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In the section on the overall structure of the patterns, the state of the diagonal cells was not important, only whether the cell was a descendant of the seed or not. When looking at the ground, the state of the diagonal cells is an essential part of patterns 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 state as the field:

Rule 000

Rule 023

Rule 255

The following sections look at the types of ground patterns generated, grouped by field conditions, and the states of the diagonals. The diagonals in all field groups are either:

1. Both diagonals are passive
2. One diagonal is active
3. Both diagonals are active

The diagonals in the Block 0 Passive Block 7 Passive and Block 0 Passive Block 7 Active groups display distinctly, emanating cleanly from the seed cell. The diagonals in the other two filed groups, in some cases, do no display cleanly with active and passive cells intermixing which will be discussed in their sections.

A second feature in all patterns is a center column emanating from the seed cell. In some cases, the column is a single row of active cells against a paasive ground (see Rule 004); in other cases, it is a random mix of active and passive cells (see Rule 030); in yet others, it's a mix of the two. The particulars will be discussed in each section.

Block 0 Passive: a total of 128 patterns
The field condtion for Block 0 Passive Block 7 Passive and Block 0 Passive Block 7 Active is the same, so these two groups are combined into one section. A couple of features that show up in the ground patterns throughout these two field groups are:

1. Diagonals: the display of the active diagnonals is very distinct, emanating cleanly from the seed cell.
2. Passive Diagonals: the ground area, up to the center column, is passive when the diagonal is passive
3. Active Diagonals:
1. Simple diagonal: the corners of the cells touch but the cells adjacent to the side and bottom faces - see Rules 002 and 018
2. Stepped-line diagonal: an active cell is adjacent to the side or bottom face of every other simple diagonal cell - see Rules 006
3. Double-line diagonal: an active cell is adjacent to the side or bottom face of every simple diagonal cell - see Rules 014
4. Single Active Diagonal: the ground adjacent to an active diagonal is either:
1. Passive to a passive center column - see Rule 002
2. Mixed active and passive cells to and active center column - see Rule 028
3. Solid active cells to the center column - see Rule 206
5. Both Active Diagonals:
1. Mixed active and passive cells between the diagonals - see Rule 018
2. Solid active cells between the diagonals - see Rule 222
6. Center Column: the column conditions are:
1. Passive cells
1. All cells enamating from the seed are passive - see Rules 000, 018
2. The first cell from the seed is active, the rest are passive - see Rules 006 and 094
2. Active cells: all cells enamating from the seed are active
3. Mixed cells
1. The mixed column requires both diagonals to be active
2. Regular pattern of active and passive cells - see Rules 050 and 054
3. Non-regular pattern -- see Rules 030

The types of patterns generated in the ground area are:

• Both diagonals passive
• 16 patterns where the center column is passive
• 16 patterns where the center column is active
• Left side active
• 16 patterns with a simple diagonal, passive ground, and passive center column
• 8 patterns with a double-line diagonal, passive ground, and passive center column
• 8 patterns with a patterned ground and active center column
• Right side active
• 16 patterns with a simple diagonal, passive ground, and passive center column
• 8 patterns with a double-line diagonal, passive ground, and passive center column
• 8 patterns with a patterned ground and active center column
• Both diagonals active
• 8 patterns with simple diagonals, a full patterned ground, and passive center column
• 8 patterns with simple diagonals, a full patterned ground, and regular mixed center column
• 1 patterns with double-line diagonals, a full patterned ground, and passive center column
• 15 patterns with double-line diagonals, a full patterned ground, and regular mixed center column
• 15 patterns with double-line diagonals, a full patterned ground, and non-regular mixed center column
• 15 patterns with double-line diagonals, a full patterned ground, and active center column

Of note, there are 6 patterns where the ground is fully active - 2 each with the left diagonal, right diagonal, and both diagonal.

Sortable Chart of Rules: Block 0 Passive
Chart of Patterns: Block 0 Passive Field

The patterns associated with the Block 0 Passive field groups are cleanly delineated, which is not the case with the other field groups. The reason is found by looking at the blocks involved during pattern generation. As shown below, only four blocks are invoked in Row 1 where there is a single seed.

		Row 1
	0	1	2	4	0

In the Block 0 Passive field groups, by definition, Block 0 will produce a passive cell, so the only blocks which can produce active cells in Row 1 are Blocks 1, 2, and 4. Three examples are analyzed to show why the Block 0 Passive patterns are cleanly delineated:

			Diagonal							Vertical							Both
		0	1	2	4	0			0	1	2	4	0			0	1	2	4	0
	0	1	2	4	0			0	1	3	6	4	0		0	1	2	5	2	4	0

Diagonal: In Row 1, Block 1 produces an active cell and Blocks 2 and 4 produce passive. At Row 2, the only possible blocks which can be invoked are Blocks 0, 1, 2, and 4, and the states produced by those blocks have already been defined as passive, active, passive, and passive, respectively. That means in Row 2, only Block 1 will produce and active cell, essentially creating a single seed condition for Row 3. Therefore, the possible blocks in Row 3 will be the same as Row 2, and so on and so on. The result is a single left diagonal. This will be the pattern generated even if remaining blocks, e.g. Blocks 3, 5, 6, and 7 are active. Each of these blocks requires at least two active cells in their neighborhood, and here, there is only one. These blocks are ignored and the resulting pattern will be a left diagonal. The same analysis holds on the right side, with Block 4 active and Blocks 0, 1, and 2 passive.

Vertical: In Row 1, Blocks 1 and 2 produce active cells and Block 4 produces a passive one. At Row 2, Block 1 generates the left diagonal pattern. The combination of the two adjacent active cells provides an opportunity for Block 3 or 6 (can generate active or passive) to invoke an active cell in Row 3 and generate a patterned ground. On the right, however, Block 4, which generates passive cells, clearly defines the passive wall at the center vertical. As above, the same analysis holds for the right side, with Block 1 generating the passive wall at the center vertical.

Both: With Block 1 and 4 active, both diagonal patterns are generated and the field becomes a pattern where Blocks 1 and 4 form a Sierpinski-type dispaly or, when Block 5 produces active cells, a checkerboard (see Rules 018 and 050). When all three blocks are active, the patterns become more complicated between the two diagonals as Blocks 3, 6, and 7 come into play. It is not possible to generate a passive ground when both diagonals are active.

Block 0 Active / Block 7 Active: a total of 64 patterns
One would expect that the patterns in this group would also be clearly delineated; the field is solid active cells and the features of Block 0 Passive would prevail, perhaps with patterns reversed. This, however, is not the case.

• While most of the digaonals are delineated, they are not as clearly defined visually.
  • In many cases, an active diagonal is flanked by a passive diagonal
  • In some cases, the first cell of an active diagonal is passive
• An active diagonal does not necessarily produce an active ground adjacent to it
• The center vertical, while present, does not emanate only from the seed cell, and, in two cases, is defined by passive, not active, cells

As mentioned above, there are cases where the first cell of an active diagonal is a passive cell. In the examples below, Rule 173 has a passive cell to the left and an active diagonal to the right, where the first cell of the active diagonal is passive. Rule 233 has two active diagonals where the first cell of both is passive:

Rule 173

Rule 173 detail

Rule 233

Rule 233 detail

The groupings of patterns based on the condition of the diagonal are:

• Both diagonals active - total of 36 patterns
  • 4 patterns where the first cell of both diagonals is passive
  • 8 patterns where the first cell of the left diagonal is passive
  • 8 patterns where the first cell of the right diagonal is passive
• Left side is active - total of 12 patterns
  • 4 patterns where the first cell of the diagonal is passive
• Right side active - total of 12 patterns
  • 4 patterns where the first cell of the diagonal is passive
• Both diagonals passive - total of 4 patterns

Sortable Chart of Rules: Block 0 Active / Block 7 Active
Chart of Patterns: Block 0 Active / Block 7 Active Field

An analysis of blocks invoked will give a sense of why the patterns in this field group are not cleanly delineated. By definition, Blocks 0 or 7 will produce an active cell. In Row 1, the field cells generated by Block 0 are all active which have leads to different neighborhood definitions with Blocks 1, 2, and 4 than when Block 0 is passive.

		Row 1
	0	1	2	4	0

Analysis of the same three examples, used above, will give an indication why the Block 0 Active Block 7 Active patterns are not cleanly delineated:

			Diagonal									Vertical										Both
		0	1	2	4	0					0	1	2	4	0						0	1	2	4	0
		7	6	4	1	3	7				7	7	6	5	3	7					7	6	5	3	7

Diagonal: In Row 1, Block 1 produces an active cell and extends the active field into the ground; Blocks 2 and 4 produce passive. At Row 2, the left diagonal is continued by Block 7 which is invoked by the extension of the field in Row 1. Where the only possible blocks to be invoked are Blocks 0, 1, 2, and 4 in the Block 0 Passive field groups, here Blocks 1, 4, 3, 6, and 7 can be invoked. Blocks 1 and 7 are defined as active and Block 4 as passive; Blocks 3 and 6 can be either, depending on the pattern rule.

• If Block 6 is passive, then the condition {diagonal the same state as the field results in a ground of the same state} is broken
• If Block 3 is active, the right diagonal becomes one where the first cell is passive
• If Block 6 is active or Block 3 passive, then Block 5 will be invoked in Row 3, leading to all blocks being invoked and no blocks being ignored

Vertical: By Row 2, all blocks have been invoked. In both cases, Blocks 3, 5, and 6 can be either active or passive, leading to a wider variety of relationships between field and ground:

• If Block 6 is passive, then the condition {diagonal the same state as the field results in a ground of the same state} is broken
• If Block 6 is active and Block 5 is passive, there is the beginning of a strong vertical; however if Block 5 is passive, the condition {diagonal the same state as the field results in a ground of the same state} is broken
• If Block 3 is active, the right diagonal becomes one where the first cell is passive

Both: Again, Blocks 3, 5, and 6 can be either active or passive. It is noteworthy that the Vertical and Both examples will result in similar patterns, shifted on cell over.

Because the patterns aren't cleanly delineated, the grouping of patterns is based on the appearance of the passive diagonals; this is somewhat consistent with the Block 0 Passive groupings based on active diagonals, though is more of a visual organization than one based on structure.

• No passive diagonal - total of 22 patterns
• Passive vertial - total of 7 patterns
• Left diagonal - 14 patterns
• Right diagonal - 14 patterns
• Both diagonals - 14 patterns

Block 0 Active / Block 7 Passive: a total of 64 patterns
The first thing to note is that when a diagonal is apparent, it is made up of a step pattern of active and passive cells. Another thing to notice that this group is the only one where a continuous steep diagonal can form – two steps down, one across versus the one down, one across of the boundary diagonals. Finally, when there’s a horizontal active stripe within the ground pattern, it is shifted vertically one row so that ground active stripe lines up with the field passive stripes.

The breakdown of patterns is:

• 36 where the diagonal and field are the same on both sides. Of these:
• 14 have a steep diagonal on one side
• 2 have step diagonals on both sides
• 24 where the one diagonal is active. Of these:
• 4 have a steep diagonal on the other side
• 4 where the diagonals are active on both sides

			Diagonal									Vertical										Both
		0	1	2	4	0					0	1	2	4	0						0	1	2	4	0
		7	6	4	1	3	7				7	7	6	5	3	7					7	6	5	3	7

 

Rule 011	Rule 011 detail	Rule 011 detail
Rule 015	Rule 015 detail	Rule 015 detail
Rule 091	Rule 091 detail	Rule 091 detail
Rule 013	Rule 013 detail	Rule 013 detail
Rule 025	Rule 025 detail	Rule 025 detail

 

Rule 001 20 patterns	Rule 017 7 patterns	Rule 025 7 patterns
Rule 057 2 patterns	Rule 069 10 patterns	Rule 015 10 patterns
Rule 045 2 patterns	Rule 089 2 patterns	Rule 073 4 patterns

 

Sortable Chart of Rules: Block 0 Active / Block 7 Passive
Chart of Patterns: Block 0 Active, Block 7 Passive Field