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When constructing space, is a "meta-space" needed?

Posted 1 month ago
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Hello,

In the Wolfram model, physical space is a network of points (or nodes). But usually points are located somewhere, just like we draw points on a piece of paper. So where are those points in Wolfram model located? In a "meta-space"? If yes, what is the nature of this meta-space? (Is it continuous?) If no, how should the construction of the network of points be understood?

Thanks.

Wyman

5 Replies

My understanding is that points (vertices) are defined only in terms of their connections (edges). Apparent multi-dimensional space arises by taking paths (geodesics) through the graph.

e.g. Given {1 -> 2, 1 -> 3, 1 -> 4} from point 1 you can get to point 2, 3, or 4, thus having local dimensionality 3. Similarly {1 -> 2, 1 -> 3} has local dimensionality 2 at point 1.

Note that point 2 has local dimensionality 1, and that therefore the graph as a whole has fractional dimensionality.

Posted 1 month ago

We indeed don't need to embed the vertices in anything, however, computing dimensionality is slightly more complicated than you described. You need to consider balls (graph neighborhoods) of different radii starting from the same vertex, and see how quickly their volumes (vertex counts) grow. If they grow as a third power of the radius, for example, we have a 3-dimensional space.

Posted 1 month ago

Thanks. I take what you mean is that the vertices in Wolfram model are in fact not embedded in anything. But I just want to note that the graphs (or hypergraphs) actually have the vertices embedded in a space (i.e. the space on which the graphs are drawn).

Posted 1 month ago

That's purely for visualization purposes, it does not have any physical significance.

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