Dumb Dimension question...

[Royston]

This is probably a stupid question, but I was wondering if a point is 0 dimensions, and a line with no width is 1 dimension, can you generate 1 dimension from 2 points of 0 dimensions ?

[matt grime]

explain what you think generate means. A line in R^n is specified uniquely by any two (distinct) points on it, if that's what you mean. given x and y the equation of the line passing through them is (parametrically) x+t(y-x), though i have no idea if that satisfies your notion of generation.

[Royston]

I guess I used 'generate', because 1 dimension (to me) requires 2 points of reference. I was just wondering how you could have 0 dimensions (x) + 0 dimensions (y) = 1 dimension. x (0) + y (0) = 1

 

Or does 1 dimension not require 2 points of reference...I'm probably not visualizing dimensions correctly here. I'm assuming with the above that you can't have 1 dimension until you establish two points to plot the direction (for want of a better word) of 1 dimension. Rather than having 1 dimension and establishing 2 points on that dimension.

 

With regards to another thread I havn't studied maths for 13yrs, and I've only just started studying the basics again about 2 days ago, so please excuse me if this is nonsense.

[matt grime]

Sit back, take a breath, and read what you typed. What are the formal mathematical definitions of the words you're using? What does 'require 2 points of reference' mean?

 

What I'm trying to make you realize is that you're asking questions that in and of themselves, whilst seeming reasonable, are not actually answerable. You're attempting to make concrete your initial vague notions of what these things are in a way that they probaby weren't intended to be used. The phrases just don't make sense to me, for example:

 

"1 dimension (to me) requires 2 points of reference"

 

"0 dimensions (x) + 0 dimensions (y) = 1 dimension. x (0) + y (0) = 1"

 

"you can't have 1 dimension until you establish two points to plot the direction"

 

 

don't make sense to me because you're being, oh, what's the word, too 'physical' about these things.

 

Let us presume you're talking about R^3 as the ambient space in which you're thinking of your points living.

 

Points are 0 dimensional by definition, lines are 1 dimensional, planes 2 dimensional. They take 0,1, and 2 parameters (not points) to describe them, that is where the dimension comes from, if we let x,y,z be position/direction vectors then

 

a point is just x,

a line is x+ry, r in R being one parameter

a plane is x+ry+sz for two parameters r,s in R.

 

each of the last two is naturally the same as R and R^2 respectively (x+ry maps to r, and x+ry+sz maps to (r,s)

 

There is nothing about points of reference generating anything and no one is saying anything about addition of dimensions.

[Royston]

Thanks for setting me straight - on retrospect I was (as you stated) asking the question purely on a physical basis. I guess it was an 'off the top of my head' kinda question, without having the knowledge to express it mathematically.

 

Thanks for explaining the nature of dimensions as well, I was thinking of them in completely the wrong context...as in 'building' dimensions from nothing. Which was really what I was trying to clear up.

[MacroQuantum]

Is there a disparity between math and cosmology or just the present and when I went to school.

 

As I recall a point established one dimension, a line established two dimensions, and a second line from the point of origin establish depth. And movement of a point in the established dimensions creates a 4D world having height, width, depth and time.

 

Although, if I understand your definitions correctly, we would essentially live in a three plane world. Can time be expressed mathematically?

 

Or is there simply no relating the two?

[matt grime]

What does any of that mean?

 

'Established' is not as far as I am aware a well defined mathematical term. Seriously, that is almost totally incomprehensible. Don't confuse a firm definition in abstract geometry with a hand-wavy idea in physics. If you want to make it rigorous then let's talk atlases on manifolds which is the proper setting for this, if you are familiarwith them. Dimension is a very loose term with many meanings in different contexts that all haver roughly the same idea.

[MacroQuantum]

Well, excuse me for asking... or for not making myself clear which ever is the case.