August 5th, 2012, 06:59 PM
I was wondering exactly why there are dimensions. I understand that causal dynamical triangulation attempts to explain how they came to be, but I find it confusing. Could someone help explain it? Are there any other attempts to describe how dimensions came to be? Or is it just sort of assumed that dimensions have to exist?
Last edited by guymillion; August 7th, 2012 at 12:12 PM. Reason: Spelling error
Well, it's just the minimum number needed to define a point.
Actually, defining a point requires the amount of dimension plus an origing. Do 4 points in 3D, 3 in 2D, 2 in 1D and yes 1 in 0D, however 0D only has one point, being this origin. Dimensions are and almost seemingly intrinisc given thing in this world. But it is a rather far going concept. Physics, even the philosophical backgrounds of science and all its laws are different per dimension. It isn't so much as we 'need' dimension. It is more the world where we live in. We 'need' the earth, but we are part of a universe. We live in a 4D world, 3 spatial dimensions, and one time dimension. Does that require 5 points yes it does. And from this on I would like to explain more, but in the past this very subject wasn't considered as 'hard science' enough by the moderators. So I will refrain from it. Even though it makes up entirely different worlds of physics.
Also, most people that practice science and haven't been educated in the principles have no idea of the differences that a 2D or 3D world can imply. Let's just say, that you, as a human, cannot live in less the 4D. Can you live in more than 4? I don't know, It isn't something I can observe and hence never answer.
CDT suggests that space-time is fundamentally discreet, with the basic building block being the 4-simplex, which is geometrically flat. These simplexes can now be combined in a variety of ways, each of which will produce a different space-time continuum. The trick here is that CDT says if you align the "edges" of the simplex which correspond to the time coordinate, you automatically arrive at a causally connected space-time, i.e. causality is a direct result of basic geometric principles.Originally Posted by guymillion
The other interesting is that such a model produces our usual 4-dimensional universe on a macro-scale, but microscopically you get a 2-dimensional vacuum with fractal properties. Most fascinating.
One should also mention that, if our universe was anything else than 3+1, we would find different laws of gravity.
To stop everything happening at the same place (and time).
I disagree. The origin is part of the coordinate system and not something that needs to be specified with the point itself. E.g. if you really needed to specify the origin then 3-space would really be a 6-space, three degrees of freedom to specify the origin and three degrees of freedom to specify the point But it isn't a 6-space but truly a three space, the origin being specified with the system and not the point.
As to why there are dimensions, nobody knows.
I would tend to disagree as well. In maths the dimension is defined as the minimum number of coordinates needed to uniquely specify a point. The origin of the coordinate system used is irrelevant, since this definition is invariant under arbitrary coordinate transformations.
I refer also to the Wikipedia entry for "dimension" ( quote ) :
"In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it.[1][2] Thus a line has a dimension of one because only one coordinate is needed to specify a point on it (for example, the point at 5 on a number line)."
Thanks! This helped me a lot! Let's say that the building block was a 5-simplex. Wouldn't that mean a different number of dimensions and different laws of gravity?
Last edited by guymillion; August 7th, 2012 at 07:06 AM. Reason: Adding a question
Yes, absolutely !
Yay! Thanks!
Try and imagine at such:
Look in front of you, to an empty table. Pick any point on the table. and imagine it to be infinitely large plane and there is no pre-determined axis present. How many point do you require to assume in order to uniquely define the random point you chose in the beginning.
If we'd choose one point and the distance from that chosen point to the random point. We'd have a circle of possibilities. So one is not enough. If we choose to values, like an x and y value. We have a distance from two points, that means only the points that both respective circles cover. This is two possible points (unless it is exactly in the middle of the two chosen values, but this is physics not mathematics, unless you define things, there is no 'exact' )
So we require a 3'd point, and origin you might say. to decide which of these other 2 points is the right one. The reason I call it is simple, it cannot be regarded as another circle. It must be a point that is singular. So that when our spot to determine is that singular point there is only one possible description.
Hence you need 3 points to describe all point in a 2D plane.
Also from a mathematical point of view, you cannot construct a system of numbers without the use of an origin. I think it is actually how dimensions are defined. Hence invariant for coordinate transformations. But the argument is still the same, you need 3 points to describe it. Where one is just for direction basically.
My understanding is that CDT is dealing purely with spacetime and not with matter. However, I saw in a short youtube film that there was a man named Erik Verlinde who hypothesized that not only gravity, but all of the other forces and matter also were a result of CDT. However, I could not understand the source that they cited. Here is the source:
Everything is Emergent | Not Even Wrong
Erik Verlinde uses the holographic principle and string theory for that. But in a remarkably simple way. I however do not have the details, its been over 2 years since I spoke to him about that, and that was after 2 pints of guiness :P.
Also you must understand that String theory is trying to use the shadow of 6 dimensional elephant herd, to desribe a mouse. Sure it is possible, but so are describing the monkey, and the goose and the ape, and the rat and the horse. In any way, don't lean too much on this higher dimensional speach. But Dimensions are important. Magnetism for instance only exists in 3D. not in 2D, nor 1D. Unlike for instance coulomb repulsion. Then again Magnetism isn't classical at all.
You are wrong. The dimension is the number of independent coordinates, not their values. The origin of the coordinate system is needed only to specify the values of those coordinates, but not how many of them there are.Hence you need 3 points to describe all point in a 2D plane.
Why are you arguing this ? You have already been given references to show how this works, even if you don't believe us. The number of dimensions has nothing to do with which coordinate system is used.
Good grief. Did you actually look at the references we have already provided ?Also from a mathematical point of view, you cannot construct a system of numbers without the use of an origin. I think it is actually how dimensions are defined. Hence invariant for coordinate transformations. But the argument is still the same, you need 3 points to describe it. Where one is just for direction basically.
<deleted boo boo>
Last edited by pmb; August 8th, 2012 at 07:25 AM. Reason: deleted boo boo
Because of the infinit pool of gravity the partical get zero size and multiply its dimention in time revers , and its eqvivilet to the force of gravity that multiply one dimention to tree
Good grief? I am shocked of your answer to be honest. You take up Wikipedia as your main source of reference?
I believe we are talking about different things here. You are talking about dimensionality, I talk about how one deals with defining point depending on the dimension one is trying it in.
So let's look at things more formal then. From wolfram (reference 2 in the wikipedia article, and a good one too);
' The concept of dimension is also used in algebra, primarily as the dimension of a vector space over a field. This usage stems from the fact that vector spaces over the reals were the first vector spaces to be studied, and for them, their topological dimension can be calculated by purely algebraic means as the cardinality of a maximal linearly independent subset. In particular, the dimension of a subspace ofis equal to the number of linearly independent vectors needed to generate it (i.e., the number of vectors in its basis). Given a transformation
of
, '
In other words the dimensionality is the minimum amount of linearly independent vectors that are required to generate it. And for a 3D space this is 3, for a 2D space this is 2, and for a 1D space this is one. Point let's look at it more physically. I am looking to understand dimension in our world, the physical world. Not the mathematically defined world.
In this world, when I want to describe something I need a point of reference. There is a very neat theory about requiring points of reference, it usually calls it frames of reference.
So If I want to define any point I need to refer it to some points of reference. Say that these points are the vectors that describe the dimensionality of wolfram. One vector from one point, and another vector from another point, and another from a third. All linearly independent.
Mathematically I'd be done. But Physically I am not. I pre-assumed something that I shouldn't have. Namely that the vectors I used to describe the point, have the same origin as point itself. This has to do how a subspace of any set is defined. Any set, even in mathematics must have the zero, the origin. The empty set, is just the origin. 0D is one point. But it is still one point!
One point to describe zero dimensions. Not a vector required, but I am not talking about vectors, I am talking about points.
Yes, I only require 3 linearly independent vectors to describe a 3D space. But I need 4 points to describe those 3 vectors. All vectors must have an origin. Even mathematically this holds.
The reason I think in points is rather simple. I, and everyone else for that matter, observe in points. If I could observe vector fields, I wouldn't need weird and confusing quantum-mechanical interpretations.
Wikipedia is convenient, but if you need other sources I can get them for you; I think that will not be necessary though because the Wolfram one you yourself provided is just fine, IMHO.
Precisely. That is all I was trying to say, in more mathematical language. The number of linearly independent vectors is always the same regardless of a point of origin, it is just the values of the coordinates that change. Hence the origin is not required to define the number of dimensions of a manifold.In particular, the dimension of a subspace of
To this I agree.So If I want to define any point I need to refer it to some points of reference.
Ok, I think I can agree to this also, so long as there is no confusion as to the dimensionality of the manifold.Yes, I only require 3 linearly independent vectors to describe a 3D space. But I need 4 points to describe those 3 vectors.
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