Result 00: Basic Multi-Seed
Starting from the single-seed condition, all the rulesets in Result 00 produce a blank slate, even as more and more rules become active as the ruleset number goes higher (see the chart). The distinguishing feature of all the rulesets in Result 00 is that only rules which reference at least two active predecessor cells -- 7:111 , 6:110 , 5:101 , 3:011 -- or no active cells -- 0:000 -- are included.
As indicated elsewhere, only three rules are invoked in the first row of any pattern. These three rules reference only one active cell -- 1:001 , 2:010 , and 4:100 . None of these rules are included as active rules in the Result 00 rulesets. When starting from a single seed, there aren't any cases where any of the Result 00 rulesets can produce an active target cell.
From this we can conclude that patterns require a ruleset with at least one active rule which references only one active cell.
Let's next look at the seven rulesets with the rules which reference only single active cells:
- Wolf-002: single diagonal to the left
- Wolf-004: single vertical stripe
- Wolf-006: stepped diagonal to the left
- Wolf-016: single diagonal to the right
- Wolf-018: classic nested triangles
- Wolf-020: stepped diagonal to the right
- Wolf-022: nested ziggaurats
From this we can make two concluions. The rulesets with only one rule which references one active cell (Wolf-002, -004, and -016) produce patterns with monotonous repeat of a single cell. Two of the rulesets (Wolf-006 and -020) produce stepped line patterns while two (Wolf-018 and -022) produce nested patterns. We can extrapolate that random and complex patterns are only generated by rulesets with a combination of active rules which reference one active cell and multiple active cells. We will look at the impact of the rule which references three active cells ( 7:111 ) later on.
Rulesets and Seeds
As the the number of seed cells increase, the majority of patterns generated by Result 00 ruleseets either remain blank or display one cell in the first non-seed row. There are five cases when there are three seeds which display a cell in the second non-seed row and there are six cases where the pattern becomes a vertical stripe.
2 Seed Separated: The defining factor of this seed condition is whether Rule 5 ( 5:101 ) is active or not. Rule 5 matches the seed condition and is the only rule which can be invoked by this seed condition. However, when Rule 5 produces an active cell the result mimics the single-seed condition and the pattern dies out.
2 Seed Together: This condition produces a bit more diversity because two rules can potentially be invoked ( 6:110 and 3:011 ). When only one of the rules is active, the result is similar to the 2 Seed Separated condition. The ruleset produces a single active target cell and the pattern dies out.
When both rules are active, the result is a cascade of 2 active cells together which grows into a two-cell vertical stripe.
The sixteen rulesets in Result 00 break into four groups based on whether Rule 3 or Rule 6 is active:
| active | neither | rule 3 | rule 6 | both |
| 000 | 008 | 064 | 072 | |
| 032 | 040 | 096 | 104 | |
| 128 | 136 | 192 | 200 | |
| 160 | 168 | 224 | 232 |
Rules and Seeds
The sequences for each seed condition are:
Two seeds together: 0, 1, 0, 1, 1, S, 1, S, 0, 1, 0, 1, 1, S, 1, S
Two seeds separated: 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1
Three seeds: 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, S, 2, S
This chart compares the patterns displayed based on the four seed conditions with the use of each rule in generating the pattern.
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | ||||||
| value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | # of displayed rows | ||||
| rule | 111 | 110 | 101 | 100 | 011 | 010 | 001 | 000 | seed | 1 | 11 | 101 | 111 |
| 000 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 008 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | |
| 032 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | |
| 040 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | |
| 064 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | |
| 072 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | CS2 | 0 | 1 | |
| 096 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | |
| 104 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | CS2 | 1 | 2 | |
| 128 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | |
| 136 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 2 | |
| 160 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | |
| 168 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 2 | |
| 192 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 2 | |
| 200 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | CS2 | 0 | CS3 | |
| 224 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 2 | |
| 232 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | CS2 | 1 | CS3 | |
The first observation is that while three rules (7:111, 6:110, and 3:011) are required to form a column stripe (CS), a column can be formed with two rules (6:110 and 3:011). See the difference in rules used with the two-cell seed and the three-cell seed and the and the cases where a column stripe appears.
A second observation is the nature of the repetion within each seed sequence.
- The single-cell sequence has a monotonous repetition.
- In the two-cell, the sequence repeats after 8 rulesets. There is a minor repeat within the major repeat sequence where the seond set of four increments (0 -> 1, 1 -> CS).
- In the separated two-cell, the sequence repeats every four rulesets.
- In the three-cell, there is not a precise repetition. There is a general repetition after eight rulesets. There is an anomoly at Wolf-072: two cells become active, but only in the first row. Agreye from this, the two-cell and three-cell sequences follow a similar pattern, with the three-cell incremented one step up from the two-cell.
