Friday, February 15, 2013

The Joy of X

Steven Strogatz’s latest book is The Joy of X: A Guided Tour of Math, From One to Infinity. I have discussed books by Strogatz in previous entries of this blog, here and here. The preface defines the purpose of the Joy of X.
“The Joy of X is an introduction to math’s most compelling and far-reaching ideas. The chapters—some from the original Times series [a series of articles about math that Strogatz wrote for the New York Times]—are bite-size and largely independent, so feel free to snack wherever you like. If you want to wade deeper into anything, the notes at the end of the book provide additional details, and suggestions for further reading.”
My favorite chapter in The Joy of X was “Twist and Shout” about Mobius strips. Strogatz’s discussion was fine, but what I really enjoyed was the lovely video he called my attention to: “Wind and Mr. Ug”. Go watch it right now; it is less than 8 minutes long. It's the most endearing mathematical story since Flatland.

Of course, I always am on the lookout for medical and biological physics, and I found it in Strogatz’s chapter called Analyze This!, in which he describes the Gibbs phenomenon. I have written about the Gibbs phenomenon in this blog before, but not so eloquantly. Russ Hobbie and I introduce the Gibbs phenomenon in Chapter 11 of the 4th edition of Intermediate Physics for Medicine and Biology. When talking about the fit of a Fourier series to a square wave, we write
“As the number of terms in the fit is increased, the value of Q [a measure of the goodness of the fit] decreases. However, spikes of constant height (about 18% of the amplitude of the square wave or 9% of the discontinuity in y) remain…These spikes appear whenever there is a discontinuity in y and are called the Gibbs phenomenon.”
It turns out that the Gibbs phenomenon is related to the alternating harmonic series. Strogatz writes
“Consider this series, known in the trade as the alternating harmonic series:
1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + … .
[…] The partial sums in this case are
S1 = 1
S2 = 1 – 1/2 = 0.500
S3 = 1 – 1/2 + 1/3 = 0.833 …
S4 = 1 – 1/2 + 1/3 – 1/4 = 0.583…

And if you go far enough, you’ll find that they home in on a number close to 0.69. In fact, the series can be proven to converge. Its limiting value is the natural logarithm of 2, denoted ln2 and approximately equal to 0.693147. […]

Let’s look at a particularly simple rearrangement whose sum is easy to calculate. Supposed we add two of the negative terms in the alternating harmonic series for every one of its positive terms, as follows:

[1 – 1/2 – 1/4] + [1/3 – 1/6 – 1/8] + [1/5 – 1/10 – 1/12] + …

Next, simplify each of the bracketed expressions by subtracting the second term from the first while leaving the third term untouched. Then the series reduces to

[1/2 – 1/4] + [1/6 – 1/8] + [1/10 – 1/12] + …

After factoring out ½ from all the fractions above and collecting terms, this becomes

½ [ 1 – 1/2 + 1/3 – 1/4 + 1/5 – 1/6 + …].

Look who’s back: the beast inside the brackets is the alternating harmonic series itself. By rearranging it, we’ve somehow made it half as big as it was originally—even though it contains all the same terms!”
Strogatz then relates this to a Fourier series

f(x) = sinx – 1/2 sin 2x + 1/3 sin 3x – 1/4 sin 4x + …

This series approaches a sawtooth curve. But when he examines its behavior with different numbers of terms in the sum, he finds the Gibbs phenomenon.
“Something goes wrong near the edges of the teeth. The sine waves overshot the mark there and produce a strange finger that isn’t in the sawtooth wave itself….The blame can be laid at the doorstep of the alternating harmonic series. Its pathologies discussed earlier now contaminate the associated Fourier series. They’re responsible for that annoying finger that just won’t go away.”
In the notes about the Gibbs phenomenon at the end of the book, Strogatz points us to a fascinating paper on the history of this topic:
Hewitt, E. and Hewitt, R. E. (1979) The Gibbs-Wilbraham phenomenon: An episode in Fourier analysis. Archive for the History of Exact Sciences 21:129-160.
He concludes his chapter
“This effect, commonly called the Gibbs phenomenon, is more than a mathematical curiosity. Known since the mid-1800s, it now turns up in our digital photographs and on MRI scans. The unwanted oscillations caused by the Gibbs phenomenon can produce blurring, shimmering, and other artifacts at sharp edges in the image. In a medical context, these can be mistaken for damaged tissue, or they can obscure lesions that are actually present.”

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