We explore graph properties of the Ruliad by changing parameters of modular arithmetic functions. We explored simple rules applying the Modulo operation on linear, quadratic and cubic functions. We found Isomorphic Graphs and Cycles of many sizes for every function tried. A general framework to study the isomorphisms and cycle size needs to be built to further study and properly classify the structure of these rules.

Linear Functions

We begin by exploring the graphs formed by the rules derived from applying a simple linear function, such as x -> Mod[10 x + 1, i], {x, 0, 9}, {i, 1, 10}

Isomorphisms

The first feature we explore is the isomorphic nature of the graphs generated by this simple rule.
First, we generate the graphs by applying the rule
p1=Table[Graph[Table[xMod[10x+1,i],{x,0,9}]],{i,1,10}]
Out[]=

,
,
,
,
,
,
,
,
,

Then we group together the graphs by isomorphisms:
g1=GroupBy[p1,CanonicalGraph]
Out[]=


,
,
,
,

,
,

,

,

,


And we summarize our findings in a table
Show[​​GraphicsGrid[​​Prepend[​​Transpose[​​{Keys[g1],Length/@Values[g1],​​Table[​​Flatten[Position[IsomorphicGraphQ[#,Keys[g1][[i]]]&/@p1,​​True]],{i,1,Length[Keys[g1]]}]}],​​{"Graph","Number of Isomorphisms","Index"}]​​],​​PlotLabel​​HoldForm[IsomorphismsinxMod[10x+1,i]{i,1,10}Rule],​​LabelStyle{GrayLevel[0]}​​]
Out[]=
Finally, we explore the nature of the cycles formed
Show[​​GraphicsGrid[​​Prepend[​​Transpose[​​{Keys[g1],Length/@Values[g1],​​Table[​​Flatten[Position[IsomorphicGraphQ[#,Keys[g1][[i]]]&/@p1,​​True]],{i,1,Length[Keys[g1]]}]}],​​{"Graph","Number of Cycles","Index"}]​​],​​PlotLabel​​HoldForm[CyclesinxMod[10x+1,i]{i,1,10}Rule],​​LabelStyle{GrayLevel[0]}]
Out[]=

Quadratic Function

We then explore the quadratic function Mod[27 x^2 + 3, i], {x, 0, 28}, {i, 1, 29}
And obtain the following tables:

Cubic functions

We finish our exploration with a cubic function of the form x -> Mod[2 x^3 + 1, i], {x, 0, 3}, {i, 1, 54}

Concluding remarks

We explored some properties, like isomorphisms and cycle size, of graphs obtained by a linear, a quadratic and a cubic function. There is a hint that the further application of these kind of rules on other functions may give rise to a structure in the distribution of isomorphisms and the size of the cycles formed by these graphs. Further investigation is needed.

Keywords

◼
  • Ruliad
    • ◼
  • Modular Algebra
    • ◼
  • Graph Theory

    • Acknowledgment

    We wish to thank Nikolay Murzin and Xerxes Arsiwalla for the help and guidance of this project, and Stephen Wolfram, who defined the project.

    References

    ◼
  • A project to find the Fundamental Theory of Physics. Stephen Wolfram, 2022.

    • CITE THIS NOTEBOOK

    Explorations on the ruliad via modular Arithmetic​
    by Juan Arturo Silva-Ordaz​
    Wolfram Community, STAFF PICKS, July 12, 2023
    ​https://community.wolfram.com/groups/-/m/t/2959627