Result Group Sequences
While working on the pages grouping the patterns by result, two additional observations came to the fore in terms of sequences of numbers based on the realtionship of rulesets and results groups. To save space, the tables below first 128 patterns generated, although the observations hold for the full complement of 256 rulesets.
Ruleset to Ruleset
The first sequence is the difference of ruleset number between the rulesets within a results group. The difference between the first and second ruleset, between the third and fourth, and so on, is 8. The difference between the first and third, between the second and fourth, and so on, is 32.
Converting these differences to binary values, the difference between the first and second is 1000. This value correlates with the fourth rule, or 3:011 . The difference between the first and third is 100000 which correlates to the sixth rule, or 6:101 . These will probably become significant when analyzing how these rules affect the patterns generated by rulesets within a results group.
| Result | Ruleset | |||||||
| 00 | 000 | 008 | 032 | 040 | 064 | 072 | 096 | 104 |
| 08 | 001 | 009 | 033 | 041 | 065 | 073 | 097 | 105 |
| 01 | 016 | 024 | 048 | 056 | 080 | 088 | 112 | 120 |
| 09 | 017 | 025 | 049 | 057 | 081 | 089 | 113 | 121 |
| 02 | 004 | 012 | 036 | 044 | 068 | 076 | 100 | 108 |
| 10 | 005 | 013 | 037 | 045 | 069 | 077 | 101 | 109 |
| 03 | 020 | 028 | 052 | 060 | 084 | 092 | 116 | 124 |
| 11 | 021 | 029 | 053 | 061 | 085 | 093 | 117 | 125 |
| 04 | 002 | 010 | 034 | 042 | 066 | 074 | 098 | 106 |
| 12 | 003 | 011 | 035 | 043 | 067 | 075 | 099 | 107 |
| 05 | 018 | 026 | 050 | 058 | 082 | 090 | 114 | 122 |
| 13 | 019 | 027 | 051 | 059 | 083 | 091 | 115 | 123 |
| 06 | 006 | 014 | 038 | 046 | 070 | 078 | 102 | 110 |
| 14 | 007 | 015 | 039 | 047 | 071 | 079 | 103 | 111 |
| 07 | 022 | 030 | 054 | 062 | 086 | 094 | 118 | 126 |
| 15 | 023 | 031 | 055 | 063 | 087 | 095 | 119 | 127 |
Results Group to Ruleset
The relationship between ruleset number and results group number generates its own sequences. The figure below lists the rulesets in ascending order and charts the difference between the associated results group number. Ruleset 000 is in Results Group 00 and Ruleset 001 is in Results Groups 08. The difference is 8. The full sequence:
8, -4, 8, -10, 8, -4, 8, -14, 8, -4, 8, -10, 8, -4, 8, -13, 8, -4, 8, -10, 8, -4, 8, -14, 8, -4, 8, -10, 8, -4, 8, -15, ...
I'm not sure if this sequence has any meaning; and whether extending the sequence beyond Ruleset 255 makes any sense. Obviously the rules of the games set a boundary, but the sequence goes on. Does the ruleset boundary limit the sequence, or can the extension of the sequence become an additional realm to elementary automata?
Below is the chart of differences, up to Ruleset 127:
| Results | Ruleset | Diff | Results | Ruleset | Diff | Results | Ruleset | Diff | Results | Ruleset | Diff |
| 00 | 000 | 00 | 032 | -15 | 00 | 064 | -15 | 00 | 096 | -15 | |
| 08 | 001 | 08 | 08 | 033 | 08 | 08 | 065 | 08 | 08 | 097 | 08 |
| 04 | 002 | -4 | 04 | 034 | -4 | 04 | 066 | -4 | 04 | 098 | -4 |
| 12 | 003 | 08 | 12 | 035 | 08 | 12 | 067 | 08 | 12 | 099 | 08 |
| 02 | 004 | -10 | 02 | 036 | -10 | 02 | 068 | -10 | 02 | 100 | -10 |
| 10 | 005 | 08 | 10 | 037 | 08 | 10 | 069 | 08 | 10 | 101 | 08 |
| 06 | 006 | -4 | 06 | 038 | -4 | 06 | 070 | -4 | 06 | 102 | -4 |
| 14 | 007 | 08 | 14 | 039 | 08 | 14 | 071 | 08 | 14 | 103 | 08 |
| 00 | 008 | -14 | 00 | 040 | -14 | 00 | 072 | -14 | 00 | 104 | -14 |
| 08 | 009 | 08 | 08 | 041 | 08 | 08 | 073 | 08 | 08 | 105 | 08 |
| 04 | 010 | -4 | 04 | 042 | -4 | 04 | 074 | -4 | 04 | 106 | -4 |
| 12 | 011 | 08 | 12 | 043 | 08 | 12 | 075 | 08 | 12 | 107 | 08 |
| 02 | 012 | -10 | 02 | 044 | -10 | 02 | 076 | -10 | 02 | 108 | -10 |
| 10 | 013 | 08 | 10 | 045 | 08 | 10 | 077 | 08 | 10 | 109 | 08 |
| 06 | 014 | -4 | 06 | 046 | -4 | 06 | 078 | -4 | 06 | 110 | -4 |
| 14 | 015 | 08 | 14 | 047 | 08 | 14 | 079 | 08 | 14 | 111 | 08 |
| 01 | 016 | -13 | 01 | 048 | -13 | 01 | 080 | -13 | 01 | 112 | -13 |
| 09 | 017 | 08 | 09 | 049 | 08 | 09 | 081 | 08 | 09 | 113 | 08 |
| 05 | 018 | -4 | 05 | 050 | -4 | 05 | 082 | -4 | 05 | 114 | -4 |
| 13 | 019 | 08 | 13 | 051 | 08 | 13 | 083 | 08 | 13 | 115 | 08 |
| 03 | 020 | -10 | 03 | 052 | -10 | 03 | 084 | -10 | 03 | 116 | -10 |
| 11 | 021 | 08 | 11 | 053 | 08 | 11 | 085 | 08 | 11 | 117 | 08 |
| 07 | 022 | -4 | 07 | 054 | -4 | 07 | 086 | -4 | 07 | 118 | -4 |
| 15 | 023 | 08 | 15 | 055 | 08 | 15 | 087 | 08 | 15 | 119 | 08 |
| 01 | 024 | -14 | 01 | 056 | -14 | 01 | 088 | -14 | 01 | 120 | -14 |
| 09 | 025 | 08 | 09 | 057 | 08 | 09 | 089 | 08 | 09 | 121 | 08 |
| 05 | 026 | -4 | 05 | 058 | -4 | 05 | 090 | -4 | 05 | 122 | -4 |
| 13 | 027 | 08 | 13 | 059 | 08 | 13 | 091 | 08 | 13 | 123 | 08 |
| 03 | 028 | -10 | 03 | 060 | -10 | 03 | 092 | -10 | 03 | 124 | -10 |
| 11 | 029 | 08 | 11 | 061 | 08 | 11 | 093 | 08 | 11 | 125 | 08 |
| 07 | 030 | -4 | 07 | 062 | -4 | 07 | 094 | -4 | 07 | 126 | -4 |
| 15 | 031 | 08 | 15 | 063 | 08 | 15 | 095 | 08 | 15 | 127 | 08 |
