Q. Is there really a 4th dimension?

A. If you want a 4th spatial dimension, the jury is still out. There are theoretical physicists who think there is, and others who think there isn't, and so far nobody knows who is right. You're not going to find a 4th dimension that you can touch and perceive directly, but someone might be able to figure out if there's one out there even though we can't see it or touch it.

If you'd be satisfied with a less geometric 4th dimension, then the answer is yes... there's a 4th dimension. It just might not be what you think. Physicists say that spacetime is four dimensional: 3 spatial dimensions+1time dimension. Time is rather different than a spatial dimension, because we can't go backwards in time, but it is another degree of freedom. If I want to specifically locate something, I need to say not only where it is, but also when it is there. Any time you use 4 variables to describe something, it's a fourth dimensional problem (spacetime: (x, y, z, t)).

Q. So what good is doing geometry in 4 dimensions, if you don't have 4 spatial dimensions?

A. Geometry is simpler in many ways than physics is, and the math is easy to apply and find analogies for, once you understand it well enough (4th dimensional calculus isn't much harder than 3rd dimensional calculus), so figuring out math tools to help you measure something in a geometric 4th dimension isn't too hard. (My brother, who is also a mathematician, figured out a formula for the volume of an n-dimensional sphere during church one day using calculus, so going from 3 dimensions to any dimension you want took less than an hour).

Once you have the 4D geometry, then you can apply it to spacetime physics--all you need is the right equations... (OK--that's simplifying a lot--those physics equations are really pretty complex and different, but they needed the simple 4D math stuff to get it started). So 4D geometry helps theoretical physicist figure out relativity problems in spacetime.

4D math helps theoretical physicists figure out things like: if a light wave goes near a gravity well (a place where spacetime is curved), what path will it take (it always takes the straightest path possible, but going straight on a curved surface is hard to compute).

Q: Anything else 4D math does?

A. Sure--anytime anyone has a math equation with 4 variables, it's a 4D problem. Most of the things you are used to are 2D problems, so first I'll give you an example of 3D problems:

How fast (v=velocity) sound travels through an ideal gass, when it has pressure p and density d

That's 3 variables: v, p and d. If you want to graph the equation, you need a 3D graph;

is the (approximate) surface area of a human body, if the person has height H (in cm) and weight W (in kg).

That's 3 variables: A, H and W, so it's a 3D problem, and gets graphed in 3D:

A 4D equation would be:

F is the force due to gravity between two objects, M is the mass of one object, m is the mass of the other object, and r is the distance between them. I'd need 4D graph paper to graph this equation, you don't get a picture here, but if you knew some 4D math you could find out things like: what different masses an r's give you the same force.

Q. What sorts of things does 4D math do?

A: Well, if you have a curved spacetime--like around a gravity well--then there's math that will help you figure out the straightest path a light wave could take going near the gravity well. If you have a s

Some questions and answers we didn't make it to this semester:

2. Physics: A. When a physicist says we live in 4-dimensional spacetime, what dimensions are they talking about?

B. What does 4 dimensional geometry have to do with 4-dimensional spacetime?

3. Generalized dimensions: A. Give an example of a situation where you might have a 3-dimensional function. Explain what the 3 dimensions are.

B. Give an example of a situation where you might have a 4-dimensional function. Explain what the 4 dimensions are.

2. A. The four dimensions of space time are the 3 usual spatial dimensions (height, width, length) and time. In physics, we use four numbers (x, y, z, t) to specify a point: where it is, and at what time.

B. Physicist use what mathematicians have figured out about the geometry of higher dimensions to solve physics problems. For example, physicists can write down an equation with the 4 variables x, y, z, t that represents what gravity does. If they think about it geometrically, it makes a gravity well in space-time, so they think of space-time as being curved. When they use geometry to show what gravity does, they can use what mathematicians have figured out about straight paths on a curved surface to figure out exactly what a light wave/particle will do near a gravity well.

3. In class we used the examples of : and

is the equation that tells how fast (v=velocity) sound travels through an ideal gass, when it has pressure p and density d (k is a constant). There are 3 variables in this equation: v, d, and p, so it is a 3D function. The 3D graph that goes with this function is:

is the equation that tells the (approximate) surface area of a human body, if the person has height H and weight W. There are 3 variables in the equation A, H, W, so it is a 3D function. The 3D graph that goes with this function is :

You don't need to use a known function like these for your answer, you can come up with your own. Your answer should make sense (it should make sense to use two of the variable values to figure out the value of the third), and you do need 3 variables.

B. In class we used the example of the force of gravity equation as an example: . This is a four dimensional function because it has 4 varaibles: F (force), M (mass of one object), m (mass of the other object) and r (the distance between the objects) (g is a constant).