Try to glue a small mirror to the end of a bent piece of wire fixed to a stable platform and let the laser beam of a laser pointer reflect on it. Entangled spires of an ephemeral red dragon will perform a hypnotic dance on the wall of your room. This voluptuous dance results from two mutually perpendicular harmonic oscillations produced by the oscillations of the elastic wire. 

The curved patterns are called Lissajous-Bowditch figures and named after the French physicist Jules Antoine Lissajous who did a detailed study of them (published in his Mémoire sur l’étude optique des mouvements vibratoires, 1857). The American mathematician Nathaniel Bowditch (1773 – 1838) conducted earlier and independent studies on the same curves and for this reason, the figures are also called  Lissajous-Bowditch curves. Lissajous invented different mechanical devices consisting of two mirrors attached to two oriented diapasons (or other oscillators) by double reflecting a collimated ray of light on a screen, produce these figures upon oscillations of the diapasons.  The diapason can be substituted with elastic wires, speakers, pendulum or electronic circuits. In the last case, the light is the electron beam of a cathodic tube (or its digital equivalent)  of an oscilloscope. This article is about the mathematical theory behind these curves introduced with a demonstration program and an example of education application proposed in 1827 by C. Wheatstone . Finally, we will give a look to the equivalent of the L-B curves in 3D by exploring the spherical L-B curves.