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ResourceFunction["WolframModel"][{{x,y},{y,z}}->{{x,y},{y,x},{w,x},{w,z}},{{0,0},{0,0}},13,"FinalStatePlot"]
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Graph[Rule@@@ResourceFunction["WolframModel"][{{x,y},{y,z}}->{{x,y},{y,x},{w,x},{w,z}},{{0,0},{0,0}},13,"FinalState"]]
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DiscretizeRegion[Disk[]]
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First thing to study: spatial hypergraphs

Local models?

Euclidean space
Algebraic varieties (cf tropical algebra)

What is a hyperfold?

Limiting object ; analogy: fractals/nested structures are also limiting objects from neighbor independent substitution systems
Random graphs: grown with an algorithm

What is the relevant lowest-level metric?

Definitions

[Initially consider an undirected graph]

Point

Node in graph

Line

Shortest path between two points [which may not be unique]
or the set of shortest paths between points

Distance between points

Length of shortest path
(To make this a path metric space: we need points, and distances)
[We are implicitly assuming that the distance between adjacent points is 1 unit]

Path Deviation

[ This is a test for whether it is “really one path” ] (i.e. in the limit, is it one path or several?)
[ ?? the “number of shortest paths” ]
< Related to injectivity radius >
Going along all geodesics between two points, what is the maximum distance between any two corresponding points?
[[ The following is wrong; 11 should be 10, but then it won’t be interesting ]]
We can imagine some kind of clustering/segregation of different geodesics [trying to see which ones will be distinct in the hyperfold limit]

Tangent space [limiting concept]

Geodesic ball

[Volume (“content”) of the geodesic ball]

A count of the number of distinct nodes
[ basically this defines measure ]

Geodesic sphere

At least for finite cases, the sphere may not be connected

Triangle

Given 3 points A, B, C, draw the geodesics between them
[ special case of a loop ]
It is not true that 3 points necessarily define a unique triangle ... because there may be multiple geodesic paths between the points
(On a sphere, 3 points do define a unique triangle, except when two of the points are antipodal)

(Injectivity radius)

Roughly: how far apart can two points get, while still having a unique geodesic between them?
[In general, will vary with position]
[Consider also a fuzzy injectivity radius, where the geodesic isn’t unique, but at least doesn’t get too far apart]

Antipodes

What point(s) are furthest from a given point?
This attempts to find points that are GraphDiameter apart:
We want the antipode of a given point:

Webbed regions

Given a point and a geodesic not through that point, find the “webbed region” formed by all geodesics from the point to the given geodesic
Geodesic from 50 to 100 [of which there are several], and point 150 [ “geodesic fan” ]
How can we compute (a) area and/or (b) angles? [Related by curvature]
area ; side length ; angle deficit

Interior of region ?

Topology

Boundary

Given a collection of points, we can look at all geodesics between pairs of points
Is the whole of a geodesic ball around a point contained in the region? [For unit geodesic balls: for a given point, are all its neighbors in the subgraph?]
[ We need to back off from ordinary topology, because we won’t get nontrivial “ordinary open sets”]
[Can we weaken the set axioms? All only unions of “hopen” sets which have nonzero intersections ... ]

Open set

Discrete topology: each point is an open set
​

Functions on the hypergraph

Paint a scalar value at every node

E.g. put the distance from a given node as the value at each node

[ Generalized analog of ruler and compass constructibility ]

We have a “marked ruler” + “marked compass”
Inability to duplicate a cube / trisect an angle
Relativistic analog: the only thing you have are light cones

First level: things that work on a given hypergraph

Next level: things that are somehow correlated in evolving hypergraphs

[ Intermediate? : painting functions on hypergraphs ]