What is a dimension?

[[ infinite ]]

Hello all-

This is something that has been on my mind ever since I first learned about the fourth dimension and beyond, sometime back in fifth grade. This is simply one of those things that I never really 'got'.

What is a dimension? How does an n-th dimension (where n>3) exist? What exactly are the n-th dimensions (where n>3)? What is the evidence for their existence? How are dimensions relevant to scientific thought?

[HallsofIvy]

A "dimension" is the number of "labels" (normally numbers but can be other things) necessary to identify something of interest.

If I am working with points on a given line or curve, I could choose any point on the line as a starting point and identify all other points by their distance (positive in one direction, negative in the other) from that point. I can identify each point by a single number- one dimension.

If I am working with points in the plane, surface of a sphere, or some other surface, I can set up a Cartesian coordinate system, or polar coordinates and, either way, identify each point by two numbers-two dimensional.

If I am working with points "in space", I will need three numbers- space is three dimensional.

If I am doing research on spheres, I might label each sphere by its center (I will need 3 numbers to label the center point) and its radius (one more number). That's a "four dimensional" problem.

Going to physics, the basic thing physicists work with is an "event"- something that happens at a specific point at a specific time. Identifying an event requires the 3 numbers to identify the point and one number to identify the time- four numbers, four dimensions. That's why physicists say they work in a "four dimensional space-time continuum".

On the other hand, if I am working on thermodynamics, with a gas of, say N molecules, I might wish to identify the position (3 numbers) and the momentum vector (another 3 numbers) of each molecule as well as the time. I would be working in "6N+1" dimensional space.

[pmb_phy]

Originally posted by [ infinite ]
Hello all-

This is something that has been on my mind ever since I first learned about the fourth dimension and beyond, sometime back in fifth grade. This is simply one of those things that I never really 'got'.

What is a dimension? How does an n-th dimension (where n>3) exist? What exactly are the n-th dimensions (where n>3)? What is the evidence for their existence? How are dimensions relevant to scientific thought?

I wanted to address this question once and for all some time back so I created this web page. See
http://www.geocities.com/physics_world/ma/coord_system.htm

[[ infinite ]]

Thanks for the reply, it was really comprehensive and made things much more lucid. Also thanks for the site pmb_phy.

Reading HallsofIvy's reply gave rise to yet another question.

Originally posted by HallsofIvy
Going to physics, the basic thing physicists work with is an "event"- something that happens at a specific point at a specific time. Identifying an event requires the 3 numbers to identify the point and one number to identify the time- four numbers, four dimensions.

...I understand the 'mattress' analogy of space-time, and how it can be curved. However, can a specific time be 'identified'? Isn't time individual for all particles? How can "one number... identify the time"?

[Pergatory]

Originally posted by [ infinite ]
Thanks for the reply, it was really comprehensive and made things much more lucid. Also thanks for the site pmb_phy.

Reading HallsofIvy's reply gave rise to yet another question.



...I understand the 'mattress' analogy of space-time, and how it can be curved. However, can a specific time be 'identified'? Isn't time individual for all particles? How can "one number... identify the time"?

It can't, all things are relative. The dimension is used as a way to assign a value to something inherintly infinite. There is no 'one number' to define our position in any dimension unless a zero-point is designated. For example, if I asked you where we were in the X-axis of space, what would you say? Well, assuming you knew the orientation of the X-axis, you would ask where the zero-point is and reply with your distance from that point. To answer your question more specifically, if you define the zero-point of time as the beginning of the universe, we are at approximately 14.5 billion years in the "time" dimension.