Gulliver Knew His Physics ...

One of the best illustrations of the influence of Isaac Newton, and of the intellectual capability of the eighteenth century readers, pops up in Jonathan Swift's Gulliver's Travels (1726, amended 1735). The official name is Travels into Several Remote Nations of the World, in Four Parts. By Lemuel Gulliver, First a Surgeon, and then a Captain of several Ships.

Most people have heard of Gulliver's visit to Lilliput, home of the little people. Many don't realize that gulliver travelled to several other lands, too. Swift wrote the book as a satire and condemnation of the injustices of society in the century before the great democratic revolutions. But Swift had to be careful. He couldn't just come right out and say, for example, that the king was an ass, or that the British House of Lords was composed of imbeciles, or that justices of the peace were corrupt. Doing so could have got him thrown in jail, or worse.

Instead, he had Gulliver visit lands where the king was an ass and the appointed governing body was composed of imbeciles. When challenged, he could always say that he'd written a book of fiction. He wasn't talking about home, for goodness sake.

Isaac Newton had revolutionized science. He showed that the universe runs on simple mathematical laws. He broke down the distinction between heaven and earth by showing that the same laws of physics that govern the actions of objects on Earth apply to the planets. Prior to Newton, nature was mysterious, and scientists (or natural philosophers, as they were more properly called) were limited to explaining Earthly phenomena. The Heavens were governed by their own laws, and why not? Heaven was the realm of God, and God should not be subject to the same laws that constrained the behaviours of Earthly objects.

A generation before Newton, Kepler found that the planetary orbits follow mathematical rules. The planets travel on elliptical paths at varying speed. (Their speed increases as they get closer to the sun.) And Kepler found a fascinating, but mysterious relationship: the cube of the a planet's average radius of orbit divided by the square of the planet's period of orbit is the same for all planets. In math lingo, T-squared varies directly with R-cubed.

Newton used this relationship, Kepler's elliptical orbits, and the calculus that Newton invented while on summer holidays when he was an undergrad at Cambridge University to come up with the Law of Gravity. He could demonstrate that the same force that makes apples fall keeps the planets in orbit around the sun. When he published his discoveries (some twenty years after he made them) he became an instant celebrity.

So, back to Gulliver. On his voyage, he landed at Laputa, where astronomers studied the heavenly bodies through great telescopes. In Swift's time, no moons of Mars had been discovered, so he decided to pretend that Laputian astronomers made a fascinating discovery. Read this:

They have discovered two lesser stars, or satellites, which revolve about Mars; whereof the innermost is distant from the centre of the primary planet exactly three of his diameters, and the outermost, five; the former revolves in the space of ten hours, and the latter in twenty-one and a half; so that the squares of their periodical times are very near in the same proportion with the cubes of their distance from the centre of Mars; which evidently shows them to be governed by the same law of gravitation that influences the other heavenly bodies.

Did you see what he said? Kepler's T-squared/R-cubed law holds for the moons of Mars, showing them to follow Newton's Law of Universal Gravitation!

I think this speaks volumes about Swift's society. Swift counted on his readers knowing Kepler's and Newton's Laws. He knew that his characters had instant credibility if they were included in Newton's great intellectual revolution.

How many authors of fiction today assume their readers would be as intellectually accomplished?

How Many Dimensions Are There?

We have three physical dimensions: length, width, and height. A mathematician or physicist might represent them by x, y, and z.

In his Theory of Relativity, Einstein used time as a dimension, expressing the position of an object in space-time with four co-ordinates: x, y, z, and t. Hence the phrase "time is the fourth dimension."

Remember that the number of dimensions is just a convenience. Einstein's formulas, using four dimensions, described the universe better than Newton's equations.

You may have heard of string theory using eleven dimensions. What does this mean? Why eleven? Here's a quick explanation.

An application of the laws of conservation of energy and conservation of momentum, in senior high school physics, is calculating the final velocities of two colliding balls, given the balls' initial masses and velocities. (And the angle between them, technically included in the word "velocity".) There are two unknowns, the final velocities of each object, and two equations to use. You always need the same number of equations as unknowns.

What if there are three balls? You need three equations to find the three final velocities. But we don't have a third equation. This is the famous Three Body Problem. It's unsolved: physicists can't compute an exact answer.

But nature can! How does nature figure out what happens to three simultaneously interacting objects? It's clear that nature does know, because this situation comes up all the time. The sun, moon, and Earth are simultaneously interacting. (An interaction is one object exerting a force on another, and all three objects have gravity, which extends to infinity.)

Physicists attack the problem by dealing with the bodies in pairs, or approximating the situation by saying that the smallest object doesn't influence the others very much. But it would be nice to have a third equation, to get absolute answers instead of numerical approximations.

A special three-body question that can be solved by students is if one object comes in and hits two identical objects, like two balls touching and the third arriving on the mid-line between them. Because the situation is symmetrical, you can find an answer. (The third equation is that the final speed on one ball equals the final speed of the other, through symmetry.)

So you can solve more complex questions if symmetry is involved. Remember this fact.

Now picture a circle. It looks the same from all angles. Perfect symmetry. Even a small circle is the same as a large circle, in one respect, because a small circle is the large circle viewed from farther back.

How about a square and a diamond? Are they the same? Sure: a diamond is a square rotated.

How about a square and a hexagon? (A hexagon has six equal sides.) You can't rotate a square or view it from a different angle and direction and see a hexagon. So in two dimensions, a square and a hexagon are different.

Now use your imagination. If you illuminate a cube with a light directly overhead, the shadow is the shape of a square. But if you turn the cube, you can get a shadow the shape of a hexagon. (To convince yourself, draw a hexagon, and add the "missing lines" to make it look like a 3-D cube viewed at an angle.)

So, if you think in three dimensions, a square and a hexagon are the same thing. They're both 2-D shadows of the same 3-D object. The lesson here is that if you include an extra dimension in your considerations, you can sometimes find symmetries that didn't exist when you were working in fewer dimensions.

And symmetry allows you to solve otherwise unsolvable equations, remember?

The string theorists use eleven dimensions. Their equations are so complex, apparently, that they need eleven dimensions to give them enough symmetries to solve them.