@Jonathan Gorard I explained the 4D version of it better in the paper linked in this new thread: https://community.wolfram.com/groups/-/m/t/2381759

Probably one can save the isotropic hard limitation on the speed of light, making resistances associates with timelike grid edges negative.

I leave you a post that you might be interested in that might have a connection with your work. https://community.wolfram.com/groups/-/m/t/2027996

That would obviously be right for a small number of grids, but the point of my post is that it would work with a very large number of interweaving grids, all pointing in different directions, constructed by simple local rules.

Another guess: a wave equation modelled on a square grid looks more or less isotropic. Its finite-difference scheme can be seen as a sort of local update rule (and maybe combined with branching). However it implies storing values on graph nodes, and possibly has relativity issues.

If you want macroscopically isotropic measure on the square grid, it's simple. Imagine that your graph is a network of identical electrical resistances, and define the distance between two points A and B as a resistance between two leads connected to these points. When the points are far enough from each other, the resistance becomes a function of the Euclidean distance.

Neat idea, but I'm not sure that I like the concept of having hypergraphs with such large edge density! (not least because it leads to an unbounded Hausdorff dimension for space...)

See my comment on this related post (https://community.wolfram.com/groups/-/m/t/1950834), or read my Q&A answer (https://www.wolframphysics.org/questions/spacetime-relativity/why-do-you-get-a-euclideanriemannian-metric-as-opposed-to-a-taxicab-metric-induced-on-your-hypergraphs/) for an explanation of how we actually derive Riemannian (and, indeed, Lorentzian) metrics in the context of our models - it is, unsurprisingly, not just the combinatorial distance metric. Or have a look through my relativity paper (http://wolframphysics.org/technical-documents/) for some more complete mathematical details :)

Blockquote Are you sure that the underlying topology is still a 2d grid. If you tie infinitely many grids together in that way. You loose the property of dimensionality as Wolfram defined it since a single step adds infinitly many neighbors making the dimension undefined.

Depending on how exactly you connect the grids, the number of direct neighbors per node can go towards infinity but it can also stay a small number despite the number of grids going towards infinity. But anyways, as far as I remember the way Wolfram defined dimensionality is based on how the volume of a ball grows w.r.t. its radius, where ball is a set of nodes that are at most at the graph distance of the radius and the volume is the number of nodes in the ball. But this is just one kind of definition. You could instead take a set of random nodes and measure all the distances between them. Based on this set of long range distances you can see how many dimensions there are. It's like a complicated trigonometry exercise. A simple example: take four points on the graph, measure the 6 distances. From this information construct a tetrahedron in a real 3D space. If the tetrahedron looks flat, then the graph probably represents a 2D space.

The ratios such as 1:3 indicate how Two girds are connected to each other such as A and B. Since arcsin(1/3) is irrational w.r.t pi you could just keep on using 1:3 to connect grid C to B and so on to keep the rotation going. Due to the irrationality, the orientations of the grids would approximately fill out all the angles from 0 to 2pi. But this is just an simple example, you can also let the ratio change as you noted.

In my examples only a minority of nodes is shared between grids for simplicity. But a further option would be to let the grids share half of their nodes or even the majority and you could in principle still achieve the desired distances and symmetries but it's more difficult to visualize.

If you choose random starting points that aren't on the same grid you can't walk in parallel at maximal precision but neither can you move real elementary particles at sub-plack precision. You can however take a path through multiple different grids until you find a grid that approximately points in the direction you want to move which you can use for approximate parallel motion.

In graph theory the distance is the length of the shortest path measured in the number of edges you have to walk over. But the Lorentz distance on the other hand is the length of the Longest time-like path.

Yes it is integer valued, think of very large integers. If you use this distance to for instance measure the 4 sides and 2 diagonals of a very large Quadrilateral, then the ratios between these 6 lengths will approximate what you'd expect from Euclidean geometry, if you let the size of the Quadrilateral as well as the number of grids goes towards infinity.

The Spacetime is a directed acyclic graph (DAG), so there exists a longest path along the arrows. Paradoxically the Lorentz distance is the longest path between two nodes. This longest path is the one where the largest amount of time passed between two events, which is also a geodesic of this flat spacetime.