Spastik_Relativity said:
Ive been reading a couple of threads concerning the 4th(non-time) dimension. This has got me thinking wether we are closer to realising the 4th dimension than we actually think.
Say for instance a point A in the 1st dimension can move left to right in its dimension. In the 2nd dimension an identical point can move left to right, and up and down. In the 3rd an identical point can move left to right, up and down, and back and forth. But what if that particle or point wasn't to move at all. Is it possible to define what dimension it lies in?
Just a thought that confused me for a while.
Dimension is in essence a topological concept. This means that the concept of "open balls", the set of points "close to" a given point, determine the dimensionality of a space. It's not necessary for any points to actually move, but it is necessary to have some concept of what points are close to other points in order for the space to have a topology and hence a dimension.
The usual way to define a topology is to have a metric, a way of measuring the distance between any two points. Then one can say that points within some given (small) distance delta are "close to" or "in the neighborhood" of some point. These sets of points that are close to other points are called "open balls" or "open sets" in topology.
Given a topology, a space naturally has a "Lebesque covering dimension". (This is something I learned about from this forum, btw). This notion of dimension - the "Lebesque covering dimension" requires (IMO) the least amount of structure to define - the more usual notions of dimension, the notion of the dimension of a vector space for instance, require more axioms to define what a vector space is. (A manifold requires even more structure to define than a vector space - every manifold has a tangent vector space at any given point, and all the tangent vector spaces have the same dimension, so the dimension of a vector space defines the dimension of a manifold).
Toppology is important to the concept of dimension for the following reason. It is possible to map all the points in a line to all the points in a plane with a 1:1 mapping!
Such mappings, however, are not _continuous_ - they do not preserve the structure of the neighborhood (what points are close to other points).
Mappings which preserve the neighborhood structure are homeomorphisms. In order to tell a homeomorphism from a non-homeomorphism, one needs a concept of neighborhood. This makes the concept of neighborhood _necessary_ to define dimension - with no concept of neighborhood, we can't distinguish between a line and a plane. The existence of the Lebesque covering dimension makes the concept of neighborhood _sufficient_ to define dimension.
See
this past thread
for more - another poster here (Chronon) introduced me to the concept.
