Originally posted by twhitehead It is not so much a question of angle, but rather a question of relationship. Can an object move in one direction without moving in the other?
So as long two dimensions are not totally parallel they are independent from eachother and can be regarded as two distinctively different dimensons, right.
A time dimension is a dimension with an arrow. Perhaps that is what defines the time dimension as time, and why space dimension as space because you can move any direction in space?
When you mention 'an object', do you then mean only an object or do you mean anything, objectlike or not?
I don't discuss this because I know, but because I want to know.
Originally posted by FabianFnas When you mention 'an object', do you then mean only an object or do you mean anything, objectlike or not?
I mean anything that can be given a location in space/time.
As adam warlock rightly points out, dimensions are typically man made constructs and what we really should be talking about are degrees of freedom, and I am not sure those have directions as such. In the spacial dimensions there are 3 degrees of freedom, but one cannot really say that they point in any given direction.
Originally posted by KazetNagorra A dot would be zero-dimensional, a line one-dimensional, and a surface two-dimensional. Mathematically, there is no restriction and one can formulate any number of dimensions including an infinite number and a fractional (or fractal) number.
Originally posted by twhitehead For spacial dimensions, one can have different axis constituting different reference frames and for most purposes the choice of direction seems arbitrary. Is this possible with time? ie can you have a reference frame in which a dimension is not along the traditional time axis, but rather is a combination of time and space?
If one uses polar co-ordinates for space, are they considered orthogonal?
The standard formulation of 2D polar coordinates uses orthogonal eigenvectors. One of the vectors is directed outwards from the origin, and the other one is orthogonal to it (two choices are possible, but one of them is chosen by convention).
Originally posted by twhitehead I mean anything that can be given a location in space/time.
As adam warlock rightly points out, dimensions are typically man made constructs and what we really should be talking about are degrees of freedom, and I am not sure those have directions as such. In the spacial dimensions there are 3 degrees of freedom, but one cannot really say that they point in any given direction.
The standard three-dimensional Cartesian space is spanned by three orthogonal eigenvectors: one pointing in the positive x-direction, one in the y-direction, and one in the z-direction.
Originally posted by FabianFnas So as long two dimensions are not totally parallel they are independent from eachother and can be regarded as two distinctively different dimensons, right.
A time dimension is a dimension with an arrow. Perhaps that is what defines the time dimension as time, and why space dimension as space because you can move any direction in space?
When you ment ...[text shortened]... anything, objectlike or not?
I don't discuss this because I know, but because I want to know.
If there are two degrees of freedom then it is always possible to find two directions which are othogonal. But there is a difference between local and global structure. Imagine a piece of A4 paper with one arrow drawn from the centre of the page to the top and one arrow drawn from the centre to the side of the page. Now, we can fold (without making a crease) the paper so the two arrows are more or less parallel, but at the centre they are still orthogonal.
Originally posted by twhitehead For spacial dimensions, one can have different axis constituting different reference frames and for most purposes the choice of direction seems arbitrary. Is this possible with time? ie can you have a reference frame in which a dimension is not along the traditional time axis, but rather is a combination of time and space?
If one uses polar co-ordinates for space, are they considered orthogonal?
Is this possible with time? ie can you have a reference frame in which a dimension is not along the traditional time axis, but rather is a combination of time and space?
I don't know if this answers your question but in special relativity you define a reference frame whose axis are space-like and time-like and you can rotate between different reference frames so that you end with reference frames that have dimensions that are kind of a combination of space and time.
Besides in special relativity there already is an a priori mix of the space and time dimensions.
If one uses polar co-ordinates for space, are they considered orthogonal?
Even tough polar coordinates are curvilinear the basis vectors have a 0 dot product in all points. Hence the basis vectors are indeed orthogonal. Remember that two vectors are defined to be orthogonal whenever their dot product is zero.
Originally posted by twhitehead Can one have fractional degrees of freedom?
Other than just counting the number of degrees of freedom there are at least two other ways of defining dimension.
One is called the Hausdorf dimension, and it gives the space filling property of the object, for two dimensional quantum gravity without any matter present it comes out as 4, add enough matter fields and that drops to 2. Roughly for a sphere the volume goes as r^d where d is the number of dimensions. For a fractal the amount of space it fills goes as r^d_H where d_H is the Hausdorf dimension.
The other option is spectral dimension, which is related to the return probability of a random walker. The higher the dimension the less likely it is to find it's way back. The spectral dimension for 2D gravity with no matter fields present is 2 (from memory I didn't do anything on this so I could have remembered it wrong). Add enough matter fields (3 species of electron is enough) and the spectral dimension comes out as 4/3 (again from memory so don't quote me).
How many dimensions are there ?
02 Dec '14 09:00
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