# Epicycles & Visual Music

I spent some time this evening looking again at the motion of particles moving along epicyclic trajectories. The motion of a single particle q in the complex plane, z, is given by the following parametric function of t:

$z_{q}(t) = Re^{i\omega_{q} t} - re^{i\Omega_{q} t} \;\;\; with \; \Omega_{q} > \omega_{q}, \;\; R > r$

The animation above is generated using 50 particles moving according to the same parametric equation, with angular velocities given as integer multiples of a fundamental value:

$\omega_{q} = q \omega_{1}, \;\;\; \Omega_{q} = q \Omega_{1}, \;\;\; q = 1, 2, 3 \cdots$

With the velocities distributed in this way, the angular positions attain various kinds of harmonic relations, whence the arrangement of the particles in the complex plane falls into simple symmetric patterns. The animation bears some resemblance to the visual music of James and John Whitney. In his later work John Whitney made use of digital computer programs to create visual music animations (Whitney, 1981), using particle systems related to the one described here.

[1] John Whitney, Digital Harmony: On the Complementarity of Music and Visual Art , New York, McGraw-Hill, Inc., (1981).