Discreteness in space

Many systems with continuous underlying rules generate discrete cellular structures in space. One common mechanism is for a wave of a definite wavelength to form (see page 988), and then for some feature of each cycle of this wave to be picked out, as in the picture below. In Chladni figures of sand on vibrating plates and in cloud streets in the atmosphere what happens is that material collects at points of zero displacement. And when a stream of water breaks up into discrete drops what happens is that oscillation minima yield necks that break.

Superpositions of waves at different angles can lead to various 2D cellular structures, as in the pictures below (compare page 1078).

Various forms of focusing and accumulation can also lead to discreteness in continuous systems. The first picture below shows a caustic or catastrophe in which a continuous distribution of light rays are focused by a circular reflector onto a discrete line with a cusp. The second picture shows a shock wave produced by an accumulation of circular waves emanating from a moving object—as seen in wakes of ships, sonic booms from supersonic aircraft, and Cerenkov light from fast-moving charged particles.