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.

Extra dimensions
Current cutting edge physics talks about the universe being made of eleven or so dimensions, not the three physical and one temperal (x,y,z,t) of Einstein. This concept is very hard to visualize, of course.
One of the best books I have read that helped me get an intuitive grasp of the many dimension idea is Galileo's Finger, by Peter Atkins. I highly recommend the book for thinkers, for people who want to be taken past what they already know.
In my high school classroom, I marry an idea from his book with my own examples to show why extra dimensions arise in physics. In a nutshell:
1. In senior physics, students learn how to solve 2-D collisions. Ball A of a certain mass and velocity hits ball B, with its mass, and velocity, at some angle. The laws of conservation and momentum allow the two final velocities (including direction) to be calculated. Basically, two equations are used to calculate the two unknowns (the two final velocities) from the initial data.
2. What if there were three bodies hitting at EXACTLY the same time? Now there are three unknowns, but still only two equations. So, exact solution can't be calculated. This is the 3-body problem, still unsolved. (We can get approximate solutions by numerical methods, but the exact solution is not known.)
3. How about if one object comes in from the west and hits two identical objects balls lined up in contact north-south? (i.e. ball at 12 o'clock, ball at 6 o'clock touching in the centre of the clock. Other ball comes in from 9 o'clock) This 3-body problem is solvable because of the symmetry. The two stopped balls will pick up the same velociy, different directions but equal angles from 3 o'clock direction.
4. Thus if there is symmetry, we can calculate answers to problems which would otherwise be unsolveable.
5. Atkins points out that a line segment horizontal line segment is equivalent to a vertical line segment. We just rotate it (or turn our head!) Similarly a square and a diamond are equivalent, because one is a rotation of the other. (Turn the square 45 degrees.) Are a square and a hexagon equivalent? No. You can't rotate a square and make it a hexagon. Except that...
6. Draw a square and a cube. The cube is the 3-D object whose shadow is a square. But if you rotate a cube, its shadow can be a hexagon. (To see this, draw a box whose corner is rotated toward you, viewed from slightly above.) Therefore a hexagon and a square can be considered equivalent, or symmetrical, if you think in 3-D instead of 2-D.
7. We have already seen that if you find symmetries you can solve previously unsolveable problems. And we just saw that by viewing a concept from a higher dimension, you can find symmetries that were not apparent (or not there!) in lower dimensions.
8. Physicists are attempting to find formulas to describe the fundamental subatomic particles, and the behaviour of radiation and matter at the very small and large, and quantum theory, and relativity, and gravity (as discussed in general relativity). They haven't succeeded in finding exact solutions to the various equations that combine to describe things. BUT...
9. By working in more directions, symmetries were found that enable solutions.
10. Apparently, solutions exist if you use eleven dimensions.
So, there you are. Like all concepts in science, we find explanations that satisfy us and are useful. Eleven dimensions proves to be useful. Hence, proposing eleven dimensions becomes useful.