Hey, can I ask a really stupid question? I should wiki it, I guess, but can somebody here give me an idiot's guide answer to 'What is a dimension?'
Athon asks.
Part of figuring out if I've understood this stuff is trying to explain it to others in layman's terms, so I hope those of you who are more experienced will indulge me.
So here goes Jimbo's first layman's lesson in linear algebra... aherm...
...
i)
A dimension is simply a direction.
The rest:
Stick your arm out in any direction and look down it and you've just defined a dimension by defining the direction. Now, let's define 2 dimensions (or directions):
y
|
|
|
|
|
-------------------- x
These two
dimensions are
directions that will define any point in a 2 dimensional
space. It goes like this, the point (x,y) = (3,4) is 3 units along in the x direction and 4 units along in the y direction.
Now you can ask, why these directions and not, say, something between y and x? It is because once you have defined these two lines, any other line can be expressed as some combination of them. So a point 1 unit along a line 45 degrees between them is actually (0.707 x, 0.707 y). You can check this using Pythagoras (or 1 sin(45)). The mucky linear algebra term is to say that this line is
linearly dependent on x and y. Now... you
could take any two arbitrary lines (so long as they are not right on top of each other), but x and y as shown here have the nice property of orthogonality (that means they're 90 degrees apart). The reason math uses orthogonal, rather than 90 degrees is that when your
vectors are equations or something else, it doesn't make sense to express their separation in spatial degrees.
ii)
A dimension is often taken as a value and a direction
If you say "1 unit in the direction of x" you have defined a
unit vector. It can be expressed (x,y) = (1,0). The y unit vector is (0,1). These two unit vectors
span the space in that every other vector can be expressed by multiplying and adding these guys, so (3,4) = (3x(1,0), 4x(0,1))
iii)
A dimension is merely a number
It is the number of these unit vectors that will define a point in a space. So a 1D space has the unit vector (1) and is 1 dimensional. A 2D space has the unit vectors (1,0),(0,1). A 3D space has the unit vectors (1,0,0),(0,1,0), and (0,0,1) or (x,y,z). Now we're in the "space" of everyday experience! How about a 4D space?
(x,y,z,ct) or (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1)
You see that I have easily described 4 orthogonal vectors mathematically in a 4D space. However, just try to
draw an everyday picture of 4 lines, each crossing all the others at 90 degrees!
Try writing out the unit vectors for a 5D space yourself. Since we are largely 3D creatures, try to visualize a 5D object in 2D. I can't, although I saw some neato attempts at 4D objects in Wikipedia (see tesseract).
iv) Physics!
The dimensions (x,y,z,ct) might be thought of as (length, width, depth, time) and now you're working in the realm of mechanics! “ct” is a representation of the speed of light times time. This is what they call 4D spacetime. It makes sense, actually, if you represent x, y and z in units of metres, then ct gives (m/s times s), the seconds cancel out and you're measuring all of your 'dimensions' in units of metres! When they say that the 5th dimension (or whatever) is only a few millimetres long, I'm assuming (without any reference to the article) that they've done this sort of correction. (For those in the know... is this true?) so, if you haven't already guessed, the units in 5D are:
(m, m, m, (m/s)s, something something equalling metres

) and the unit vectors are:
(1,0,0,0,0), (0,1,0,0,0), (0,0,1,0,0), (0,0,0,1,0) and (0,0,0,0,1). Your first point in 5D space might be (1,1,1,1,1) which means 1 unit along in every direction. Note that this still accounts for everyday experience! (1,1,1,0,0) would mean that you are at 1 unit of length, width and depth, now is time zero and zero in the fifth dimension!
v) Advanced physical theory
I'm not qualified to say anything about this, but I have to ask the question... why aren't there (physically) some number
n arbitrary dimensions? Math allows for (1,0,0,0,0,0,0,0,0,... etc.) I
assume that something particular in the individual theories constrains or dictates the number of dimensions they are working with.
… whew…
Easy enough?
