## What is the notion of an integral?

What is the notion of an integral?

Does it still compute area?

d^d x etc.

Vector analysis.... Gauss’s theorem, etc.

#### What is Div[XXX]?

What is Div[XXX]?

## Derivative is just change between graph nodes

Derivative is just change between graph nodes

## Divergence is just change of function as a ball expands

Divergence is just change of function as a ball expands

## Grad just looks at geodesics to find maximum change

Grad just looks at geodesics to find maximum change

## Laplacians / PDEs on graphs...

Laplacians / PDEs on graphs...

https://arxiv.org/pdf/math/0509193.pdf

# Non-Flat Sierpinskis

Non-Flat Sierpinskis

#### Definitions

Definitions

In[]:=

Vols[graph_,start_:All,max_:Automatic]:=MeanAround/@Transpose[Values[ResourceFunction["GraphNeighborhoodVolumes"][graph,start,max]]]

In[]:=

RandVols[graph_,n_Integer,max_:Automatic]:=Vols[graph,"Random"n,max]

In[]:=

Dims[vols_]:=ListLinePlotTable[(Log[vols[[r+1]]]-Log[vols[[r]]])/(Log[r+1]-Log[r]),{r,Length[vols]-1}],

## For Sierpinski

For Sierpinski

In[]:=

HypergraphToGraph[WolframModel[{{1,2,3}}{{1,4,6},{2,5,4},{3,6,5}},{{0,0,0}},5,"FinalState"]]

Out[]=

In[]:=

UndirectedGraph[HypergraphToGraph[WolframModel[{{1,2,3}}{{1,4,6},{2,5,4},{3,6,5}},{{0,0,0}},8,"FinalState"]]];

In[]:=

Dims[RandVols[%,200]]

Out[]=

In[]:=

UndirectedGraph[HypergraphToGraph[WolframModel[{{1,2,3}}{{1,4,6},{2,5,4},{3,6,5}},{{0,0,0}},9,"FinalState"]]];

In[]:=

Dims[RandVols[%,500]]

Out[]=

In[]:=

SierpinskiMesh[4]

Out[]=

In[]:=

Graphics[%250,AxesTrue]

Out[]=

Cf. limit sets of Kleinian groups.....

## Model Spaces

Model Spaces

Integer dimensional lattice:

Projection function (cf. space-filling curve)

Selector of which points are allowed (e.g. BitXor on digits, or Cantor set)

Limiting to continuum when allow infinite numbers of digits, but e.g. Cantor set

Ultimately, points are labeled by their provenance in the graph rewriting system. But how does one get a good family of model spaces?

Given n digit sequences, find some consistent set in these digit sequences.

## Cayley Graphs

Cayley Graphs

Add generators, then find canonical element applying all possible replacements