View Full Version : What is a Dimension?

Maddad

2006-Feb-21, 08:57 PM

We think our world has three, but it also has time, a fourth. I hear discussions about what it would be like to live inside the event horizon of a five dimensional star, and whether string theory uses ten dimensions or eleven. People scratch their heads trying to explain how that could be, so they tell you that the extra dimensions are tiny and curled up. Of course, it's difficult to see how that would not simply be an extension of our existing dimensions. That got me to thinking: what exactly is one?

I think of a dimension as a bi-directional direction in which to travel. Going to the left on a line keeps you in the same dimension as going right on the same line. I see it as unbounded, meaning that it can be great big on little tiny. If you curl a dimension, like a line, then you are making use of a second dimension, a plane.

How do you define a dimension, and does your definition rule out or allow more than the usual four?

Fortunate

2006-Feb-21, 11:48 PM

Hi Maddad,

Let's say you want to put coordinates on a Euclidean plane. You can use so-called rectangular Cartesian coordinates (the ones we usually used in high school) or polar coordinates or a set of coordinates in which the axes are oblique to each other. Of course, there are many other options also. But whichever one you choose, you will need {b]two[/b] of them. If you try to use fewer than two coordinates, you will not be able to include every point, and if you try to use three of them, they will be redundant, some of them will be calculable from the others. So the first definition of dimension is "the number of coordinates necessary for each point to have a unique local representation."

Thus a point in three-dimensional space is specified by three coordinates. If we add time, we have four dimensions (t,x,y,z). Time is different from the other three, in that it cannot have a neg

Edit: I got kicked off line and half the post was lost. I don't have the time or enthusiasm to rewrite it all now.

01101001

2006-Feb-22, 12:33 AM

In case you want to break the confining restraints of the humble physical space in which you exist:

Mathworld: Dimension (http://mathworld.wolfram.com/Dimension.html)

Fortunate

2006-Feb-22, 01:24 AM

I left off in my other post on the verge of saying that the time coordinate is different from the spatial coordinates in that it cannot have a negative value, and events cannot flow backwards in time. This introduces an element of asymmetry. We can add one or more extra spatial coordinates, and their values can cycle like a the numbers on a clock. One can describe this colloquially by saying that they are "rolled up."

The OP also raises the question of whether adding a rolled up dimension actually requires two extra dimensions because, for instance, a circle drawn on a sheet of paper extends into both dimensions of the page. The short answer is no. The "problem" is that by even imagining yourself standing outside the drawing looking at it, you are presupposing added dimensions. In imagining yourself viewing a straight line drawn on a page, you are still requiring the page. We need to imagine the universe from the inside, not from the outside. Viewed that way, there is no difficulty imagining travelling in a direction and eventually returning to our starting point.

snarkophilus

2006-Feb-22, 04:22 AM

I left off in my other post on the verge of saying that the time coordinate is different from the spatial coordinates in that it cannot have a negative value, and events cannot flow backwards in time.

Really? I see no reason why not, especially if you accept time reversal invariance and physical determinism. We sort of define forward time as a general increase in entropy, but there's no reason why time couldn't stop, go backward, then go forward again and end up in the same state (and do this all the time). There are some QM interpretations that claim this isn't the way the world works, but it's debatable, at the least.

I think what is often meant by rolled up dimensions is that the values associated with those properties are part of a bounded (possibly finite) set. There's some argument that all dimensions might be bounded, but on our scale, we can't see the overall curvature of space (in the same way that a flea on a merry-go-round might not see that the outer rim is curved), except under high gravity.

As long as we're talking about spatial dimensions, we may as well mention that sometimes you use other dimensions in calculations, too. For instance, you might consider momentum and space, giving you a six dimensional equation. And you don't necessarily need to use real numbers. Using complex numbers for spatial dimensions can eliminate the need for time variables, for instance. The same goes for things like quantum numbers... you might consider those to be dimensions of your equation, but their values are limited to integers (and often even a finite set of them). One might even consider something like neutrino flavours to be dimensions if the calculation required it.

Huevos Grandes

2006-Feb-22, 04:49 AM

EDIT: Sorry- link is gone now, due to an allegation of copyright infringement. I don't wish to violate any potential forum rules... the reason stated below is not my words. Censorship is a nasty, nasty thing... :(

EDIT #2: Entire post + any and all images have been removed for fear they may or may not be copyrighted. I sincerely apologize that Post #6 (http://www.bautforum.com/showpost.php?p=687369&postcount=6) in this thread has become distinctly not fun. That was certainly not the original intent. Sorry again :)

Fortunate

2006-Feb-22, 04:49 AM

I think what is often meant by rolled up dimensions is that the values associated with those properties are part of a bounded (possibly finite) set. There's some argument that all dimensions might be bounded, but on our scale, we can't see the overall curvature of space (in the same way that a flea on a merry-go-round might not see that the outer rim is curved), except under high gravity.

I never really read about the proposed extra dimension of string theory, Kaluza-Klein theory, etc. Maybe I'm wrong. When I heard the term "rolled up," I thought that meant "circular" or "spherical."

As long as we're talking about spatial dimensions, we may as well mention that sometimes you use other dimensions in calculations, too. For instance, you might consider momentum and space, giving you a six dimensional equation. And you don't necessarily need to use real numbers. Using complex numbers for spatial dimensions can eliminate the need for time variables, for instance. The same goes for things like quantum numbers... you might consider those to be dimensions of your equation, but their values are limited to integers (and often even a finite set of them). One might even consider something like neutrino flavours to be dimensions if the calculation required it.

True. I thought the OP referred to spatial dimensions.

snarkophilus

2006-Feb-22, 05:24 AM

http://img134.imageshack.us/img134/7523/advofbuckaroobanzaifrt9ks.jpg

Wow, that looks like a great movie. I think I will have to hunt it down.

What's the name of the actor in the top right?

Wolverine

2006-Feb-22, 05:35 AM

All you need to know is that the black Lectroids are good (and talk like rastafarians), and that the red Lectroids are bad, and are led by super-maniacal genius, John Whorfin:

Image (http://img134.imageshack.us/img134/7523/advofbuckaroobanzaifrt9ks.jpg)

Hotlinking an 800kb+ file is frowned upon here, as not everyone has the benefit of broadband. Moreover, there are copyright issues involved which are not sidestepped by hosting a file on ImageShack. Please don't do that again.

The Bad Astronomer

2006-Feb-22, 07:52 PM

EDIT: Sorry- link is gone now, due to an allegation of copyright infringement. I don't wish to violate any potential forum rules... the reason stated below is not my words. Censorship is a nasty, nasty thing... :(

This is not a potential rule: it is specifically listed as its own rule: no copyright violations. Err on the side of caution.

This is not censorship, either. This is a legal issue. If you think we censor you, then try a forum where they change what you say and/or delete whole posts if they disagree with the admin.

The rules here about copyright violation and censorship are clear. Read them again.

01101001

2006-Feb-22, 09:31 PM

This is not a potential rule: it is specifically listed as its own rule: no copyright violations. Err on the side of caution.

Good idea. Google was just enjoined (no doubt to be appealed) from using simple thumbnailing within Google Images as being a violation of an adult-content site's copyrights. C-Net: story (http://news.com.com/Nude-photo+site+wins+injunction+against+Google/2100-1030_3-6041724.html)

I hope Google's great tool isn't crippled, but it's under fire -- the kind of fire a place like this doesn't need.

01101001

2006-Feb-22, 09:36 PM

What's the name of the actor in the top right?

John Lithgow as Dr. Emilio Lizardo/Lord John Whorfin (IMDB: Buckaroo Banzai (http://www.imdb.com/title/tt0086856/))

trinitree88

2006-Feb-24, 12:35 AM

We think our world has three, but it also has time, a fourth. I hear discussions about what it would be like to live inside the event horizon of a five dimensional star, and whether string theory uses ten dimensions or eleven. People scratch their heads trying to explain how that could be, so they tell you that the extra dimensions are tiny and curled up. Of course, it's difficult to see how that would not simply be an extension of our existing dimensions. That got me to thinking: what exactly is one?

I think of a dimension as a bi-directional direction in which to travel. Going to the left on a line keeps you in the same dimension as going right on the same line. I see it as unbounded, meaning that it can be great big on little tiny. If you curl a dimension, like a line, then you are making use of a second dimension, a plane.

How do you define a dimension, and does your definition rule out or allow more than the usual four?

Try this. 1. A line

2. a plane

3. a cube

4. a line of cubes

5. a plane of cubes

6. a cube of cubes

7. a line of cubes of cubes

8. a plane of cubes of cubes...etc

easier to picture and describe to the novice, but the math is need to make it work. :shifty:

gzhpcu

2006-Feb-24, 05:42 PM

Abbott's "Flatland" book is an entertaining way to get an introduction to extra spatial dimensions.

http://www.ibiblio.org/eldritch/eaa/FL.HTM

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