e c a f i e l d s & g r o u n d s
Overall Structure
The basic structure of a time-phase pattern is based on the relationship of any cell in the pattern to the initial seed cell or cells. There are descendant cells which can trace a lineage back to the seed and those that can't. The illustration below of a single seed three-cell block structure shows the difference.
Start with Row 1. Moving in from the left side, blocks [a-b-c] through [e-f-g] do not contain the seed. The first block to contain the seed is [f-g-h] which then determines the state of cell 7.
The same occurs on the other side. From the right blocks [m-n-o] through [i-j-k] do not contain the seed. The first block from the right to contain the seed is [h-i-j] which determines the state of cell 9. In the middle, block [g-h-i] also contains the seed and determines the state of cell 8. Cells 7, 8, and 9 have a direct lineage back to the seed.
| Seed | a | b | c | d | e | f | g | h | i | j | k | l | m | n | o |
| Row 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| Row 2 | A | B | C | D | E | F | G | H | I | J | K | L | M | M | O |
| Row 3 |
The same analysis happens in Row 2 with a slight modification: we are looking for blocks which contain cells which have a lineage to the seed. As shown in the illustration, on the left, block [1-2-3] through [4-5-6] do not contain a lineage cell, but block [5-6-7], which determines the state of cell F, does. On the right, block [9-10-11], which determines the state of cell j, contains a lineage cell while the rest do not. In the middle, blocks [6-7-8, [7-8-9], and [8-9-10] also contain lineage cells.
Whether these lineage cells are active or passive is not the concern. The point is there is a defined area within each pattern where cells either have a lineage to the seed or don’t. In an automaton with a 3-cell block, these areas are clearly delineated by a single step diagonal emanating out from the seed. If it is a single cell seed, as shown above, then the diagonals form a triangle with the seed as the peak. It a multiple cell seed, the diagonals originate at the outer active cells of the seed. The diagonals and the cells between them make up the ground. The cells outside the diagonals make up the field.
The division between ground and field exists with other neighborhood conditions. If the blocks are made up of 5 cells, the diagonal shifts over two cells and down one. On the left, the first block to include the cell is [d-e-f-g-h] and this determines the state of cell 6. On the right it’s [h-i-j-k-l] which determines the state of cell 10.
| Seed | a | b | c | d | e | f | g | h | i | j | k | l | m | n | o |
| Row 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| Row 2 | A | B | C | D | E | F | G | H | I | J | K | L | M | M | O |
| Row 3 |
This method of analysis works for all automata though may be much more convoluted based on the design of the seed wiring diagram and the definition of blocks.
Fields
The pattern of the field cells are determine by Block 0 and Block 7. Passive cells spread out left and right of the seed cells all produce all Block 0s. If Block 0 is defined to generate a passive cell, then the field in the next row will be filled with passive cells. In this case, it doesn’t matter if Block 7 is active or passive; all the neighborhoods are Block 0 and Block 7 is never invoked in the field.
If Block 0 is active, then Row 1 field cells are at active and the resulting neighborhoods will all be Block 7. If Block 7 is passive, then Row 2 will be all passive Block 0 neighborhoods which will generate active cells in Row 3, and so on, leading to a striped field. If Block 7 is active, then Row 2 will be all active Block 7 neighborhoods which will generate active cells in Row 3, and so on, leading to an active field.
 Rule 022 Block 0 Passive Block 7 Passive |
 Rule 126 Block 0 Passive Block 7 Active |
 Rule 073 Block 0 Active Block 7 Passive |
 Rule 147 Block 0 Active Block 7 Active |
The decimal numbers of all patterns where Block 0 is passive, or is assigned the digit “0”, are even. When Block 0 is active /1, the numbers are odd. When Block 7 is passive / 0, the decimal numbers are less than 128; when Block 7 is active / 1, the decimal numbers are greater than 128.
To summarize:
| Block 0 | Block 7 | Field | Numbering |
| Passive | Passive | Passive | Even < 127 |
| Passive | Active | Passive | Even > 127 |
| Active | Passive | Striped | Odd < 128 |
| Active | Active | Active | Odd > 128 |
Rule Patterns: Field Groups - Active Passive
Rule Patterns: Field Groups - Color
Chart of Rule Blocks - All
Grounds
In the section above on the overall structure, the state of the diagonal cells was not important. When looking at the ground, the states of the diagonal cells is an essential part of the pattern generated.
The first observation is that one or both of the diagonals may be in the same state as the field cells leading to a bleeding of the field pattern into the ground. In the extreme, the diagonals and ground patterns are the same as the field:
 Rule 000 |
 Rule 023 |
 Rule 255 |
I went through and tried to find a set of coherent groupings of patterns within each field group – a noble effort that really didn’t produce much useful information. Here's a link to that rabbit hole. In the end it became clear that a simple change. such as a different seed. would require a whole new round of pattern analysis and that it is better to look at the mechanisms making the patterns for insight into the patterns themselves.