Amos Satterlee

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

 

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