The Rules Groups show that many of the patterns within a group which are run on a single seed are the same. In some groups, like Group 00, the patterns are all the same. In others, like Group 05, there are two patterns. Groups 01 and 04 have the same patterns which are also a mirror of each other.
And then there’s Rules 060 and 102 which are singlets but mirror each other. This immediately prompted the question: are there any patterns that are unique and not mirrors of another pattern?
I went through the rules group maps for the range of seeds (1-, 3-, 5-, 7-, 9-, 11-, 13-, and 15 seed plus m1-seed) to identify all the patterns that were either the same or mirrors of another. (Links to the maps below). I came up with:
| 1-seed | x | 27 uniques |
| 3-seed | xx | 12 uniques |
| 5-seed | xox | 40 uniques |
| 7-seed | xxx | 26 uniques |
| 9-seed | xoox | 30 uniques |
| 11-seed | xoxx | 182 uniques |
| 13-seed | xxox | 180 uniques |
| 15-seed | xxxx | 16 uniques |
| m1-seed | xoxxxoxoooxx | all unique* |
* The patterns for the m1-seed are all unique. If you ignore the first couple of rows, which some researchers suggest, then there are some couplets and triplets indicated on the map - link below.
The first observation is asymmetric seeds generate a lot more uniques that symmetric seeds. Most of these uniques are patterns which would otherwise be grouped as mirrors around the vertical axis. When that vertical axis is not symmetrical, the patterns do not mirror.
I created a chart listing the uniques for each of the seed conditions (except for the 11- and 13-seed since there are so many uniques – see link below). Of all the patterns, only ten are unique across all seed conditions, sorted by rule group:
| Rule 022 | Rule Group 07 |
| Rule 054 | RG 07 |
| Rule 150 | RG 07 |
| Rule 182 | RG 07 |
| Rule 073 | Rule Group 08 |
| Rule 105 | RG 08 |
| Rule 077 | Rule Group 10 |
| Rule 109 | RG 10 |
| Rule 147 | Rule Group 13 |
| Rule 179 | RG 13 |
As expected, these 10 patterns are symmetrical around the vertical axis – yet there are plenty of other rules which generate such symmetry. These 10 seem to have a “super symmetry” which makes them stand on their own. The path to understand what this “super symmetry” might be and the implications it may have for understanding the mechanisms starts with a deeper dive into symmetry itself within the eca.
Maps and charts: