What are dimentions: Philosophy Forums

Philosophy Forums » Philosophy of Science » Science » What are dimentions » What are dimentions

What are dimentions
& space filling curves

Page: 1 2

What are dimentions

Cadrache
Tenured Poster

Usergroup: Members
Joined: Dec 09, 2006
Location: AB, Canada

Total Topics: 104
Total Posts: 2644

Posted 08/18/09 - 11:50 AM:	#11
the rate of distortion is dependant upon the factor of Conversion... In some textbooks, it claims that the value of Y at any X-component is infinite.
"...There was a writer who asked why it was that when we find positive experiences we say that only the physical facts are real, but in negative experiences we believe that reality is subjective. He made an example of those who say that in birth only the pain is real, the joy a subjective point of view, but that in death it is the emotional loss that is the reality." - Tony Ballantyne, Recursion. _____________________________________________ Truth is want. - The internal state of matters. Truth is Need. - The external state of affairs.

enkidu
Tenured Poster

Usergroup: Members
Joined: Jul 27, 2006

Total Topics: 22
Total Posts: 1448

Posted 10/04/09 - 01:42 PM:	#12
Machiveli wrote: What exactly do we mean when we say that something (e.g. the universe) has n dimensions? Taking a naive view and going with Wikipedia dimensionality is defined as: "the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it." yet if we fill an n-dimensional space with a 1-d space filling curve then we can uniquely specify a point with a single coordinate. The problem with the assertion that a n-dimensional space can be filled up by a space filling curve is that it only applies to a very particular type of spaces and only to them, it actually is the Hahn-Mazurkiewicz Theorem: "A nonempty Hausdorff topological space can be completely filled up by a continuous curve if and only if the space is compact, connected, locally connected and metrizable." Within that context, your remark is not so interesting given that dimensionality is defined for a larger class of spaces, and especially for infinite and non-compact spaces. Consider a classic euclidean plan (2d space) and it can't be filled up with a space-filling curve, so you really need 2 coordinates to specify each point within it. Now, more formal definition of dimension can be given, using the notion of covering, it is so for the topological dimension, or for the Hausdorff dimension.
Tight toy night, streets were so bright. The world looked so thin and between my bones and skin there stood another person who was a little surprised to be face to face with a world so alive. I fell. (Tom Verlaine)

throng
Profester.

Usergroup: Members
Joined: Aug 12, 2008
Location: Downunder.

Total Topics: 43
Total Posts: 803

Posted 10/04/09 - 05:50 PM:	#13
I often wonder what is the common primary fundamental of all the definitions of dimension. Is there a fundamental property that all definitions rest on?
I know that I don't know, so I don't know if I do.

enkidu
Tenured Poster

Usergroup: Members
Joined: Jul 27, 2006

Total Topics: 22
Total Posts: 1448

Posted 10/07/09 - 02:38 PM:	#14
throng wrote: I often wonder what is the common primary fundamental of all the definitions of dimension. Is there a fundamental property that all definitions rest on? I don't know. It's a difficult question. Personally, when I want to grasp intuitively the idea of dimension, I tend to interpret it as a measure of the degrees of freedom in a system. I found this approach useful to work with the various fractal dimensions for instance.
Tight toy night, streets were so bright. The world looked so thin and between my bones and skin there stood another person who was a little surprised to be face to face with a world so alive. I fell. (Tom Verlaine)

Page: 1 2

Sorry, you don't have permission to post. Log in, or register if you haven't yet.