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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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baseobj11=Graph[UndirectedEdge@@@ResourceFunction["WolframModel"][{{x,y},{x,z}}->{{x,y},{x,w},{y,w},{z,w}},{{0,0},{0,0}},11,"FinalState"]]

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DiscretizeRegion[Disk[]]

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### First thing to study: spatial hypergraphs

First thing to study: spatial hypergraphs

#### Local models?

Local models?

Euclidean space

Algebraic varieties (cf tropical algebra)

Algebraic varieties (cf tropical algebra)

#### What is a hyperfold?

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?

What is the relevant lowest-level metric?

## Definitions

Definitions

### [Initially consider an undirected graph]

[Initially consider an undirected graph]

#### Point

Point

Node in graph

#### Line

Line

Shortest path between two points [which may not be unique]

or the set of shortest paths between points

#### Distance 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]

[We are implicitly assuming that the distance between adjacent points is 1 unit]

#### Path Deviation

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]

Tangent space [limiting concept]

#### Geodesic ball

Geodesic ball

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

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

A count of the number of distinct nodes

[ basically this defines measure ]

[ basically this defines measure ]

#### Geodesic sphere

Geodesic sphere

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

#### Triangle

Triangle

Given 3 points A, B, C, draw the geodesics between them

[ special case of a loop ]

[ 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)

(Injectivity radius)

Roughly: how far apart can two points get, while still having a unique geodesic between them?

[In general, will vary with position]

[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

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

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 ?

Interior of region ?

### Topology

Topology

#### Boundary

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

Open set

Discrete topology: each point is an open set

### Functions on the hypergraph

Functions on the hypergraph

#### Paint a scalar value at every node

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 ]

[ 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

First level: things that work on a given hypergraph

#### Next level: things that are somehow correlated in evolving hypergraphs

Next level: things that are somehow correlated in evolving hypergraphs

#### [ Intermediate? : painting functions on hypergraphs ]

[ Intermediate? : painting functions on hypergraphs ]