Need quick answer on spatial dimensions

[CylonHybrid]

I have just a quick question I was wondering about and I was wondering could someone answer it here.
Is it true that for example the third dimension is composed of an infinite number of 2 dimensional planes on top of each other which give rise to width, the third dimension? If this is true then how can a dimension with 0 width give rise to width even if there are an infinite number of planes since 0 times infinity is still 0?
Thanks for your help.

 

[ghwellsjr]

Where'd you get the idea that 0 times infinity is 0? It's indeterminate, meaning it can have any value at all. Does that help you understand the answer to your first question?

 

[CylonHybrid]

So you're saying then it is possible for an infinite number of planes with 0 width to somehow get width because an infinite number of planes of 0 euclidean distance along a direction X has an indeterminate euclidean distance along direction X? How is this accomplished? I know this maybe has something to do with infinity being a concept and not a number maybe?

 

[ghwellsjr]

Let's take the more common example of an indeterminate value. What value(s) of X satisfy this equation?

X = 0/0

To see the answer, multply both sides of the equation by 0 and you get:

0*X = 0

[NOTE: * means multiply]

X can take on any value and the equation remains true. That means that the value of X is indeterminate and must be solved some other way.

So getting back to your equation:

X = 0 * infinity

Divide both sides by 0 and you have:

X/0 = infinity

Now you can see that this equation is satisfied with any value of X, meaning that it is indeterminate and must be solved some other way.

Does that help?

 

[pervect]

Infinite sets are tricky.

You might consider that your question is equivalent to asking "if I have an infinite number of points, how can they form a line?"

You might also ask "do an infinite number of points always form a line"?. The answer is no, by the way.

I'm not sure where you can find a good introductory book on infinite sets. You might try asking in the math forum.

Some rather disorganized tidbits:

Infinite sets are said to be of equal size (cardinality is the technical term) if there is a 1:1 mapping between their elements. There is more than one "size" of infinity, the set of natural numbers (0,1,2,3, etc) cannot be put into a 1:1 correspondence with the set of real numbers, for instance.

However, there is such a mapping between natural numbers and even numbers, so the "size" of the infinite set of even natural numbers is the same as the "size" of the infinite set of natural numbers. Try a google for "Hilbert's Hotel"

One consequence of this : if you had a countably infinite set of points, there wouldn't be enough of them to form a line - or even a line segment. You need an uncountably infinite number of points to make up a line.

Another interesting tidbit. You can find a 1:1 mapping between the uncountably infinite number of points on a line, and the uncountably infinite number of points on a plane. So in some sense, they both have the same number of points.

See for instance http://en.wikipedia.org/w/index.php?title=Space-filling_curve&oldid=447972617

This makes defining the concept of dimension trickier than it might appear. You're unlikely to stumble upon a good answer without doing some serious reading.

 

[CylonHybrid]

Yes I understand that but is this something purely mathematical and how/ does it apply to the real world? At which point is there a change over from dimensions being a mathematical concept to being real things in the physical world?

The indeterminate distance would have to be solved another way? Can you elaborate on how it would be solved?
After all the width of the universe is infinite so these infinite two dimensional planes with 0 width would need to give rise to infinite width.

 

[yoron]

You can't get from a two dimensional existence to a three dimensional. You can assume what it would look like for a two dimensional creature to meet a three dimensional object (on a plane) so it works one way. Which means that even though you can assume that we live in some multi dimensional reality we only can see three plus one. All as I see it.

String theory works from the assumption of one dimensional 'strings' and loops, as well as 'branes' that are one dimensional 'surfaces', but as they somehow will have to incorporate a 'volume'? And as far as I understand that, it's their 'vibrations' that is expected to do that. And as any 'volume' will to us act as a three dimensional object, possibly :) I'm not saying that I understand this, or even got it right, but it seems that way from what I've read.

Dimensions mathematically is about how many coordinates you need to define something on it. Like a line only need one coordinate to define any point, and a paper will need two. Then you have the way the room looks to us to consider too, we see it from four 'points', like looking at the 'endpoints' defining the wall in front of you, a line we see as two, So maybe the next dimension should be eight 'endpoints'?
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Eh, it is actually :) That's the room geometry we have if you count all endpoints in your room. So the next should then, adding 'dimensions' that way, be sixteen.
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But the volume we occupy in here is definitely three dimensional to me, with 'times arrow' as the flavor making me able to percieve them, or if you like, becoming the fourth dimension. And they are my SpaceTime.

 

[CylonHybrid]

Aren't sifferent sized infinites and sets just a mathematical concept only to make it easier for us to coceptualise but in reality all infinities are the same size?Anyway the question I'm asking is what is the relationship between dimensions? Which is something I would have thought essential to something like string and M theory etc

Heres a video I found on you tube about spatial dimensions. In the first 20 seconds it mentions an infinite number of planes.

 

[yoron]

And here you can see how they define 'string theory.' And they too mention 'endpoints' as some sort of mathematical 'anchors' for a room geometry. And no, different sized infinities are mathematical descriptions that 'exist' as real mathematical objects. How to translate it into the room we exist in though?

Makes you wish that they taught that in school instead. Would have made it more interesting :)

 

[CylonHybrid]

And here you can see how they define 'string theory.' And they too mention 'endpoints' as some sort of mathematical 'anchors' for a room geometry.

Makes you wish that they taught that in school instead. Would have made it more interesting :)

I string theory it talks about point particles, the building blocks of three dimensional atoms being 1 dimensional strings. Yet somehow 1 dimensional constructs give rise to higher dimensions. No one yet has given what is the relationship between spatial dimensions.

 

[yoron]

Tell me, how many dimensions have light?

 

[CylonHybrid]

Tell me, how many dimensions have light?

I'm not a physicist but I don't see why matter and energy would only be constrained to three dimensions. I'd imagine all matter and energy exist in 10 spatial dimensions but we just can't see the other 6.

 

[yoron]

The 'volume' I discussed, that 'strings' are thought to 'vibrate' in is what we call SpaceTime. You could also define such as there is two sets of ideas. One is background independent, there what is the smallest 'bit's' also becomes the background, and that one is rather tricky, Smolins loop quantum gravity does it by reducing spacetime to loops that then build 'networks' creating our dimensions where SpaceTime comes from the 'edges' joining them. As a guess I would say that it still is vibrations involved creating that 'volume' needed. And that implies a arrow, or indeterminacy, as in HUP. Look it up and see what you think.

 

[yoron]

We don't see energy :)

Assuming we're not shamans that is. 'Energy' is a description of transformations. Entropy speaks about it as useful energy, and energy that is used, reaching some 'lowest state' or equilibrium from where you can't use it any more. We do see light though.

 

[CylonHybrid]

Anyway back to the point.

Consider a line segment. A line is one dimensional but it is made of an infinite number of 0 dimensional points. But a point has 0 dimensions. So how do an infinite number of points make up a line segment?

 

[yoron]

A dimension is defined through coordinates. A point only need one, same as on a line. A 'point' on a cone (two dimensionally seen) needs two coordinates. That is assuming that you're not talking about dimensionless points?

In SpaceTime you need four, three of which is - width, length, height- plus one - the time-, and the system will be observer dependent, meaning that the coordinate system you use most probably will differ from your neighbor.
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In mathematics you can define dimensionless points too. And using HUP you can find that defining one value of two make the other existing 'everywhere'. like a photon becomes 'smeared out' as you exactly define its momentum. Or a electron 'cloud'. But it's more of pure mathematics what you want to discuss now than it is Relativity.

 

[CylonHybrid]

A dimension is defined through coordinates. A point only need one, same as a line. A 'point' on a cone needs two coordinates.

In SpaceTime you need four, three of which is - width, length, height- plus one - the time-, and the system will be observer dependent, meaning that the coordinate system you use most probably will differ from your neighbor

Thats dancing around the maypole. The coordinate system defines a dimension but doesn't describe a dimension or its relationship to other dimensions. What is the nature of direction, of dimension? There can't be a coordinate without a space for a coordinate to exist.

 

[CylonHybrid]

Where'd you get the idea that 0 times infinity is 0? It's indeterminate, meaning it can have any value at all. Does that help you understand the answer to your first question?

Also you are mistaken. In indeterminacy it says that infinity x 0 is undetermined but 0 to the power of infinity is mistakenly thought of as undetermined but is actually 0.

 

[yoron]

Okay, let's try this then. A dimensionless point is as a value used in mathematics, it describes something but have no dimensional characteristics. Points making up a defined line, like on a paper, will always have coordinates.

If you take light it is supposed to be able to be superimposed, and it also has no defined size, only a 'energy', that will to my eyes define it as dimensionless. So maybe you can use a photon to illustrate it. A photon only exist in its annihilation, so from that definition you might want to say that this dimension less photon will, as it transforms into electro chemical energy on your retina get a shape, and so some coordinates.

There is a lot of stuff that is conceptual in mathematics, and hard to pinpoint as their counterpart in our normal day to day living. That concept must be one of the hardest.
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You know, thinking of it, space is filled with dimensionless points, but we can put coordinates to them, even if we don't agree on what they are :)

 

[ghwellsjr]

Anyway back to the point.

I thought you were wondering how three dimensional space could be modeled mathematically as an infinite set of zero-width two-dimensional planes stacked one next to the other.

Let me explain it this way:

Consider a loaf of bread which occupies a three-dimensional volume. Now you slice the bread and each slice represents a two-dimensional plane with a finite thickness. Now consider slicing each slice in half so that they are half the original thickness. Now you have twice as many slices. You repeat this a great many times. In the limit, as you approach an infinite number of times, the slices will have zero width and you will have an infinite number of them. Now extrapolate this to a volume containing all of space where you do the same process of repeatedly slicing it into thinner and thinner slices.

Can you see how you can have an infinite number of zero-width slices that represents all of space instead of the zero space as you originally speculated?

 

[CylonHybrid]

So you're saying its a function N where x approaches 0 as N tends towards infinity. However in mathematics you never reach infinity and x never reaches 0.

 

[ghwellsjr]

Also you are mistaken. In indeterminacy it says that infinity x 0 is undetermined but 0 to the power of infinity is mistakenly thought of as undetermined but is actually 0.

True, but we're not talking about multiplying zero by itself an infinite number of times, we're talking about multiplying a single zero by a single infinity and since a single infinity is the same as any number divided zero, the answer to what is zero multiplied by infinity can be any number. (Look back at post #4.) If it has a meaningful answer, you have to deduce it by some other means. In other words, you need more information. But you cannot just declare that the answer is zero any more than I can declare that the answer is 3, although those are possible valid answers, until we have more information.

 

[ghwellsjr]

So you're saying its a function N where x approaches 0 as N tends towards infinity. However in mathematics you never reach infinity and x never reaches 0.

That is what calculus is all about. Calculus makes it valid.

 

[yoron]

Not happy?

Can't blame you, points are weird. But so are lines, if they are made out of dimensionless points they shouldn't be there should they? But they exist, and if they as you define it comes from objects without a dimension, then those object still have the property of defining a 'one dimensional' line :)

Now, just for the fun of it, imagine a, not one dimensional, but two dimensional object inside space time :) It has two properties a length and ahhh, a height :)

So we rotate it, admiring its beauty.. But hey, it disappeared?
Well, it had too, it doesn't have any width, but don't give up. I'm sure you're still holding it, so keep on rotating..

It's a mathematical property, doesn't necessarily mean that we meet it at the pub, although, after some beers? Maybe?
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This one is friendly to read. But this one is better.