Mathematical Photography by Justin Mullins
MUSIC
For Simon, my piano teacher, 4 October 2002
Do math equations like being photographed? We don’t know, but Justin Mullins might…
The Art of Justin Mullins
Mathmatical Photography–Images from another world.
From Justin Mullins’ web site: If mathematicians are explorers, then my role is that of a photographer who retraces their steps. During my journey, I photograph what I find. By that I mean frame it, record it and later present it.
There is nothing particularly special about this process. In the same way that an ordinary photograph is a snapshot of an area of outstanding natural beauty, a mathematical photograph is a snapshot of mathematical beauty.
The MUSIC equation caption reads:
Add together two simple waves and you produce a third, more complicated wave. Add another simple wave and the waveform becomes even more complicated and so on. In fact, the process of adding waves together allows you to produce a waveform of almost any shape and in principle any sound can be created like this, even complex pieces of music.
But the reverse process of analysing complex pieces of music is much more difficult. When starting with a very complicated sound it is not immediately clear what simple waves have been added together to create it. But the mathematical process shown here does this for you. No matter how intricate the sound, this equation teases apart its waveform to reveal its basic components.
The equation is a called a Fourier Transform after the 19th century French mathematician and engineer Joseph Fourier. It takes a complicated wave represented by x(t) and breaks it into simple waves of frequency ω. The entire spectrum of these waves is represented by X(ω). The symbol e is a constant number, i is an imaginary number, t is time, ∞ is infinity and ∫ … dt represents a mathematical process called integration which is related to addition.
FURTHER READING
A derivation of the Fourier transform is at:
http://mathworld.wolfram.com/FourierTransform.html