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We've seen how the fields are governed by Blocks 0 and 7 and are delineated by the cells running diagonally from the seed. The diagonals and the cells they bound make up the ground of the pattern. Every cell in the ground is a descendant of the seed; a path can be traced back, time step by time step, to a seed cell. Even when the diagonal and field may be the same state, the diagonal is part of the pattern ground, not the field.

The patterns of diagonals are directly influenced by the field, and this impacts the possible structures of ground patterns. The charts below show the the blocks invoked for the three field conditions. The numbers reference the possible blocks that are invoked in the field or can be invoked in the diagonal cells. The circumflex (^) indicates either multiple possible blocks can be invoked or, as in this case, the block / state is unimportant.

Block 0 Passive
Seed  
Step 1 0 0 0 1 2 4 0 0 0
Step 2 0 0 0/1 ^ ^ ^ 0/4 0 0
Step 3 0 0/1 ^ ^ ^ ^ ^ 0/4 0
Step 4 0/1 ^ ^ ^ ^ ^ ^ ^ 0/4
Block 0 Active / Block 7 Passive
 
0 0 0 1 2 4 0 0 0
7 7 6/7 ^ ^ ^ 3/7 7 7
0 0/1 ^ ^ ^ ^ ^ 0/4 0
6/7 ^ ^ ^ ^ ^ ^ ^ 3/7
Block 0 Active / Block 7 Active
 
0 0 0 1 2 4 0 0 0
7 7 6/7 ^ ^ ^ 3/7 7 7
7 6/7 ^ ^ ^ ^ ^ 3/7 7
6/7 ^ ^ ^ ^ ^ ^ ^ 3/7

The other key feature of ground patterns is a center column in line with the seed cell(s). Where there is a single seed, the column is governed by the state generated by Block 2. While Block 2 must be active to for an active center column, it is not a sufficient condition. Where the seed is made up of multiple cells, the state of the column is determined by the arrangement of the seed cells and the blocks invoked.

Block 0 Passive - Passive Field. The ground patterns are clearly categorized depending on diagonal and center column conditions.

Diagonals

  • If Block 1 or 4 generate an active state, then the diagonal cells will continue to be governed by Block 1 or 4 and the diagonal wil be in the active state
  • If Block 1 or 4 generate a passive state, then after the first time step, those diagonal cells will be governed by Block 0 and continue in the passive state
  • The diagonals are either all active or all passive cells; the diagonals can not be a mix of active and passive cells

There are three distinct active diagonals: single, stepped, and double-line. The chart below tracks the left diagonal and assumes Block 4 generates passive - the same mechanics work on the right using Blocks 4, 2, and 6.

  • Single: Block 1 generates active and Block 2 generates passive as noted
  • Stepped: Both Blocks 1 and 2 generate active. Block 3 passive is noted
  • Double-line: Blocks 1, 2, and 3 generate active. Note that Block 2 is only invoked at the first time step
Single Diagonal
Seed
Step 1 0 0 0 1 2 4 0 0 0
Step 2 0 0 1 2 4 0 0 0 0
Step 3 0 1 2 4 0 0 0 0 0
Step 4 1 2 4 0 0 0 0 0 0
Stepped Diagonal
0 0 0 1 2 4 0 0 0
0 0 1 3 6 4 0 0 0
0 1 2 4/5 0/2 0/4 0 0 0
1 3 6 ^ ^ 0/4 0 0 0
Double-line Diagonal
0 0 0 1 2 4 0 0 0
0 0 1 3 6 4 0 0 0
0 1 3 6/7 4/6 0/4 0 0 0
1 3 6/7 ^ ^ 0/4 0 0 0

When looking at all the patterns, there's an equal number of active and passive diagonals. There is also an equal number of left and right diagonals. Further, for each type of diagonal, there is an equal number of left and right but there are more single diagonals than there are stepped or double-line.

Center column: for simplicity, the state of column when there's a single seed.

  • Passive: Block 2 invokes a passive state so all cells are in the passive state. Occurs when diagonals are either passive or single active
  • Active / Passive: Block 2 invokes an active state. The first time step cell is active, then the cells are passive. This only occurs when a diagonal is either stepped or double-line, though not all stepped or double-line diagonals generate active / passive center columns
  • Mixed: Block 2 invokes an active state. The center column is a mix of active and passive cells. This occurs only when both diagonals are active
  • Active: Block 2 invokes an active state. All cells in the center column are active. This occurs when either both diagonals are passive or one diagonal is either stepped or double-line

Pattern sturctures

  • Passive diagonal: the pattern up to the center column will be passive
  • One active diagonal
    • Center column passive or active/passive: the rest of the pattern will be passive
    • Center column active: the pattern between the diagonal and column will be mixed
  • Both diagonals active: the pattern will be mixed

Ground Patterns: Block 0 Passive
Ground Definitions: Block 0 Passive Sort Tables

 

Block 0 Active / Block 7 Passive - Striped Field. The diagonal and center column structures are more complicated. Rather than clear active or passive diagonals, the diagonals in this group alternate between active and passive cells.

  • The passive diagonals, like the Block 0 Passive group diagonals, follow the same pattern as the field, the difference being this is a striped pattern, not a solid passive pattern.
  • When the diagonal is active, the stripes are shifted on step from the field stripes.
  • The center columns are more varied and do not provide such strong guidance and they do with the Block 0 Passive group patterns.
  • This group of patterns is the only one which exhibits a steep diagonal

Diagonals: the diagonals are characterized by the starting condtion (shown below for the left diagonal)

  • Passive Passive Active (PPA): Step 1 and 2 are passive, Step 3 is active
  • Passive Active Passive (PAP): Step 1 and 3 are passive, Step 2 is active
  • Active Passive Active (APA): Step 1 and 3 are active, Step 2 is passive
  • PPA and APA result in passive diagonals; only PAP results in active diagonals

The starting conditions are not equally distributed across the patterns. There are more passive diagonals than there are active ones.

  • PPA: 32 instances, producing passive diagonals
  • PAP: 32 instances, producing active diagonals
  • APA: 64 instances, producing passive diagonals
Passive Passive Active: Block 2 Passive
Seed  
Step 1 0 0 0 1 2 4 0 0 0
Step 2 7 7 6 4 0 1 3 7 7
Step 3 0 0 0 1 2 4 0/2 4/0 0
Step 4 7 7 6 4 0 0/1 ^ ^ 3/7
Passive Active Passive: Block 2 Passive
 
0 0 0 1 2 4 0 0 0
7 7 6 4 ^ 1 3 7 7
0 1 2 4 ^ ^ ^ 0/4 0
6 4 ^ ^ ^ ^ ^ ^ 3/7
Active Passive Active: Block 2 Passive
 
0 0 0 1 2 4 0 0 0
7 7 7 6 4 1 3 7 7
0 0 0/6 ^ ^ ^ ^ 0/4 0
7 6/7 ^ ^ ^ ^ ^ ^ 3/7

Passive Passive Active: Block 2 Active
Seed  
Step 1 0 0 0 1 2 4 0 0 0
Step 2 7 7 6 5 2 5 3 7 7
Step 3 0 0 0/1 1/3 2/7 ^ ^ 4/0 0
Step 4 7 6/7 4/6 ^ ^ ^ ^ ^ 3/7
Passive Active Passive: Block 2 Active
 
0 0 0 1 2 4 0 0 0
7 7 6 5 2 5 3 7 7
0 1 2/3 5/7 2/7 4/6 ^ 0/4 0
6 4 ^ ^ ^ ^ ^ ^ 3/7
Active Passive Active: Block 2 Active
 
0 0 0 1 2 4 0 0 0
7 7 7 7 6 5 3 7 7
0 0 0 1/7 ^ ^ ^ 0/4 0
7 7 6/7 ^ ^ ^ ^ ^ 3/7

 

Steep diagonals: the pattern is generated by a cell remaining active for two time steps, then the active cell shifting on space to either left or right

  • All three starting conditions can procude a steep diagonal
  • The two patterns with both left and right steep diagonals are generated by the APA condition

Center column: the majority of center columns are mixed, either created by the stripes or by the pattern. In some caes, the center column is solid active; there are no passive center columns, however there are a number of cases where the active column is flanked by a passive column.

Ground Patterns: Block 0 Active / Block 7 Passive
Ground Definitions: Block 0 Active Sort Tables

 

Block 0 Active / Block 7 Active - Active Field. The diagonal and center column structures are also complicated.

Diagonals: there are three diagonal conditions.

  • Active: all the cells in the diagonal are active
  • Passive Active: the cell in Row 1 is passive, but the rest of the diagonal cells are active
  • Passive: all the cells in the diagonal are passive
Active & Passive Diagonals
Seed
Step 1 0 0 0 1 2 4 0 0 0
Step 2 7 7 7 6/7 4/6 1/5 3 7 7
Step 3 7 7 6/7 ^ ^ ^ 1/5 3 7
Step 4 7 6/7 ^ ^ ^ ^ ^ 1/5 3
Passive Active Diagonal
0 0 0 1 2 4 0 0 0
7 7 6 4 1/3 3/7 7 7 7
7 7 7 6/7 4/5 ^ 3/7 7 7
7 7 7 6/7 ^ ^ ^ 3/7 7

 

Center column: the center column patterns are consistent with the diagonal patters

  • All active cells
  • All passive cells
  • Mixed active and passive
  • First one or two cells passive, the remainder active

Ground Patterns: Block 0 Active / Block 7 Active
Ground Definitions: Block 0 Active Sort Tables