Dimensions

[leveni]

Dimensions.

I was browsing Wikipedia the other day and had a look at Dimensions.

I now have questions about dimensions that Wikipedia didn't cover.

I just want to go through step by step in a slow tedious logical manner and obtain a reasonable answer to my questions.

Hope someone can help.

 

Q1. Is the 1st dimension just a perfectly straight line on a plane of infinite length that never bends.

Or can it also be a bendy line that curves all over the place in 3 dimensions?

 

Personally I think it can only be a perfectly straight line. But need conformation.

 

Q2. This line, could also never have any width or height it could only have length. Is this true?

If it did have width it would then be 2 dimensional.

If it had height again it would be 2 dimensional.

If it had both width and height it would be 3 dimensional.

Are these three statements also true?

[ajb]

Q1. Is the 1st dimension just a perfectly straight line on a plane of infinite length that never bends.

Or can it also be a bendy line that curves all over the place in 3 dimensions?

 

One dimensional topological spaces are either the infinite real line or the circle.

 

If you represent them as sub-manifolds of larger dimensional manifolds then they can look "bendy" as you put it. Look up Knot theory for a whole branch of mathematics concerning embedding the circle into 3-d manifolds.

 

Q2. This line, could also never have any width or height it could only have length. Is this true?

If it did have width it would then be 2 dimensional.

If it had height again it would be 2 dimensional.

If it had both width and height it would be 3 dimensional.

Are these three statements also true?

 

Yes, these statements are true.

[leveni]

Hi ajb,

thanks for the quick reply.

I'm going to change Q2 a little, to:

Q2. This line, could also never have any depth or height it could only have width. Is this true?

If it did have depth it would then be 2 dimensional.

If it had height again it would be 2 dimensional.

If it had both width and depth it would be 3 dimensional.

 

So if we are looking at the line in front of us, it has width but no depth or height. Just to make the visualization easier.

 

Also for the time being I'm going to limit these questions to an infinite real line.

 

My next step is a little strange. So I'll explain what I'm thinking. I'm thinking how something in the first dimension would perceive it's surroundings. I guess this is entering Metaphysics, and may not belong here, but I'm just trying to look at the 1st dimension from the 1st dimensions point of view. Not my 3 dimensional point of view. This is purely hypothetical because nothing can exist if it has no height or depth, and this is also speculative because we can't really know what or how something would perceive it's surroundings if it existed in the first dimension. Why am I doing this? I am doing this because I have the habit of looking at things from other peoples points of view. In this case it's a kind of mathematical/dimensional empathy, if that is possible.

 

Anyway;

Would the equation for the infinite real line be

 

–∞ < x < +∞ (x being any given point on the real line)

 

(noting that the y-axis and the z-axis do not exist in this equation)

Edited September 29, 2010 by leveni

[ajb]

... but I'm just trying to look at the 1st dimension from the 1st dimensions point of view. Not my 3 dimensional point of view.

 

This is fine, we do not have to think of spaces (manifolds in this case) as being embedded in others. In particular the circle and infinite line are "entities in their own right".

 

Anyway;

Would the equation for the infinite real line be

 

–∞ < x < +∞ (x being any given point on the real line)

 

Any point on a one dimensional manifold requires just one coordinate to be specified. In essence you identify every point with a number.

 

 

 

(You can also think of the real line as a vectors space over itself, again of dimension one)

[leveni]

Hi ajb,

Thanks for the reply.

I'm just reading through all the terminology on Wikipedia. I just looked up 'line' on Wikipedia and have now decided to change the meaning of line back to my original statement, which was the same as Wikipedia's. 'Lines ... have no width or height at all and are usually considered to be infinitely long'

[leveni]

Hi Ajb,

Thanks for your time.

I've given up on what I was thinking before.

The reason is I think the 2nd dimension doesn't really exist.

eg:

Something in the 3rd dimension has the volume: x times y times z. If it has volume it exists.

But in the 2nd dimension there is no z-axis, z equals zero.

So everything in the 2nd dimension will have a volume of zero. x times y times zero equals zero.

And the 2nd dimension itself has no z-axis. So it also doesn't exist.

So the 2nd dimension is not something that exists, it's just a something mathematicians thought up and use to explain and calculate surface area and vectors.

Is this correct?

Sorry for taking up your time.

Edited December 21, 2010 by leveni

[Xerxes]

The reason is I think the 2nd dimension doesn't really exist.

 

Now, beware assigning a "hierarchy" to dimensions. In mathspeak, the number that defines a dimension is cardinal, not ordinal.

 

 

If it has volume it exists.

 

Exists in what sense? You mean that 2-dimensional objects don't exist?

 

So the 2nd dimension is not something that exists, it's just a something mathematicians thought up and use to explain and calculate surface area and vectors.

Is this correct?

 

No.

 

Look, and I am sorry to be boringly technical (you may well feel it proves your point!)

 

Consider a line of finite length, as lines are usually understood. We will assume that this line can be infinitely sub-divided. Then we will assume that each element in this this sub-division corresponds to a real number. Thus our line is a "segment" of the real line [math]R^1[/math]

 

We now ask, how many real numbers do I need to uniquely identify a point on this line? The answer is, of course 1. Let us call this line as 1-dimensional, by virtue of this fact. Let's take our line segment, and join it head-to-tail. We instantly recognize this as a circle.

 

Obviously, the same applies; any point on this circle can be uniquely described by a single real element, and accordingly I will call this geometric object as 1-dimensional. Mathmen use the symbol [math]S^1[/math] for this character, and call it the 1-sphere. Now let's try and think about the "2-dimensional line", or 2-line. What can this mean (if anything)?

 

Well, using the above, we may assume that this is the "line" that requires two numbers to uniquely describe a point. From which we infer that the "2-line" is the plane. We may also infer, from the above, that the 2-sphere [math]S^2[/math] is, in some weird and abstract sense, a head-to-tail "joining" of a part of this plane. .

 

The 2-sphere is merely some sort of jazzed up plane, that is, it knows nothing about the area/volume it may or may not enclose, any more than does the 2-plane. The same applies to any n-sphere.

 

With a grinding of gears, let's now consider the area enclosed by the 1-sphere as defined above. Intuition tells us, in this case quite correctly, that it is part of the "2-line" i.e the plane. This part of the plane is usually referred to as the "disk" [math]D^2[/math]. It is, in fact, the 2-ball.

 

Similarly, the "area" enclosed by the 2-sphere is a part of the 3-plane and so on. Obviously, this latter "area" is the volume (as it normally understood) enclosed by the 2-sphere, from which we conclude that, provided we are allowed to think of area as a 2-volume, then, as a generalization, the n-sphere encloses an n + 1 volume.

 

But the next thing we have to think about is whether or not, for any "n-volume", I need to have an enclosing n - 1 space.

 

Well, it largely a matter of definition; as a geometric object, the n + 1 volume enclosed by the n-sphere may or may not include the n-sphere. If it does, one says that the n + 1 ball is closed. Otherwise, they say that the n + 1 ball is open.

 

Intuition says exactly this; a set is closed iff it includes its boundary, it is open otherwise (note bene; this is not the topologist's definition)

 

So the boundary of an n-ball [math]D^n[/math] is precisely the n - 1 sphere that encloses it. A concise notation might be that [math]\partial D^n = S^{n-1}[/math] where the [math]\partial[/math] denotes the boundary

Edited December 21, 2010 by Xerxes

[Bignose]

So the 2nd dimension is not something that exists, it's just a something mathematicians thought up and use to explain and calculate surface area and vectors.

Is this correct?

 

Any physical object will have three dimensions -- even if it is as small as angstroms (10^-10 m) such as the diameter of an atom. But that doesn't mean that 2-D objects don't exist. The surface area of a 3-D object is still two-dimensional. You can locate things on that 2-D space, such as using the latitude and longitude on the Earth's surface.

[Xerxes]

Any physical object will have three dimensions

This is a very bold assertion; can you justify it? For a counter example, any fundamental particle such as, say, the electron is zero-dimensional. Are these not "real" physical objects?

 

Anyhoo, as this is a Math forum, I had assumed that the objects under discussion were mathematical objects, hence my seemingly inappropriate last post

 

But just to nit-pick

 

The surface area of a 3-D object is still two-dimensional.

Area is a number, it is 1-dimensional. You probably meant the surface itself, as I tried to show in my previous (and obviously failed)

[khaled]

Q1. Is the 1st dimension just a perfectly straight line on a plane of infinite length that never bends.

Or can it also be a bendy line that curves all over the place in 3 dimensions?

 

the 1st Dimension, is an infinite line, or a Circle, just like what mister ajb said ...

 

Q2. This line, could also never have any width or height it could only have length. Is this true?

If it did have width it would then be 2 dimensional.

If it had height again it would be 2 dimensional.

If it had both width and height it would be 3 dimensional.

Are these three statements also true?

 

Your statements are wrong if we take them together .. let me explain:

 

1. because in 2Dimenional, a plane can have a width & a height ...

and if you go into 3Dimensional, a space can have width, height, and depth !

 

2. in 1Dimensional, a Line can have a length, the line itself have width of Zero,

given that the line when represented in 2Dimensional Plane is from

negative infinity to positive infinity on the X-axis, but on a single value on the Y-axis

(Assuming K) the width would be |K-K| = Zero

 

Any physical object will have three dimensions -- even if it is as small as angstroms (10^-10 m) such as the diameter of an atom. But that doesn't mean that 2-D objects don't exist. The surface area of a 3-D object is still two-dimensional. You can locate things on that 2-D space, such as using the latitude and longitude on the Earth's surface.

 

true, in reality, there exist no other than 4Dimensional objects, consider time as the fourth,

 

even the surface is still 3Dimensional, but mathematically it's taken as 2Dimensioanl

as a consideration of an abstract plane ...

 

My assertion to "in reality, there exist no other than 4Dimensional objects, consider time as the fourth" as following:

 

1. since Atoms is the primary component of everything, and since Atoms have a constant set of irregular movements

-- then we can say that every object in reality is in constant movement at all times, and thus every object is 4D

 

2. since the stability of things is related to having an atom not separated from its contents

-- then we cannot have a surface consisted of what is less than atoms, and since the atom

-- have a spherical shape, it has width, height, and depth ...

 

3. moreover, even if we made a plate of electrons or so .. those parts also have width,

-- height, and depth .. so the smallest possible stable thing to that would be an Atomic Plate ...

Edited January 18, 2011 by khaled