Swingers

This post looks at Mathematical Thinking, which we define as “The belief, especially characteristic of scientists, that events can be understood, described, and predicted using mathematical equations; thinking founded on this belief.” Sometimes it’s a good thing, sometimes not so good.

Imagine you’re at the bar, and order a gin martini. The bar man serves it in the usual conical martini glass. Tell us, when the drink is halfway down the side of the glass how much is left?

Half? No, the glass is pointy at the bottom so there’s clearly less in the bottom half than in the top half. A quarter? Final answer. One quarter!

Wrong.

You may recall from school the idea of similar triangles. They are triangles with the same shape, but a different size. This is what we also see in the cone, our martini glass. The bottom half of the martini glass is “similar” to the full glass.

The problem of calculating volumes of geometric objects goes back to the ancient Greeks. Draw a square on a piece of paper and then draw another with sides twice the length. How much has the area increased by? It’s four times bigger. You can see that four of the smaller squares will fit inside the larger. Now take eight sugar cubes or dice. Put four into a square and put another four on top. You’ve now got a larger cube, each side of which is twice the length of the smaller. So for a cube halve the length, you’ve got a volume that’s a mere eighth of the original!

The shape doesn’t matter here. All that matters is that the half-full glass is “similar” in the mathematical sense to the full glass. The formula is just 1 divided by 2 raised to the power d, where d is the number of dimensions. Since we live in d = 3 dimensions – we’re leaving out spooky extra dimensions – the answer is 1 divided by 2 cubed, which is 1/8, or 12.5%.

Or in mathematical terms: it’s time to order another.

Mathematicians are full of tricks like this where pure thought can lead to answers that are not otherwise obvious or where the brain gets tricked. Say that, reaching for a bottle on an upper shelf behind the bar, the barman’s arm brushes against a light hanging from a cable, making it swing gently. What is the time taken to swing left to right then back again (i.e. the period)?

Galileo first became interested in this problem when he noticed the sway of a chandelier in the Pisa cathedral. There are a few ways of solving it mathematically. Most require some knowledge of physical laws, but there is a simple method known as dimensional analysis that gets you most of the way there. As the name suggests, it is again based on the concept of dimension.

We first ask what quantities could affect the period of the swing. A couple are obvious: the mass of the weight, and length of the string. Anything else? The colour of the string or the time of day shouldn’t come into it, but one can imagine the answer is different on the moon from here in the bar so gravity must play a role. The acceleration due to gravity, usually represented by g, is 9.8 meters per second squared. On the moon it is only about 16 percent of that number, because the moon is less massive.

Now let’s postulate that the period is given by multiplying or dividing those numbers together, where again you are allowed to raise a number to a power. We will include fractional powers, so for example a number raised to the power ½ is the square root. The trick is that, for the formula to work, the units of the expression of the period, which are seconds, have to be the same as the units of the expression that you just made up. The two sides of the equation must be made of the same stuff.

Write it out and you find that the only way to make it work is if – ta da! – the period of the swing is proportional to the square root of length divided by the gravitational constant g. So the mass doesn’t – and can’t – come into it.

There is no magic involved here, no spooky automatic writing or communication with the other side, but in a sense the result is magical, because it shows us something very counterintuitive, namely that the time of the swing doesn’t depend on the mass of the pendulum. The weight only comes in through the acceleration due to gravity. On the moon gravity is less and the swing will take longer. In fact, you can use this relationship to test tiny local perturbations in the gravitational field.

It turns out that if the length is exactly one meter, then the period T is almost exactly 2 seconds, and a single swing counts out a second. Which is why traditional grandfather clocks, first produced in the late seventeenth century, have a pendulum of this length. It also inspired the definition of the metre a century later.

This technique of dimensional analysis has been used to great effect in other circumstances. New Mexico was the site of the first explosion of an atomic bomb, at 5:29am on July 16, 1945. Codenamed “Trinity,” much of the project – including the amount of energy released by the explosion – was obviously kept secret. However one observer at the explosion was the mathematician GI Taylor. Taylor is famous for his work in fluid and solid mechanics, in particular the difficult problem of turbulent flow. He estimated the energy in the atomic explosion by applying precisely the above dimensional trick but with quantities energy, air density, radius of the explosion, and time. He estimated the energy released to be 17 kilotons of TNT. President Truman later revealed it to have been 22 kilotons. Taylor’s approximation was remarkably accurate and shows the power of quite simple mathematics, or perhaps more precisely the power of Mathematical Thinking. As we’ll see, similar thinking was behind the bomb itself – this is a dangerous kind of power.

Few people are as clever as GI Taylor. And unfortunately there are many people as gullible as Doyle. But there is also an intersection between magical thinking and mathematical thinking where the power and beauty of mathematics gets used as a decoy to confound while giving the illusion of clarity. We first discovered this happening in finance and economics. But once you’ve been alerted to the possibilities of such abuse you start seeing examples everywhere.

This is what we call Mathemagical Thinking. And in the wrong hands it can be as dangerous as nuclear weapons.