Monday, April 20, 2015

Pi -- Part Three

But there’s still more to π. After all, other famous irrational numbers, like e (the base of natural logarithms) and the square root of two, bridge different areas of mathematics, and they, too, have never-ending, seemingly random sequences of digits.

What distinguishes π from all other numbers is its connection to cycles. For those of us interested in the applications of mathematics to the real world, this makes π indispensable. Whenever we think about rhythms — processes that repeat periodically, with a fixed tempo, like a pulsing heart or a planet orbiting the sun — we inevitably encounter π. There it is in the formula for a Fourier series:

A Fourier series is a way to represent a wave-like function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The Discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series.

The Z-transform, another example of application, reduces to a Fourier series for the important case |z|=1. Fourier series are also central to the original proof of the Nyquist–Shannon sampling theorem applied, among other uses, in the encoding of CDs, DVDs, and modern "digital" television transmission. The study of Fourier series is a branch of Fourier Analysis.

The Fourier series is named in honor of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Harmonic analysis is among the most successful and applicable branches of modern mathematics. It is an indispensable tool in subjects ranging from number theory to partial differential equations and numerical analysis. All that starts with a simple circle and the relationship given the name of the most famous Greek letter.

Combining the trigonometric functions of sine and cosine, the Fourier series, among other interesting facts, embodies the concept that all complex waveforms from music to the strange waves produced in computer circuits are just made up of an appropriate combination of basic “sine waves” from simple harmonic motion, spinning in a circle, and you know about circles and pi, they go together like apple pie and ice cream. Yummy … and so ubiquitous it fits everything from the rotation of our galaxy to spinning atoms and all the sizes between.

The Fourier series is an all-encompassing representation of any process, x(t), that repeats every T units of time. The building blocks of the formula are pi and the sine and cosine functions from trigonometry. Through the Fourier series, pi appears in the math that describes the gentle breathing of a baby and the circadian rhythms of sleep as well as the wakefulness that govern our bodies. When structural engineers need to design buildings to withstand earthquakes, pi always shows up in their calculations. Pi is inescapable because cycles are the temporal cousins of circles; they are to time as circles are to space. Pi is at the heart of both.

For this reason, π is intimately associated with waves, from the ebb and flow of the ocean’s tides to the electromagnetic waves that let us communicate wirelessly. At a deeper level, π appears in both the statement of Heisenberg’s uncertainty principle and the Schrödinger wave equation, which capture the fundamental behavior of atoms and subatomic particles in quantum mechanics. In short, π is woven into our descriptions of the innermost workings of the universe.

How’s that for an inclusive statement? All from this ratio of the simplest measurements of a circle. This number that seems to have no end … either in digits or in uses. Now that is beautiful!

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