The Use of Rules
This pattern appears eight times in Results Group 05. I found this was somewhat surprising, given the variety of rules and rulesets. What became more surprising, however, is that the rules used to generate the pattern are the same for each of the eight rulesets.
The rulesets that generate the nested triangle pattern are Rulesets 018, 026, 082, 090, 146, 154, 210, and 218. When I analyzed each ruleset, it became clear what's happening: there are rules in the ruleset that are meant to turn a target cell active but which are not used in the generation of the pattern. In the table below, all the rules are listed with an indication of status. The number "1" indicates rules which are meant to turn the target cell active and "0" indicates the rules meant to be passive. Black numbers indicates the rules that are used to generate the pattern and red desingates those active rules which are not used or ignored.
| value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| rule | 111 | 110 | 101 | 100 | 011 | 010 | 001 | 000 |
| 018 | 0 | 1 | 0 | 1 | 0 | |||
| 026 | 0 | 1 | 1 | 0 | 1 | 0 | ||
| 082 | 1 | 0 | 1 | 0 | 1 | 0 | ||
| 090 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | |
| 146 | 1 | 0 | 1 | 0 | 1 | 0 | ||
| 154 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | |
| 210 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | |
| 218 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 |
The table shows that the same five rules are used by all eight rulesets in the process of generating the pattern. As the rulesets get higher in number, more and more rules are specified, but these additional rules are ignored. Finally, at Ruleset 218, all the rules are called into play -- three active rules and five passive rules -- but only the original five rules are used to generate the pattern.
In addition to the ignored active rules, there are two passive outcomes. One is where the passive rule is part of the generation pattern and the second is a passive rule no used. This is indicated in the chart above by the blank cells.
In summary, then, there are:
- 256 rulesets, each made up of
- 8 rules, which can be set in
- 2 states -- target active or target passive -- with
- 4 possible outcomes
- unused: a passive rule that is not part of the pattern generation
- passive: rule determines the "0" condition used in the pattern
- active: rule determines the "1" condition used in the generation of a pattern
- ignored: rule invokes the "1" condition, but the rule is not used in the pattern
The researchers focused on active/passive patterns generally ascribe the pattern illustrated at the top of the page as being gnerated by Ruleset 90. While Ruleset 90 does generate the pattern, the analysis of its results group and rules ignored indicates that Ruleset 018 is the fundamental ruleset to generate this pattern. Part of their reasoning for using the Ruleset 090 label is their analysis of the effects of multi-seed conditions. This raises the question of whether patterns in multi-seed conditions for Ruleset 018 and Ruleset 090 generate significant differences that justify using Ruleset 090 as the canonical ruleset for this pattern. Only a detailed analysis of the mechanics and pattern probablities can provide the answer.
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