A Class of Models with the Potential to Represent Fundamental Physics
  1. Introduction
  2. Basic Form of Models
  3. Typical Behaviors
  4. Limiting Behavior and Emergent Geometry
  5. The Updating Process for String Substitution Systems
  6. The Updating Process in Our Models
  7. Equivalence and Computation in Our Models
  8. Potential Relation to Physics
  9. Additional Material
  10. References
  11. Index
Download PDf download pdf
ARXIV ARXIV
peer-review peer review
technical introduction Typical Behaviors

3.6 Rules Depending on One Ternary Relation

There are 9373 inequivalent left-connected 13 23 rules. Here are typical examples of their behavior after 5 steps, starting from a single ternary self-loop:

GraphicsGrid[ Partition[ ParallelMap[ ResourceFunction["WolframModel"][#, {{0, 0, 0}}, 5, "FinalStatePlot"] &, {{{1, 1, 1}} -> {{1, 1, 1}, {1, 1, 1}}, {{1, 1, 2}} -> {{1, 2, 1}, {1, 2, 3}}, {{1, 1, 2}} -> {{1, 1, 3}, {2, 3, 2}}, {{1, 1, 2}} -> {{2, 2, 3}, {3, 2, 3}}, {{1, 1, 2}} -> {{2, 2, 3}, {1, 4, 2}}, {{1, 1, 2}} -> {{1, 2, 3}, {4, 5, 3}}, {{1, 1, 2}} -> {{3, 3, 1}, {3, 3, 2}}, {{1, 1, 2}} -> {{3, 3, 1}, {3, 1, 4}}, {{1, 1, 2}} -> {{3, 3, 2}, {4, 1, 2}}, {{1, 1, 2}} -> {{3, 3, 4}, {3, 1, 1}}, {{1, 1, 2}} -> {{3, 4, 1}, {1, 2, 4}}, {{1, 2, 1}} -> {{2, 2, 2}, {1, 3, 2}}, {{1, 2, 2}} -> {{1, 2, 1}, {3, 1, 2}}, {{1, 2, 2}} -> {{1, 1, 3}, {1, 3, 3}}, {{1, 2, 1}} -> {{1, 3, 1}, {3, 2, 1}}, {{1, 2, 2}} -> {{1, 3, 3}, {3, 1, 2}}, {{1, 2, 2}} -> {{1, 3, 3}, {1, 3, 4}}, {{1, 2, 2}} -> {{1, 3, 1}, {1, 4, 4}}, {{1, 2, 1}} -> {{2, 3, 2}, {4, 1, 3}}, {{1, 2, 2}} -> {{1, 3, 4}, {4, 2, 1}}, {{1, 2, 1}} -> {{3, 2, 3}, {3, 2, 3}}, {{1, 2, 2}} -> {{3, 2, 2}, {2, 1, 1}}, {{1, 2, 1}} -> {{3, 2, 3}, {2, 1, 4}}, {{1, 2, 1}} -> {{3, 2, 3}, {3, 4, 1}}, {{1, 2, 1}} -> {{3, 1, 3}, {4, 2, 1}}, {{1, 2, 2}} -> {{3, 2, 3}, {4, 2, 2}}, {{1, 2, 1}} -> {{3, 4, 3}, {2, 3, 4}}, {{1, 2, 2}} -> {{3, 4, 4}, {1, 4, 3}}, {{1, 2, 1}} -> {{3, 4, 3}, {2, 5, 4}}, {{1, 2, 3}} -> {{1, 2, 1}, {2, 3, 4}}, {{1, 2, 3}} -> {{2, 1, 1}, {3, 2, 4}}, {{1, 2, 3}} -> {{2, 3, 3}, {1, 3, 4}}, {{1, 2, 3}} -> {{3, 2, 3}, {2, 3, 4}}, {{1, 2, 3}} -> {{1, 2, 1}, {2, 4, 3}}, {{1, 2, 3}} -> {{2, 1, 1}, {2, 4, 1}}, {{1, 2, 3}} -> {{2, 3, 1}, {1, 4, 3}}, {{1, 2, 3}} -> {{3, 2, 2}, {1, 4, 3}}, {{1, 2, 3}} -> {{1, 2, 1}, {2, 4, 5}}, {{1, 2, 3}} -> {{3, 2, 1}, {3, 4, 5}}, {{1, 2, 3}} -> {{1, 2, 2}, {4, 2, 1}}, {{1, 2, 3}} -> {{2, 2, 1}, {4, 2, 3}}, {{1, 2, 3}} -> {{3, 1, 3}, {4, 1, 3}}, {{1, 2, 3}} -> {{3, 3, 2}, {4, 3, 1}}, {{1, 2, 3}} -> {{3, 2, 2}, {4, 2, 5}}, {{1, 2, 3}} -> {{3, 3, 3}, {4, 5, 3}}, {{1, 2, 3}} -> {{1, 2, 4}, {1, 3, 4}}, {{1, 2, 3}} -> {{2, 1, 4}, {1, 4, 2}}, {{1, 2, 3}} -> {{2, 2, 4}, {4, 2, 1}}, {{1, 2, 3}} -> {{3, 2, 4}, {2, 1, 4}}, {{1, 2, 3}} -> {{3, 3, 4}, {4, 1, 2}}, {{1, 2, 3}} -> {{2, 1, 4}, {4, 3, 5}}, {{1, 2, 3}} -> {{1, 2, 4}, {4, 5, 3}}, {{1, 2, 3}} -> {{3, 3, 4}, {1, 5, 4}}, {{1, 2, 3}} -> {{2, 3, 4}, {5, 3, 1}}, {{1, 2, 3}} -> {{1, 4, 1}, {2, 3, 2}}, {{1, 2, 3}} -> {{1, 4, 3}, {3, 2, 4}}, {{1, 2, 3}} -> {{2, 4, 1}, {2, 1, 3}}, {{1, 2, 3}} -> {{2, 4, 2}, {4, 3, 1}}, {{1, 2, 3}} -> {{2, 4, 4}, {4, 3, 1}}, {{1, 2, 3}} -> {{3, 4, 3}, {2, 4, 1}}, {{1, 2, 3}} -> {{1, 4, 1}, {2, 4, 5}}, {{1, 2, 3}} -> {{2, 4, 1}, {4, 3, 5}}, {{1, 2, 3}} -> {{1, 4, 3}, {2, 5, 3}}, {{1, 2, 3}} -> {{3, 4, 3}, {4, 5, 3}}, {{1, 2, 3}} -> {{2, 4, 3}, {5, 4, 3}}, {{1, 2, 3}} -> {{1, 4, 5}, {3, 4, 5}}, {{1, 2, 3}} -> {{1, 4, 5}, {2, 4, 6}}, {{1, 2, 3}} -> {{4, 1, 2}, {2, 3, 1}}, {{1, 2, 3}} -> {{4, 1, 4}, {4, 2, 1}}, {{1, 2, 3}} -> {{4, 2, 2}, {4, 1, 3}}, {{1, 2, 3}} -> {{4, 2, 4}, {4, 1, 2}}, {{1, 2, 3}} -> {{4, 3, 3}, {2, 3, 1}}, {{1, 2, 3}} -> {{4, 4, 1}, {2, 3, 1}}, {{1, 2, 3}} -> {{4, 4, 2}, {3, 2, 3}}, {{1, 2, 3}} -> {{4, 4, 3}, {3, 4, 2}}, {{1, 2, 3}} -> {{4, 1, 4}, {4, 2, 5}}, {{1, 2, 3}} -> {{4, 3, 1}, {2, 1, 5}}, {{1, 2, 3}} -> {{4, 2, 3}, {3, 5, 2}}, {{1, 2, 3}} -> {{4, 4, 3}, {3, 5, 3}}, {{1, 2, 3}} -> {{4, 3, 4}, {5, 2, 4}}, {{1, 2, 3}} -> {{4, 1, 5}, {1, 3, 2}}, {{1, 2, 3}} -> {{4, 3, 5}, {3, 2, 1}}, {{1, 2, 3}} -> {{4, 5, 2}, {2, 3, 1}}, {{1, 2, 3}} -> {{4, 5, 3}, {4, 6, 3}}}], UpTo[14]], ImageSize -> Full]

Here are results from a few of these rules after 10 steps:

ParallelMap[ Labeled[ResourceFunction["WolframModel"][#, {{0, 0, 0}}, 10, "FinalStatePlot"], RulePlot[ResourceFunction["WolframModel"][#]], ImageSize -> {200, 150}] &, {{{1, 2, 3}} -> {{4, 5, 1}, {2, 1, 3}}, {{1, 2, 3}} -> {{4, 4, 2}, {4, 1, 3}}, {{1, 2, 3}} -> {{3, 4, 2}, {4, 3, 1}}}]

The number of relations in the evolution of 13 23 rules can grow in a slightly more complicated way than for 12 n2 rules. In addition to linear and 2t growth, there is also, for example, quadratic growth: in the rule

{{x, x, y}} -> {{y, y, z}, {x, y, x}}
RulePlot[ResourceFunction[ "WolframModel"][{{x, x, y}} -> {{y, y, z}, {x, y, x}}]]

each existing “arm” effectively grows by one element each step, and there is one new arm generated, yielding a total size of = :

ResourceFunction["WolframModelPlot"][#, ImageSize -> Tiny] & /@ ResourceFunction[ "WolframModel"][{{1, 1, 2}} -> {{2, 2, 3}, {1, 2, 1}}, {{0, 0, 0}}, 6, "StatesList"]

The rule

{{x, x, y}} -> {{y, y, y}, {x, y, z}}
RulePlot[ResourceFunction[ "WolframModel"][{{x, x, y}} -> {{y, y, y}, {x, y, z}}]]

yields a Fibonacci tree, with size Fibonacci[t+2]–1 ∼ :

ResourceFunction["WolframModelPlot"][#, ImageSize -> Tiny] & /@ ResourceFunction[ "WolframModel"][{{x, x, y}} -> {{y, y, y}, {x, y, z}}, {{0, 0, 0}}, 6, "StatesList"]

13 23 rules can produce results that look fairly complex. But it is a consequence of their dependence only on a single relation that once such rules have established a large-scale structure, later updates (which are necessarily purely local) can in a sense only embellish it, not fundamentally change it:

ResourceFunction["WolframModelPlot"][#, ImageSize -> Tiny] & /@ ResourceFunction[ "WolframModel"][{{1, 2, 3}} -> {{4, 4, 2}, {4, 1, 3}}, {{0, 0, 0}}, 10, "StatesList"]

There are 637,568 inequivalent left-connected 13 33 rules; here are samples of their behavior:

GraphicsGrid[ Partition[ ParallelMap[ ResourceFunction["WolframModel"][#, {{0, 0, 0}}, 4, "FinalStatePlot"] &, {{{1, 2, 2}} -> {{3, 3, 2}, {3, 1, 3}, {4, 4, 3}}, {{1, 2, 2}} -> {{3, 4, 3}, {4, 5, 2}, {1, 3, 2}}, {{1, 1, 2}} -> {{3, 4, 3}, {3, 5, 6}, {5, 2, 7}}, {{1, 1, 2}} -> {{3, 4, 3}, {5, 3, 6}, {2, 5, 3}}, {{1, 2, 1}} -> {{2, 2, 3}, {1, 3, 1}, {4, 2, 5}}, {{1, 2, 1}} -> {{3, 4, 2}, {4, 3, 2}, {3, 1, 4}}, {{1, 2, 3}} -> {{4, 3, 4}, {1, 3, 4}, {1, 5, 2}}, {{1, 2, 2}} -> {{2, 1, 2}, {3, 4, 1}, {5, 4, 6}}, {{1, 2, 2}} -> {{3, 1, 1}, {3, 4, 1}, {5, 1, 3}}, {{1, 1, 2}} -> {{3, 3, 4}, {4, 5, 3}, {5, 1, 2}}, {{1, 2, 1}} -> {{2, 3, 1}, {1, 3, 4}, {1, 5, 2}}, {{1, 2, 3}} -> {{4, 4, 1}, {5, 6, 4}, {3, 6, 1}}, {{1, 2, 3}} -> {{2, 4, 5}, {4, 2, 6}, {1, 2, 6}}, {{1, 2, 3}} -> {{3, 4, 3}, {4, 2, 5}, {1, 5, 5}}, {{1, 2, 3}} -> {{3, 3, 4}, {3, 1, 5}, {1, 2, 6}}, {{1, 2, 2}} -> {{3, 4, 1}, {1, 4, 5}, {6, 5, 3}}, {{1, 2, 3}} -> {{1, 3, 1}, {4, 1, 3}, {1, 4, 2}}, {{1, 2, 3}} -> {{3, 4, 2}, {4, 5, 2}, {6, 3, 1}}, {{1, 2, 2}} -> {{3, 1, 2}, {3, 4, 5}, {2, 6, 4}}, {{1, 2, 2}} -> {{3, 4, 4}, {4, 2, 3}, {3, 5, 6}}, {{1, 2, 2}} -> {{2, 3, 3}, {3, 1, 2}, {1, 1, 4}}, {{1, 2, 1}} -> {{3, 3, 1}, {1, 3, 1}, {3, 2, 4}}, {{1, 2, 1}} -> {{2, 2, 1}, {1, 3, 3}, {3, 2, 1}}, {{1, 2, 1}} -> {{3, 4, 2}, {5, 2, 4}, {1, 6, 4}}, {{1, 2, 2}} -> {{3, 4, 3}, {4, 3, 2}, {4, 5, 4}}, {{1, 2, 3}} -> {{4, 2, 4}, {4, 3, 5}, {2, 1, 3}}, {{1, 2, 2}} -> {{3, 4, 4}, {5, 6, 4}, {5, 6, 2}}, {{1, 1, 2}} -> {{1, 2, 2}, {3, 1, 4}, {4, 5, 6}}, {{1, 2, 1}} -> {{1, 1, 3}, {4, 1, 3}, {4, 1, 4}}, {{1, 2, 1}} -> {{3, 4, 3}, {2, 3, 4}, {5, 2, 4}}, {{1, 2, 2}} -> {{3, 4, 5}, {6, 4, 1}, {2, 5, 7}}, {{1, 2, 3}} -> {{4, 1, 1}, {4, 1, 5}, {5, 3, 2}}, {{1, 2, 3}} -> {{3, 3, 1}, {2, 1, 1}, {3, 4, 2}}, {{1, 2, 3}} -> {{3, 3, 1}, {2, 1, 2}, {4, 5, 1}}, {{1, 2, 3}} -> {{4, 4, 5}, {3, 4, 2}, {3, 5, 1}}, {{1, 1, 2}} -> {{3, 4, 3}, {3, 4, 3}, {2, 3, 5}}, {{1, 2, 3}} -> {{1, 4, 2}, {4, 2, 1}, {1, 4, 5}}, {{1, 2, 3}} -> {{2, 4, 1}, {4, 1, 2}, {3, 5, 1}}, {{1, 1, 2}} -> {{2, 2, 3}, {2, 3, 1}, {4, 1, 5}}, {{1, 2, 3}} -> {{4, 4, 3}, {4, 3, 2}, {2, 1, 4}}, {{1, 2, 1}} -> {{3, 4, 3}, {1, 3, 4}, {1, 3, 5}}, {{1, 2, 1}} -> {{1, 2, 1}, {2, 3, 4}, {5, 3, 1}}, {{1, 2, 1}} -> {{2, 2, 1}, {3, 2, 3}, {3, 4, 2}}, {{1, 2, 1}} -> {{3, 1, 2}, {4, 1, 5}, {6, 2, 7}}, {{1, 1, 2}} -> {{3, 1, 3}, {3, 4, 1}, {5, 4, 5}}, {{1, 1, 2}} -> {{3, 4, 3}, {2, 4, 1}, {1, 5, 2}}, {{1, 2, 3}} -> {{4, 1, 4}, {2, 4, 3}, {3, 5, 6}}, {{1, 2, 3}} -> {{4, 2, 2}, {2, 1, 4}, {4, 1, 3}}, {{1, 1, 2}} -> {{1, 3, 1}, {2, 3, 3}, {3, 4, 5}}, {{1, 2, 3}} -> {{3, 1, 1}, {3, 4, 4}, {1, 4, 3}}, {{1, 2, 3}} -> {{4, 5, 5}, {5, 1, 3}, {2, 6, 4}}, {{1, 1, 1}} -> {{2, 3, 4}, {3, 1, 5}, {6, 7, 2}}, {{1, 2, 3}} -> {{1, 4, 4}, {1, 5, 5}, {3, 4, 4}}, {{1, 2, 3}} -> {{2, 2, 4}, {4, 3, 2}, {3, 5, 2}}, {{1, 1, 2}} -> {{3, 4, 3}, {4, 2, 5}, {6, 1, 7}}, {{1, 2, 3}} -> {{3, 1, 2}, {1, 2, 4}, {5, 1, 2}}, {{1, 2, 3}} -> {{4, 1, 4}, {2, 1, 4}, {2, 1, 1}}, {{1, 1, 2}} -> {{1, 3, 4}, {3, 1, 5}, {6, 4, 3}}, {{1, 2, 3}} -> {{3, 3, 4}, {2, 4, 5}, {6, 1, 4}}, {{1, 1, 2}} -> {{2, 3, 2}, {2, 1, 3}, {3, 4, 1}}, {{1, 1, 2}} -> {{3, 1, 2}, {3, 2, 4}, {4, 1, 2}}, {{1, 2, 3}} -> {{3, 1, 2}, {3, 2, 4}, {2, 5, 1}}, {{1, 1, 2}} -> {{3, 2, 2}, {4, 2, 1}, {5, 2, 1}}, {{1, 2, 1}} -> {{3, 4, 4}, {3, 5, 6}, {2, 5, 1}}, {{1, 2, 1}} -> {{3, 3, 2}, {2, 4, 5}, {1, 6, 2}}, {{1, 1, 2}} -> {{1, 3, 4}, {4, 5, 6}, {7, 8, 6}}, {{1, 2, 3}} -> {{1, 1, 4}, {2, 3, 4}, {3, 5, 6}}, {{1, 2, 1}} -> {{2, 3, 4}, {5, 6, 4}, {7, 8, 4}}, {{1, 2, 3}} -> {{2, 1, 1}, {4, 1, 3}, {5, 3, 6}}, {{1, 2, 3}} -> {{4, 4, 3}, {3, 4, 3}, {1, 4, 2}}, {{1, 2, 3}} -> {{1, 4, 4}, {5, 4, 2}, {3, 2, 4}}, {{1, 2, 3}} -> {{1, 4, 1}, {2, 3, 4}, {2, 5, 3}}, {{1, 2, 1}} -> {{3, 2, 3}, {3, 1, 1}, {2, 1, 3}}, {{1, 2, 3}} -> {{2, 3, 4}, {3, 1, 4}, {5, 6, 1}}, {{1, 2, 3}} -> {{2, 4, 2}, {1, 4, 1}, {2, 1, 5}}, {{1, 2, 3}} -> {{4, 1, 1}, {4, 2, 3}, {3, 5, 1}}, {{1, 1, 2}} -> {{2, 2, 1}, {2, 3, 1}, {2, 4, 4}}, {{1, 2, 2}} -> {{3, 2, 4}, {4, 1, 3}, {5, 6, 2}}, {{1, 2, 3}} -> {{2, 3, 2}, {2, 4, 1}, {5, 3, 6}}, {{1, 2, 2}} -> {{2, 1, 1}, {1, 2, 3}, {3, 1, 1}}, {{1, 2, 2}} -> {{1, 1, 3}, {3, 4, 5}, {3, 4, 6}}, {{1, 1, 2}} -> {{2, 3, 3}, {3, 1, 1}, {4, 2, 5}}, {{1, 1, 2}} -> {{3, 3, 4}, {2, 3, 5}, {2, 1, 2}}, {{1, 2, 3}} -> {{4, 2, 5}, {4, 1, 6}, {2, 7, 6}}}], UpTo[14]], ImageSize -> Full]

The results can be more elaborate than for 13 23 rules—as the following examples illustrate—but remain qualitatively similar:

ParallelMap[ Labeled[ResourceFunction["WolframModel"][#, {{0, 0, 0}}, 7, "FinalStatePlot"], RulePlot[ ResourceFunction["WolframModel"][#]]] &, {{{1, 2, 3}} -> {{4, 2, 5}, {4, 5, 1}, {5, 1, 3}}, {{1, 2, 3}} -> {{4, 5, 3}, {2, 3, 5}, {6, 5, 1}}}]

One notable 13 33 rule (that we will discuss below) in a sense directly implements the recursive formation of a nested Sierpiński pattern:

{{x, y, z}} -> {{x, u, v}, {z, v, w}, {y, w, u}}
RulePlot[ResourceFunction[ "WolframModel"][{{x, y, z}} -> {{x, u, v}, {z, v, w}, {y, w, u}}]]
ResourceFunction["WolframModelPlot"][#, ImageSize -> Tiny] & /@ ResourceFunction[ "WolframModel"][{{x, y, z}} -> {{x, u, v}, {z, v, w}, {y, w, u}}, {{1, 2, 3}}, 5, "StatesList"]
  • Download Notebook
  • Open in Wolfram Cloud
  • Discuss This Page