tracing out a geometric pattern on a sheet of paper.
Since the system is damped by friction, the pattern spirals in on itself as time progresses.
Each of the GIFs above simulate the output of a 4-pendulum system (modeled after a harmonograph as configured in this video). The different outputs are generated by using different pendulum length ratios in each simulation.
Mathematica code posted here.
Additional source not linked above.