Constructing dimensions out of a graph structure?

[FlorianDietz]

I have a question to anyone experienced with graphs and topology. The question is relevant for this topic: https://www.physicsforums.com/threads/a-graph-based-model-of-physics-without-dimensions.887694/

Is it possible to construct an arbitrarily large graph (a set of vertices, a set of edges), such that the following is true:

There exists a mapping f(v) of vertices to (x,y,z) coordinates such that
For any pair of vertices m,n:
the euclidean distance of f(m) and f(n) is approximately equal to the length of the shortest path between m and n (inaccuracies are fine so long as the distance is small, but the approximation should be good at larger distances).

In other words, is it possible to construct a graph that effectively simulates a 3 dimensional space?

 

[zinq]

FlorianDietz, this is an interesting question.

But just asking for a mapping *from* the vertices of some graph G *to* 3-dimensional Euclidean space R³ is not at all the same as asking whether it's possible to construct a graph "that effectively simulates a 3 dimensional space".

So I'm not sure if you are phrasing the question in such a way as to ask what you are really interested in.

For example, I bet you can find such a mapping if a) G has 1 vertex and 0 edges, or b) G has 2 vertices and 1 edge, or c) if G has 3 vertices and 3 edges arranged in a triangle, or d) if G has n+1 vertices and n edges arranged in a straight line . . . but no one would think these come close to "simulating" even a finite portion of 3-dimensional space.

Maybe if you thought about this question a little more you might rephrase the question so that it comes closer to what you are aiming at? One possibility is that you would like to find an *infinite* graph G with a mapping of its set V of vertices into R³:

f: V → R³​

with certain properties that you might like to have? Or just let the vertices and edges of G be a subset of R³.

For instance, you could imagine the graph whose vertices correspond to all the points of R³ having integer coordinates (K, L, M), and whose edges are all the unit intervals connecting a vertex to its 6 nearest neighbors (right, left, front, back, up down).

In that case, in what precise sense would that graph "simulate" R³ ?

 

[Stephen Tashi]

In other words, is it possible to construct a graph that effectively simulates a 3 dimensional space?

A 3-D grid can be regarded as a graph. It seems to me that a fine enough grid satisfies your requirement.

 

[The Bill]

A 3-D grid can be regarded as a graph. It seems to me that a fine enough grid satisfies your requirement.

That gives a taxicab metric instead of the Euclidean metric the OP wants.

 

[Stephen Tashi]

That gives a taxicab metric instead of the Euclidean metric the OP wants.

Can we connect diagonally opposite vertices with an edge ?

 

[Auto-Didact]

This sounds an awful lot like Penrose' combinatorial graph construction of space(-time) called spin-networks. Incidentally, spin networks can be used to quantize area and volume in LQG.

Here's a link to Penrose' original paper, courtesy of John Baez: http://math.ucr.edu/home/baez/penrose/Penrose-AngularMomentum.pdf