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
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technical introduction Typical Behaviors

3.11 Rules Involving More Ternary Relations

There are about 9 billion inequivalent left-connected 23 43 rules. About 20% lead to connected results, and of these about half show continued growth. Here is a random sampling of the behavior of such rules:

rulesample = {{{{1, 2, 3}, {2, 4, 5}} -> {{6, 7, 4}, {6, 8, 4}, {4, 7, 9}, {1, 3, 2}}, {{0, 0, 0}, {0, 0, 0}}, 64}, {{{1, 2, 3}, {1, 3, 4}} -> {{1, 5, 4}, {1, 4, 6}, {7, 1, 6}, {8, 6, 3}}, {{0, 0, 0}, {0, 0, 0}}, 16}, {{{1, 2, 3}, {4, 3, 5}} -> {{6, 7, 7}, {6, 2, 8}, {9, 7, 8}, {5, 6, 10}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {4, 2, 5}} -> {{6, 7, 3}, {7, 8, 5}, {3, 8, 9}, {4, 10, 6}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {4, 3, 5}} -> {{6, 7, 8}, {7, 6, 9}, {10, 11, 6}, {4, 10, 8}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {2, 4, 5}} -> {{2, 4, 1}, {1, 6, 7}, {8, 4, 7}, {5, 3, 8}}, {{0, 0, 0}, {0, 0, 0}}, 15}, {{{1, 2, 3}, {4, 3, 5}} -> {{2, 3, 3}, {2, 6, 7}, {8, 2, 9}, {1, 10, 7}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 4, 4}, {5, 7, 6}, {5, 8, 9}, { 1, 3, 8}}, {{0, 0, 0}, {0, 0, 0}}, 56}, {{{1, 2, 3}, {4, 5, 3}} -> {{1, 6, 1}, {7, 8, 1}, {4, 9, 7}, {3, 10, 8}}, {{0, 0, 0}, {0, 0, 0}}, 17}, {{{1, 2, 3}, {4, 3, 5}} -> {{6, 6, 7}, {3, 6, 7}, {4, 4, 8}, {9, 4, 3}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {4, 5, 3}} -> {{1, 1, 6}, {7, 8, 6}, {8, 5, 9}, {8, 10, 11}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 1, 2}, {3, 1, 4}} -> {{5, 2, 2}, {2, 4, 4}, {6, 4, 4}, {1, 7, 8}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 1, 6}, {6, 5, 7}, {1, 2, 8}, { 9, 2, 3}}, {{0, 0, 0}, {0, 0, 0}}, 30}, {{{1, 2, 3}, {1, 4, 5}} -> {{6, 3, 6}, {6, 1, 7}, {3, 4, 8}, {9, 10, 1}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {2, 4, 3}} -> {{3, 3, 4}, {3, 1, 4}, {5, 3, 6}, {7, 8, 1}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {2, 4, 5}} -> {{1, 6, 1}, {7, 1, 5}, {7, 8, 2}, { 3, 9, 6}}, {{0, 0, 0}, {0, 0, 0}}, 65}, {{{1, 2, 3}, {4, 3, 5}} -> {{1, 6, 7}, {6, 4, 8}, {7, 8, 9}, {4, 10, 1}}, {{0, 0, 0}, {0, 0, 0}}, 38}, {{{1, 2, 3}, {2, 4, 5}} -> {{1, 5, 1}, {5, 6, 4}, {3, 6, 7}, {2, 8, 5}}, {{0, 0, 0}, {0, 0, 0}}, 11}, {{{1, 1, 2}, {3, 4, 1}} -> {{3, 3, 5}, {6, 4, 3}, {7, 4, 8}, {7, 9, 10}}, {{0, 0, 0}, {0, 0, 0}}, 19}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 6, 7}, {7, 7, 3}, {8, 8, 3}, {6, 1, 9}}, {{0, 0, 0}, {0, 0, 0}}, 98}, {{{1, 2, 3}, {3, 4, 5}} -> {{4, 2, 4}, {4, 6, 7}, {8, 7, 2}, {6, 9, 10}}, {{0, 0, 0}, {0, 0, 0}}, 19}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 6, 7}, {8, 7, 7}, {8, 5, 9}, {10, 11, 6}}, {{0, 0, 0}, {0, 0, 0}}, 65}, {{{1, 2, 1}, {3, 2, 4}} -> {{2, 2, 1}, {5, 2, 2}, {1, 6, 7}, {8, 1, 7}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {2, 4, 5}} -> {{5, 5, 6}, {5, 4, 7}, {8, 7, 1}, { 2, 8, 9}}, {{0, 0, 0}, {0, 0, 0}}, 7}, {{{1, 1, 2}, {1, 3, 4}} -> {{2, 2, 5}, {6, 5, 2}, {7, 5, 5}, { 2, 7, 3}}, {{0, 0, 0}, {0, 0, 0}}, 66}, {{{1, 2, 3}, {1, 4, 5}} -> {{2, 5, 5}, {2, 6, 1}, {6, 5, 7}, {8, 2, 4}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {4, 3, 5}} -> {{6, 5, 1}, {5, 3, 7}, {7, 8, 6}, { 9, 10, 3}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {3, 4, 5}} -> {{1, 6, 7}, {7, 6, 3}, {8, 6, 2}, {9, 7, 5}}, {{0, 0, 0}, {0, 0, 0}}, 26}, {{{1, 2, 3}, {4, 2, 5}} -> {{6, 6, 7}, {6, 1, 8}, {8, 8, 4}, {2, 6, 9}}, {{0, 0, 0}, {0, 0, 0}}, 98}, {{{1, 2, 3}, {2, 4, 5}} -> {{4, 6, 4}, {7, 4, 8}, {5, 7, 4}, {9, 10, 6}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {1, 4, 5}} -> {{2, 5, 5}, {6, 2, 7}, {8, 4, 6}, {1, 9, 3}}, {{0, 0, 0}, {0, 0, 0}}, 17}, {{{1, 2, 3}, {2, 4, 5}} -> {{4, 6, 7}, {6, 1, 5}, {7, 8, 1}, {5, 3, 9}}, {{0, 0, 0}, {0, 0, 0}}, 7}, {{{1, 2, 3}, {4, 5, 3}} -> {{6, 6, 2}, {7, 6, 5}, {4, 8, 7}, { 9, 4, 3}}, {{0, 0, 0}, {0, 0, 0}}, 46}, {{{1, 2, 3}, {1, 4, 5}} -> {{6, 6, 2}, {5, 6, 3}, {7, 5, 3}, {6, 8, 9}}, {{0, 0, 0}, {0, 0, 0}}, 66}, {{{1, 2, 3}, {3, 4, 5}} -> {{1, 6, 7}, {6, 8, 1}, {7, 1, 2}, {1, 3, 9}}, {{0, 0, 0}, {0, 0, 0}}, 26}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 5, 6}, {7, 4, 8}, {4, 8, 9}, {10, 11, 7}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 1, 7}, {1, 5, 6}, {2, 7, 8}, {4, 2, 1}}, {{0, 0, 0}, {0, 0, 0}}, 6}, {{{1, 2, 3}, {2, 4, 5}} -> {{4, 3, 2}, {3, 4, 6}, {1, 4, 7}, { 5, 7, 6}}, {{0, 0, 0}, {0, 0, 0}}, 8}, {{{1, 2, 3}, {3, 2, 4}} -> {{3, 5, 5}, {3, 5, 1}, {4, 6, 1}, { 7, 8, 1}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 1, 7}, {7, 1, 2}, {2, 8, 9}, { 3, 4, 2}}, {{0, 0, 0}, {0, 0, 0}}, 29}, {{{1, 2, 2}, {3, 2, 4}} -> {{5, 6, 6}, {1, 6, 5}, {7, 8, 5}, {8, 9, 4}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 1}, {1, 3, 4}} -> {{1, 1, 2}, {1, 5, 6}, {7, 8, 5}, {7, 3, 9}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 1, 2}, {3, 4, 2}} -> {{1, 1, 3}, {1, 5, 3}, {6, 3, 5}, {7, 4, 7}}, {{0, 0, 0}, {0, 0, 0}}, 17}, {{{1, 2, 3}, {2, 4, 5}} -> {{5, 5, 6}, {4, 6, 5}, {6, 4, 2}, {3, 1, 1}}, {{0, 0, 0}, {0, 0, 0}}, 6}, {{{1, 2, 3}, {4, 3, 5}} -> {{3, 1, 6}, {7, 1, 6}, {8, 9, 3}, { 5, 4, 8}}, {{0, 0, 0}, {0, 0, 0}}, 13}, {{{1, 2, 3}, {4, 5, 3}} -> {{1, 1, 3}, {3, 6, 7}, {8, 9, 1}, {2, 10, 11}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {1, 4, 5}} -> {{1, 2, 1}, {3, 2, 6}, {7, 8, 6}, { 9, 7, 4}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {1, 4, 5}} -> {{2, 1, 1}, {2, 6, 6}, {3, 1, 5}, {7, 8, 5}}, {{0, 0, 0}, {0, 0, 0}}, 7}, {{{1, 2, 3}, {3, 4, 5}} -> {{5, 2, 2}, {2, 6, 5}, {7, 1, 8}, { 9, 4, 8}}, {{0, 0, 0}, {0, 0, 0}}, 17}, {{{1, 2, 3}, {4, 3, 5}} -> {{5, 1, 6}, {5, 6, 1}, {2, 7, 8}, {9, 7, 10}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {4, 3, 5}} -> {{6, 7, 6}, {5, 7, 7}, {8, 5, 2}, {1, 4, 2}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 6, 7}, {8, 6, 6}, {1, 3, 7}, {3, 9, 5}}, {{0, 0, 0}, {0, 0, 0}}, 26}, {{{1, 2, 3}, {1, 4, 5}} -> {{2, 6, 1}, {2, 7, 4}, {6, 5, 7}, {1, 8, 3}}, {{0, 0, 0}, {0, 0, 0}}, 10}, {{{1, 2, 3}, {1, 4, 5}} -> {{6, 2, 2}, {2, 3, 3}, {2, 5, 7}, {7, 3, 8}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {4, 3, 5}} -> {{6, 6, 2}, {3, 2, 7}, {1, 3, 3}, { 8, 4, 1}}, {{0, 0, 0}, {0, 0, 0}}, 5}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 7, 3}, {7, 8, 9}, {3, 10, 4}, {2, 9, 11}}, {{0, 0, 0}, {0, 0, 0}}, 48}, {{{1, 2, 1}, {2, 1, 3}} -> {{4, 5, 4}, {5, 4, 6}, {1, 4, 3}, {7, 2, 3}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {2, 4, 5}} -> {{3, 6, 3}, {6, 7, 1}, {1, 8, 9}, {8, 7, 5}}, {{0, 0, 0}, {0, 0, 0}}, 5}, {{{1, 2, 3}, {3, 4, 5}} -> {{2, 6, 7}, {6, 2, 7}, {7, 8, 4}, { 5, 1, 9}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {4, 2, 5}} -> {{1, 4, 5}, {6, 4, 7}, {2, 8, 4}, {9, 3, 8}}, {{0, 0, 0}, {0, 0, 0}}, 11}, {{{1, 2, 3}, {3, 4, 5}} -> {{2, 6, 5}, {2, 5, 3}, {4, 6, 3}, {7, 1, 3}}, {{0, 0, 0}, {0, 0, 0}}, 4}, {{{1, 2, 3}, {3, 4, 5}} -> {{2, 5, 2}, {5, 4, 6}, {1, 6, 7}, { 8, 7, 9}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {1, 4, 5}} -> {{6, 2, 7}, {4, 7, 2}, {6, 5, 8}, {1, 3, 5}}, {{0, 0, 0}, {0, 0, 0}}, 8}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 4, 7}, {7, 8, 9}, {5, 4, 10}, {3, 10, 8}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {4, 5, 3}} -> {{6, 3, 7}, {6, 3, 1}, {7, 8, 3}, {9, 8, 1}}, {{0, 0, 0}, {0, 0, 0}}, 34}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 3, 3}, {2, 6, 3}, {2, 4, 2}, {4, 1, 3}}, {{0, 0, 0}, {0, 0, 0}}, 4}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 6, 5}, {7, 5, 4}, {8, 5, 1}, { 9, 7, 1}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {4, 3, 5}} -> {{6, 1, 1}, {6, 4, 7}, {1, 2, 7}, { 2, 3, 8}}, {{0, 0, 0}, {0, 0, 0}}, 66}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 2, 7}, {2, 4, 6}, {8, 2, 9}, {3, 8, 5}}, {{0, 0, 0}, {0, 0, 0}}, 17}, {{{1, 2, 3}, {4, 3, 5}} -> {{3, 3, 6}, {6, 4, 7}, {8, 4, 3}, {4, 8, 1}}, {{0, 0, 0}, {0, 0, 0}}, 5}, {{{1, 2, 2}, {3, 2, 4}} -> {{5, 6, 6}, {7, 6, 8}, {8, 4, 6}, { 9, 6, 1}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 2}, {1, 3, 4}} -> {{5, 4, 4}, {5, 6, 7}, {8, 7, 4}, {3, 2, 9}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 5, 3}, {5, 7, 1}, {7, 8, 6}, {9, 2, 7}}, {{0, 0, 0}, {0, 0, 0}}, 12}, {{{1, 2, 3}, {3, 2, 4}} -> {{3, 3, 2}, {2, 5, 6}, {7, 8, 5}, {9, 4, 4}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {1, 4, 5}} -> {{1, 6, 2}, {6, 7, 2}, {6, 8, 3}, {9, 4, 8}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 6, 7}, {1, 6, 3}, {3, 7, 8}, {5, 4, 1}}, {{0, 0, 0}, {0, 0, 0}}, 5}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 5, 6}, {7, 5, 8}, {3, 9, 6}, { 3, 10, 2}}, {{0, 0, 0}, {0, 0, 0}}, 29}, {{{1, 2, 3}, {4, 3, 5}} -> {{6, 6, 7}, {8, 9, 6}, {4, 10, 8}, {2, 10, 9}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 7, 8}, {7, 8, 9}, {10, 3, 6}, {1, 4, 8}}, {{0, 0, 0}, {0, 0, 0}}, 11}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 4, 4}, {4, 6, 2}, {4, 1, 3}, {7, 8, 3}}, {{0, 0, 0}, {0, 0, 0}}, 66}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 7, 8}, {6, 9, 4}, {2, 7, 4}, {10, 2, 8}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {3, 4, 5}} -> {{3, 6, 4}, {4, 7, 3}, {8, 3, 9}, {10, 9, 3}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {2, 4, 5}} -> {{1, 1, 5}, {2, 1, 6}, {7, 8, 6}, { 7, 9, 4}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {4, 3, 5}} -> {{3, 1, 6}, {1, 7, 8}, {9, 7, 5}, {4, 10, 7}}, {{0, 0, 0}, {0, 0, 0}}, 12}, {{{1, 2, 3}, {1, 4, 5}} -> {{6, 6, 7}, {7, 8, 8}, {2, 6, 5}, {4, 3, 2}}, {{0, 0, 0}, {0, 0, 0}}, 19}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 4, 7}, {4, 8, 3}, {9, 3, 7}, {1, 10, 3}}, {{0, 0, 0}, {0, 0, 0}}, 25}, {{{1, 2, 3}, {4, 5, 3}} -> {{2, 6, 7}, {6, 4, 2}, {4, 1, 8}, {9, 10, 2}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {2, 4, 5}} -> {{1, 6, 4}, {6, 2, 7}, {8, 9, 1}, {10, 11, 8}}, {{0, 0, 0}, {0, 0, 0}}, 19}, {{{1, 2, 2}, {2, 3, 4}} -> {{2, 2, 4}, {4, 3, 5}, {6, 5, 5}, {7, 8, 3}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {4, 3, 5}} -> {{3, 6, 7}, {3, 8, 1}, {9, 3, 2}, {10, 11, 7}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {4, 3, 5}} -> {{2, 1, 1}, {6, 2, 1}, {5, 7, 2}, {4, 7, 5}}, {{0, 0, 0}, {0, 0, 0}}, 5}, {{{1, 2, 3}, {4, 3, 5}} -> {{1, 6, 6}, {6, 7, 3}, {4, 3, 7}, { 4, 2, 8}}, {{0, 0, 0}, {0, 0, 0}}, 17}, {{{1, 2, 3}, {4, 3, 5}} -> {{6, 6, 7}, {8, 7, 1}, {2, 5, 7}, {9, 10, 7}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {4, 3, 5}} -> {{5, 5, 6}, {7, 8, 5}, {7, 8, 3}, {5, 9, 1}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {4, 5, 3}} -> {{3, 6, 3}, {3, 2, 7}, {8, 3, 9}, {10, 11, 9}}, {{0, 0, 0}, {0, 0, 0}}, 65}, {{{1, 2, 3}, {4, 2, 5}} -> {{6, 7, 7}, {7, 1, 2}, {8, 6, 1}, {9, 6, 2}}, {{0, 0, 0}, {0, 0, 0}}, 66}, {{{1, 1, 2}, {2, 3, 4}} -> {{5, 5, 6}, {6, 7, 4}, {7, 3, 2}, {8, 5, 2}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {4, 3, 5}} -> {{1, 6, 6}, {1, 7, 6}, {7, 3, 7}, {6, 5, 8}}, {{0, 0, 0}, {0, 0, 0}}, 66}, {{{1, 2, 3}, {4, 5, 3}} -> {{1, 1, 3}, {6, 5, 7}, {7, 5, 2}, {4, 6, 8}}, {{0, 0, 0}, {0, 0, 0}}, 17}, {{{1, 2, 3}, {4, 5, 3}} -> {{1, 4, 3}, {1, 6, 3}, {7, 4, 8}, {8, 9, 10}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {4, 3, 5}} -> {{2, 2, 6}, {2, 6, 7}, {2, 4, 4}, {5, 8, 7}}, {{0, 0, 0}, {0, 0, 0}}, 17}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 7, 8}, {7, 2, 9}, {8, 10, 11}, {1, 9, 12}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {3, 4, 5}} -> {{4, 6, 4}, {3, 6, 7}, {3, 8, 6}, {9, 1, 10}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {3, 4, 5}} -> {{6, 6, 7}, {4, 6, 6}, {8, 9, 6}, {8, 1, 4}}, {{0, 0, 0}, {0, 0, 0}}, 9}, {{{1, 2, 3}, {4, 5, 3}} -> {{2, 4, 5}, {6, 4, 5}, {7, 4, 8}, { 9, 1, 7}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {2, 4, 5}} -> {{2, 2, 3}, {5, 2, 6}, {6, 6, 4}, {7, 8, 5}}, {{0, 0, 0}, {0, 0, 0}}, 66}, {{{1, 2, 3}, {1, 4, 5}} -> {{6, 7, 6}, {6, 8, 1}, {8, 1, 2}, {9, 4, 2}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 1, 2}, {2, 3, 4}} -> {{5, 5, 4}, {6, 5, 5}, {1, 7, 6}, {7, 1, 3}}, {{0, 0, 0}, {0, 0, 0}}, 79}, {{{1, 2, 3}, {4, 2, 5}} -> {{3, 6, 1}, {1, 7, 8}, {8, 6, 9}, {7, 10, 4}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {2, 4, 5}} -> {{6, 7, 1}, {1, 8, 6}, {9, 3, 6}, {9, 10, 8}}, {{0, 0, 0}, {0, 0, 0}}, 49}, {{{1, 2, 3}, {4, 2, 5}} -> {{6, 3, 4}, {6, 7, 4}, {8, 9, 6}, {10, 9, 4}}, {{0, 0, 0}, {0, 0, 0}}, 65}, {{{1, 2, 3}, {3, 4, 5}} -> {{2, 6, 2}, {2, 7, 5}, {8, 6, 1}, {9, 10, 2}}, {{0, 0, 0}, {0, 0, 0}}, 49}}; GraphicsGrid[ Partition[ ParallelMap[ ResourceFunction["WolframModelPlot"][ ResourceFunction["WolframModel"][#[[1]], #[[2]], #[[3]] - 1, "FinalState"]] &, rulesample], 14], ImageSize -> Full]

The fraction of complex behavior appears to be no higher than for 23 33 rules, and no obvious major new phenomena are seen. Much like in systems such as cellular automata (and as suggested by the Principle of Computational Equivalence [1:12]), above some low threshold, adding complexity to the rules does not appear to add complexity to the typical behavior produced.

GraphicsGrid[ Partition[ ParallelMap[ Labeled[ResourceFunction["WolframModelPlot"][ ResourceFunction["WolframModel"][#[[1]], #[[2]], #[[3]], "FinalState"], ImageSize -> 1.1 {150, 100}], RulePlot[ResourceFunction["WolframModel"][#[[1]]], "RulePartsAspectRatio" -> 1, ImageSize -> Tiny]] &, {{{{1, 2, 3}, {2, 4, 5}} -> {{6, 7, 2}, {5, 7, 8}, {4, 2, 8}, {9, 3, 5}}, {{0, 0, 0}, {0, 0, 0}}, 18}, {{{1, 2, 3}, {1, 4, 5}} -> {{3, 2, 6}, {2, 5, 6}, {7, 4, 3}, {1, 8, 7}}, {{0, 0, 0}, {0, 0, 0}}, 19}, {{{1, 1, 2}, {2, 3, 4}} -> {{5, 5, 4}, {6, 5, 5}, {1, 7, 6}, {7, 1, 3}}, {{0, 0, 0}, {0, 0, 0}}, 79}, {{{1, 2, 3}, {3, 4, 5}} -> {{4, 6, 1}, {4, 6, 1}, {7, 6, 8}, {8, 5, 2}}, {{0, 0, 0}, {0, 0, 0}}, 13}}], UpTo[4]], ImageSize -> Full]

The trend continues with 33 43 rules, with one notable feature here being an increased propensity for rules to yield results that become disconnected, though only after many steps. The general difficulty of predicting long-term behavior is illustrated for example by the evolution of this 33 53 rule, sampled every 10 steps:

RulePlot[ResourceFunction[ "WolframModel"][{{1, 2, 3}, {2, 4, 5}, {6, 7, 2}} -> {{4, 4, 4}, {5, 4, 8}, {8, 6, 3}, {9, 10, 6}, {7, 1, 10}}]]
ResourceFunction["WolframModelPlot"] /@ Take[ResourceFunction[ "WolframModel"][{{1, 2, 3}, {2, 4, 5}, {6, 7, 2}} -> {{4, 4, 4}, {5, 4, 8}, {8, 6, 3}, {9, 10, 6}, {7, 1, 10}}, {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}, 60, "StatesList"], 10 ;; -1 ;; 10]
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