Extreme Musical Venues

No, I don’t mean this: www.extrememusic.com. But now that I’ve got your attention, I want to return to a topic I wrote about last month: the Earth’s gravity and its effect on mechanical metronomes (or, more abstractly, its effect on pendulums). That post boiled down this fact:

Pendulums near the poles will swing more quickly than pendulums of the same length near the equator.

Of course, that begs the question—how much more quickly? At the time of my previous post, I hadn’t researched the mathematics required to do this. But now I’ve crunched some numbers and have some results to share. Before that, though, let me set up some of the basic ideas behind the calculations.

1. The Earth isn’t a sphere. We know today that the Earth is an oblate spheroid, and is flatter at the poles than at the equator. The verification of this fact dates back to the 1730s, and is credited to Pierre Louis Maupertuis. It’s worth mentioning that the oblateness follows from Newton’s mechanics, of which Maupertuis was a notable advocate. [Maupertuis did many other things, and you can read about them in this book.]

2. Gravity varies according to latitude. This follows quickly from point (1) above, though the first attempt at a precise formula was made by Alexis Claude de Clairaut in his 1743 treatise, Théorie de la figure de la terre. W. W. R. Ball has a short summary of Clairaut’s work. Today, we use a more precise formula, as part of the World Geodetic System:

Screen Shot 2013-09-21 at 7.11.24 PM

Here, GE ≈ 9.78033 is gravity at the equator, e ≈ 0.0818192 is the Earth’s eccentricity (which measures the “sphereness” of the ellipsoid, ranging from  0 to 1), and k ≈ 0.00193185 is a parameter that depends on gravity at the poles. (Details can be found here on page 4-2). 

3. Pendulum formula requires calculus. I’m going to skip the derivation, and point you to Wikipedia on this one (grain of salt and all that). Here’s the formula for one “swoop” of a pendulum:

Screen Shot 2013-09-21 at 7.25.12 PM

with l being the length of the pendulum and θ0 being its initial angle.

4. Gravity decreases with altitude. The experts use something called the free-air correction to compensate for this. For example, Reynolds’ An Introduction to Applied and Environmental Geophysics states that the proper way to adjust the formula in part (2) above is by subtracting (3.086×10-7)h, where h is your elevation above sea level (I’ve converted the units from Reynolds’ version).

Now we’re ready for some results! For all calculations, I took θ0 = π/4, and l ≈ 0.229060. This way, a pendulum at sea level at the equator will beat exactly 120 times per minute. Then, I picked five extreme points on the surface of the Earth:  the summits of Chimborazo and Everest, and the Antarctic stations McMurdo, Russkaya, and Amundsen-Scott. And here’s what we have:

Location Elevation Latitude Beats per Minute
Reference Point 0 m. 120
Chimborazo 6268 m. 1.469° S 119.881
Everest 8848 m. 27.988° N 119.902
Russkaya 0 m. 74.766° S 120.296
McMurdo 0 m. 77.850° S 120.304
Amundsen-Scott 2835 m. 90° S 120.264

I’ve ignored the fact that the mass of the terrain that you’re standing on will affect the gravity at its summit (the Bouguer anomaly)—so the values for Everest and Chimborazo are probably a bit off. But, taking the numbers at their word, our hypothetical metronome will move the most quickly at McMurdo Station—which has the double-benefit of being both close to the pole (and thus closer to the center of the Earth) and having a low elevation. (In fact, McMurdo has the world’s southernmost port; you can go see what they’re up to right now.)

To conclude, it appears that a piece of music set to 120 beats per minute, that takes 5 minutes to perform, will finish almost exactly 1 beat sooner at McMurdo station than it would at the summit of Mount Everest. Not enough to mess anything up musically, but enough to notice. Extreme, right? (Not extreme enough for you? Then go read this comic.)

Convocation and Advice From Old Books

At my university, today marked the beginning of classes for the fall semester. It began with convocation (at which our University president provided a much-appreciated etymology of the word “convocation”), and classes followed quickly thereafter.

Since things are rather busy for me this week, I’m kicking the can down the road—no major posts until next week at the earliest. But for now, I’d like to advertise a blog I’ve recently discovered, Ask the Past, which describes itself as providing “advice from old books.” Here are three posts that seem appropriate for the start of the school year:

Plus ça change, plus c’est la même chose…

Bringing a Riemannian Gun to a Euclidean Knife Fight

The prime numbers have fascinated mathematicians for many centuries. As far back as 300 BCE, Euclid proved that there are infinitely many primes.

Today, part of the appeal of prime numbers is that there remain many apparently-simple questions about primes whose answers are unknown. (One famous example is Goldbach’s conjecture, posed in 1742 in a letter to Leonhard Euler, that every even integer > 2 can be written as the sum of two primes. Note, however, that Terence Tao has recently proven that every odd integer > 1 is the sum of at most 5 primes).

Over the years, the math (and its conjectures) has gotten considerably more complicated. The most famous (infamous?) example is the Riemann hypothesis; let me just drop the statement on you:

Riemann’s hypothesis: The nontrivial zeroes of the Riemann zeta function have real part = 1/2.

Don’t worry if you don’t have the mathematical background to make sense of this statement, but do note that it carries a hefty reward if it’s proven: the Clay Math Institute is offering $1 million for a solution to this problem.

In this post, I’m going to reverse the historical trend: I will begin with something complicated and deep, and use it to derive something quite simple.

First, let’s take the imaginary parts of the Riemann zeta function’s zeroes:

14.134725142, 21.022039639, 25.010857580, 30.424876126, etc.

My list goes up to the 100th one, at 236.524229665. Let Zn be the nth entry in this list, and now define the function F(x) according to the following rule:

Screen Shot 2013-09-03 at 10.23.44 AM

Clearly, F depends on our choice of N, which is where things get interesting. Here’s a graph for N = 5 (click to enlarge):

Screen shot 2013-09-03 at 8.08.26 AM

F(x) for N = 5.

Neat, but not very enlightening. (By the way, note that the vertical axis is at x=5, not x=0.) But what if we take N = 20?

Graph of F(x) for N = 20.

Graph of F(x) for N = 20.

Notice where those major dips occur: at x = 2, 3, 5, 7, etc. We’ve found some prime numbers! And it keeps getting better—here’s the graph for N = 100:

Graph of F(x) for N = 99.

Graph of F(x) for N = 100.

Notice that while the primes cause the most pronounced dips, other numbers produce more moderate dips—look at 4, 9, 8, and 16 for the best four examples of this. And some numbers don’t correspond to anything significant in the graph—see 6, 10, 12, 14, 15, and 18.

Take a second to reflect on this: we’ve used the zeroes of the Riemann zeta function (the “gun” from the title) to find the sequence of prime numbers (the “knife fight”)! And along the way, we’ve found a correspondence between the analytic properties of F(x) and some non-prime numbers! And, as you might imagine, the graph of F(x) will produce more accurate values as N increases.

Clearly, there’s a theorem lurking in the background of this. Unfortunately, I can’t offer a reference. I remember seeing someone speak about it at a conference, and I hastily wrote down the formula, but that’s all the information I have. So if anyone recognizes this, or has any insight on the non-prime dips, let me know!