Defining a Discrete Object In Terms Of Only Itself?

[hoponpop]

Given only a single discrete object, is it possible to describe that object solely in terms of its structure?

 

Let us define a discrete object P as an arbitrary piece of matter. What it is is unimportant.

 

Let P constitute a matrix P' propagating infinitely in every dimension. Let us define each point in the matrix P' as P'(x,y,z,t, ...).

 

Further, let every point in the matrix P' constitute a second similar matrix, P", also propagating infinitely in every dimension. Let us define every point in the matrix P" as, e.g., P"(x',y,z, ...), wherein x' denotes the x component of the previous matrix P' is being expanded upon.

 

Let this pattern continue infinitely for every point in every subsequent matrix.

 

Effectively, we have created as complete a description of a discrete system as (I can see) is possible.

 

So, the question comes down to this:

 

Does there exist a function that defines P solely in terms of its (infinite) constituents, even if everything is known about it?

 

I have no idea.

 

Edit: Actually, I think I only described a point.

 

Hmmm... Where from here

Edit Edit: I think that no matter how definitively you describe a point, even if you know everything about that point, you effectively know nothing about it without comparing it to something else (e.g., putting it in a coordinate system).

Edited October 15, 2015 by hoponpop

[RuthlessOptimism]

I suggest you read about the field of mathematics known as "Differential Geometry". I am not an expert on this topic, I simply took an introductory course on it a few years ago at my university and have been tinkering with it regarding a project I have been working on lately.

 

One of the more interesting and important aspects of this field is the description of surfaces in terms of intrinsic coordinates. The simplest example to explain how this works is a sphere. Basically any 3d surface can be described solely in terms of its torsional and tangential curvature, usually denoted k1 and k2 for short. A sphere has constant curvature in every direction, hence why it is the simplest example. Now there is something known as the Gauss Bonnet theorem, which states that the total curvature, if you integrate k1 and k2 over the entire surface will always equal 2pi*X, where X is the surfaces euler number. Euler numbers of surfaces are constants that define what "family" the surface belongs to. For example all of the platonic solids (including spheres) have X=2. So a sphere has four pi total curvature.

 

In order to describe how we can use k1 and k2 as a coordinate system intrinsic to the surface of a sphere it is helpful to imagine the sphere as having a "frame" of two perpendicular circles, one horizontal, the other vertical. If you were to travel around the horizontal circle you can obviously go through a maximum rotation of 2 pi radians, same for the vertical one. Thus we can define points on the sphere by associating them with different values of k1 and k2.

 

Obviously there is a whole bunch of spatial information lost, with regards to "where is this point in space?" and "how is the sphere oriented in space?". But this is one method of describing surfaces in terms of solely themselves, ie uncoupled from the space they are embedded in.