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

Posted 1 year 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

6 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 year 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 year 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 year ago

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

Posted 1 month ago

Not to be obtuse, but in physical space there is no point in space, that is a human construct. Space is nothing but a relation between objects (or more accurately events). When we humans define a point in space we are making a lot of assumptions. You cannot make the same assumptions in the WM.

What assumptions are we humans taking? Lets use as an example how you would define the point in space 1/2 of the distance between the earth and moon. How would you uniquely define it? First, you need to define when you are defining that point because the moon is orbiting the earth. The moment after you define that time the point is not 1/2 the distance between the earth an moon any more, everything has moved. So how can you define that time? Normally you would do so against a common reference. In the WM it is the step number (actually easier than in physical space because of Relativity).

Ok, we now have a time, what about location? 1/2 the way between the Earth and Moon works if you are human and live on the Earth, but you are comparing physical space to the WM, so where is the Earth and the Moon? In the Sol solar system. Where is that...Galaxy...Galaxy Cluster...Observable Universe...Universe. How would you define all of these locations with enough accuracy to be more than just a vague notion? Not very easy to do, so we don't. We just reference time and location from the planet we live on and ignore the rest.

Bottom line is in physical space we define a point in space to be relative to common reference points (in time and space). You do the same thing with the nodes. The complicating factor though is that there is no inherent geometry to work with, you have to build that up with the node and it relation to nodes around it, and the nodes around them, and so on until you get enough relationships between enough nodes to define the geometry to define a location and a common reference points.

Not an easy task.

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