Dimensions

[Kyrisch]

So, it is commonly accepted that the first three dimensions are those that we experience every day (e.g. length, width, and depth). And that the first dimension is linear, the second to do with area, and the third to do with volume. The fourth, however, is sometimes called duration, meaning that it represents "time". Often it is conceived as also linear (such phrases as "timeline" exemplify just that). So my question is as follows; is this "cyclical" nature of dimensions inherent or conceived? I imagined at first that it would be like [math]\i^n[/math] in that it repeats every four integral values but then I realised that this conception was probably largely fabricated by human thought. Is there any mathematical basis in this?

[Sisyphus]

I don't understand what you mean by "cyclical" in this context.

[iNow]

I just finished this article, which was shared by SFN member Martin, and it may help you to understand some of these issues more clearly:

 

http://www.scribd.com/doc/3366486/SelfOrganizing-Quantum-Universe-SCIAM-June-08

[ajb]

What do you mean by "linear"? This is probably just the statement that space-time is a manifold, i.e. you give it local coordinates.

[Kyrisch]

I don't understand what you mean by "cyclical" in this context.

 

What do you mean by "linear"? This is probably just the statement that space-time is a manifold, i.e. you give it local coordinates.

 

People refer to time as linear, to 'points' on the timeline. This is analogous to one-dimensional, linear verbiage. My question is whether there is any maths that back up this cyclical (dimensions having analogs every 'three layers'; this of course is also assuming that, unlike in string theory, the dimensions from 3-6 are all temporal and not extra spatial dimensions) nature.

[ajb]

Yes Kyrisch, the mathematical theory is called "Modern Differential Geometry".

 

This deals with topological spaces called "manifolds". The important property of an n-dimensional manifold [math]M[/math] is that locally (i.e. on a "small chunk") you can describe it using [math]\mathbb{R}^{n}[/math] (which is a vector space). The important thing here is that locally you can describe the manifold using the coordinates on [math]\mathbb{R}^{n}[/math], i.e. (locally) a point [math]p[/math] can be described by the collection of n numbers [math]x^{\mu}(p)[/math].

 

So to be more specific, and two dimensional manifold locally looks like [math]\mathbb{R}^{2}[/math]. Meaning it can be described using two numbers for each point, [math]x^{\mu} = (x,y)[/math].

 

I will stress this again, locally.

 

The mathematical framework for describing the nature of space and time is general relativity, which gets most of its tools form modern differential geometry.

 

Hope that helps,

 

p.s. who is "people"?