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.