THAT'S MATHS:Our universe is dominated by periodic variations, from the oscillations of sub-atomic particles to the cyclic pulsations of variable stars. In engineering, we have aircraft vibrations; in mechanics, the pendulum; in music, there are harmonic sounds; in biology, the heartbeat; in neurology there are brainwaves; and in the climate system, El Nino. Galileo first observed the regular swing of a pendulum and realised that time could be measured using its periodic oscillations.
Later, a mathematical description of periodic motions was perfected by the French mathematician and physicist Joseph Fourier. He showed that variations in time can be broken down into regular components called sine-waves, each component having a definite period of variation or frequency. Fourier’s analysis was one of the most important developments of mathematics, and it pervades all areas of applied mathematics to this day.
The basic fabric of the universe comprises oscillations. Light of different colours consists of electromagnetic waves with different frequencies. We think of the constituents of matter as particles. But quantum mechanics shows that they have a dual nature, with characteristic wave-like behaviour. Thus, oscillations underlie everything in the universe.
There is an interesting exhibition running at the Science Gallery in Trinity College, Dublin. Called Oscillator, it illustrates a multitude of wave-like phenomena, from simple mechanical and electrical oscillations to the complex fluctuations of biological systems. Several mechanical systems show a range of oscillating behaviours. A heavy metal plate called an Euler disk spins for minutes, increasing its rate of oscillation as it nears its resting state. An elongated boat-like wooden body called a rattleback rotates happily in one direction, but if spun in the other, stubbornly reverses.
A mass hanging from a spring exchanges energy periodically from vertical springing to horizontal pendular motion and back. And a set of 15 pendulums illustrate wave-like patterns as they move in and out of phase. After 60 seconds, they return as if by magic to their initial synchronised state. But it’s not magic, it’s mechanics.
A long rope, spun by a motor at each end, exhibits a range of wave phenomena. As the audience moves along, their presence influences the rope, causing it to beat and pulsate in a variety of visually attractive ways, generating beautiful 3D patterns. Another wave machine shows how a combination of two sinusoidal oscillations can produce stunning graphical effects.
Another exhibit shows the long-period oscillations in an insect population, the cicada or Magicicada septendecim, which emerges in huge numbers every 17 years to mate, fall to the ground and return to its subterranean state. The prime number 17 appears to have been naturally selected to minimise the overlap of this species with other animals likely to eat it.
Oscillations are a crucial part of digital circuitry. They govern the flow of bits – that is, information – through computer systems and synchronise multiple computations undertaken in parallel. Clock-ticks of computers occur billions of times per second. An exhibit in Oscillator shows a simple switching circuit drastically slowed to a time-scale of a second, enabling us to visualise these incredibly rapid variations. Many of the exhibits are at the interface of art and science, merging logical structure with aesthetic appeal.
Peter Lynch is professor of meteorology at the School of Mathematical Sciences at University College Dublin. His blog is at thatsmaths.com
