This is an analytical approach to CA systems with the intent to understand how and why certain configurations occur. Using input images as the variable in the study, the aim is to understand how the CA’s local rule sets in rule 2-3-3 communicate to produce a larger generative formation.
1
2
TABLE OF CONTENTS
00 GAME OF LIFE 04 01 _INITIAL TEST 06 02_METHODS OF CONTROL 14 03_POPULATION GENERATION 16 04_DEFINING CRITERIA 22 05_FAMILIES 26 DIAMOND FAMILY 01 26 DIAMOND FAMILY 02 28 SQUARE FAMILY 01 30 SQUARE FAMILY 02 32 06_BASE FAIL LAYER 34 07_AVERAGE IMAGE 36 08_DEFINING PARAMETERS 38
LIVE/DEAD RATIO
38
PROXIMITY 40 NEIGHBORS 42
09_FAMILY ANALYSIS 44
SQUARE FAMILY 44 DIAMOND FAMILY
52
CIRCLE FAMILY 60 10_FAMILY ANALYSIS_CONCLUSION 68 11_FAIL FORMATIONS 70 12_GENERATING SUCCESS 78
3
00_GAME OF LIFE lonliness if you are ALIVE and you have < 2 neighbors you DIE
overcrowding if you are ALIVE and you have > 3 neighbors you DIE
reproduction if you are DEAD and you have = 3 neighbors you come ALIVE
4
5
01_INITIAL TEST
6
7
01_INITIAL TEST
8
9
01_INITIAL TEST
10
11
01_INITIAL TEST
12
13
02_METHODS OF CONTROL
Cellular Automata works on local rule sets where a cell analyzes its neighborhood and responds according to the fixed rule set assigned to the system to control its state (on or off) By using one rule set (233) as a control, the variable in this study are the input images as initial start states
14
15
03_POPULATION GENERATION
16
PRIMITIVES
TRANSFORMATIONS
INVERSIONS
17
03_POPULATION GENERATION Square Family
Square Scale Family
18
Diamond Family
Diamond Scale Family
19
03_POPULATION GENERATION CircleinCircle Family
Cross Family
20
Circle Family01
Circle Family02
21
04_DEFINING CRITERIA
Failure = Stability
The emergence of elevators. These elevetators are the aging cells.
22
Success = Instability
An active cellular automata with a unstable core with a combination of both younger cells and aging cells within.
23
04_DEFINING CRITERIA
24
25
05_FAMILIES_DIAMOND FAMILY01
26
27
05_FAMILIES_DIAMOND FAMILY02
28
29
05_FAMILIES_SQUARE FAMILY01
int start state
fail state - lvl 30
int start state
int start state
fail state - lvl 36
30
fail state - lvl 49
int start state
fail state - lvl 38
int start state
fail state - lvl 12
int start state
fail state - lvl 19
int start state
31
int start state
fail state - lvl 41
05_FAMILIES_SQUARE FAMILY02
int start state
fail state - lvl 38
int start state
fail state - lvl 54
int start state
32
int start state
fail state - lvl 28
int start state
fail state - lvl 16
int start state
fail state - lvl 44
int start state
33
fail state - lvl 3
int start state
fail state - lvl 3
06_BASE FAIL LAYER
The layer that results in inactivity is extracted and used as input to analyze why it results in stability.
Start State
Fail State The base layer where elevators start to emerge
34
Start State
Fail State The base layer where elevators start to emerge
35
07_AVERAGE IMAGE
An average image is compiled to visually approximate how frequently each pixel is used in the initial start state of the CA
36
37
08_DEFINING PARAMETERS Live/Dead Ratio
38
39
08_DEFINING PARAMETERS Proximity
40
41
08_DEFINING PARAMETERS Neighbors
42
43
09_FAMILY ANALYSIS_SQUARE FAMILY
44
45
09_FAMILY ANALYSIS_SQUARE FAMILY
46
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47
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09_FAMILY ANALYSIS_SQUARE FAMILY 80 80
32 32
32
40 40
80
80
32
80
88 88
40
40 32
16 16
88
24
24 16
32
32
16
16
24 24
40 32 32
88
88
24 80 80
32 32
48 48
80 80
32 32
40 40
48 48
32
48
48 88 88
16 16
16 16
24 24 32 32
48 48
32 32
48 48
48 48
48 48
48 48
88 88
48
48
240 240 240 240
48 32 32
88
88
32
32
64 64 32
64
64
24 24 48
88
48 48
48
64 104 104
104
48 48 104
32 32
88 88
48 48
48
88 88
48
48
40 40
48 48
48
48
32 32
64 64
88 88
48 48
64 64
104 104 104 104
48
Proximity Fail
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09_FAMILY ANALYSIS_SQUARE FAMILY
50
Neighbors Fail 0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
4.0
4.0
3.0
3.0
0.0
4.0
4.0
3.5
3.0
0.0
3.5
4.0
2.67
4.0
4.0
4.0
4.0
2.67
4.0
3.5
0.0
3.0
3.5
4.0
4.0
0.0
3.0
3.0
4.0
4.0
0.0
0.0
4.0
4.0
3.0
3.0
0.0
4.0
3.67
4.0
0.0
3.0
4.0
4.0
3.67
3.33
4.0
4.0
3.33
3.67
4.0
4.0
3.0
0.0
4.0
3.67
4.0
0.0
3.0
3.0
4.0
4.0
0.0
0.0
3.0
3.0
0.0
0.0
0.0
0.0
0.0
3.0
3.5
4.0
3.0
3.0
3.0
3.0
0.0
0.0
3.0
3.0
3.0
3.0
4.0
3.5
3.0
0.0
0.0
0.0
0.0
0.0
3.0
3.0
0.0
0.0
3.0
3.0
0.0
0.0
0.0
3.0
3.0
0.0
4.0
4.0
0.0
3.0
2.0
1.0
0.0
0.0
1.0
2.0
3.0
0.0
4.0
4.0
0.0
3.0
3.0
0.0
0.0
0.0
3.0
3.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
3.0
3.0
3.0
0.0
0.0
3.0
4.0
0.0
1.0
4.0
4.0
1.0
0.0
4.0
3.0
0.0
0.0
3.0
3.0
3.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
4.0
4.0
0.0
3.0
3.0
0.0
3.0
3.0
3.0
0.0
4.0
2.67
0.0
0.0
4.0
4.0
0.0
0.0
2.67
4.0
0.0
3.0
3.0
3.0
0.0
3.0
3.0
0.0
4.0
4.0
0.0
0.0
4.0
3.67
0.0
3.0
3.0
3.0
0.0
3.0
3.0
3.0
3.5
0.0
3.0
0.0
0.0
0.0
0.0
3.0
0.0
3.5
3.0
3.0
3.0
0.0
3.0
3.0
3.0
0.0
3.67
4.0
0.0
0.0
3.5
4.0
3.0
0.0
3.0
3.0
3.0
0.0
3.0
3.0
2.0
2.0
3.0
0.0
0.0
0.0
0.0
3.0
2.0
2.0
3.0
3.0
0.0
3.0
3.0
3.0
0.0
3.0
4.0
3.5
0.0
0.0
3.0
0.0
3.5
4.0
0.0
3.0
3.0
3.0
0.0
2.0
3.0
2.33
0.0
0.0
3.5
3.5
0.0
0.0
2.33
3.0
2.0
0.0
3.0
3.0
3.0
0.0
4.0
3.5
0.0
3.0
0.0
0.0
0.0
3.0
4.0
4.0
0.0
0.0
3.0
3.0
2.0
0.0
3.0
2.0
1.0
3.0
4.0
4.0
3.0
1.0
2.0
3.0
0.0
2.0
3.0
3.0
0.0
0.0
4.0
4.0
3.0
0.0
0.0
0.0
3.5
4.0
3.0
0.0
3.0
4.0
3.5
2.0
3.0
3.0
0.0
1.0
2.0
3.0
3.5
3.5
3.0
2.0
1.0
0.0
3.0
3.0
2.0
3.5
4.0
3.0
0.0
3.0
4.0
3.5
0.0
0.0
4.0
4.0
3.0
3.0
4.0
2.67
0.0
2.0
2.33
2.0
1.0
0.0
3.5
3.0
4.0
4.0
3.0
3.5
0.0
1.0
2.0
2.33
2.0
0.0
2.67
4.0
3.0
3.0
4.0
4.0
0.0
0.0
2.67
3.67
3.0
2.0
0.0
0.0
3.0
3.0
0.0
1.0
2.0
3.5
0.0
4.0
0.0
0.0
4.0
0.0
3.5
2.0
1.0
0.0
3.0
3.0
0.0
0.0
2.0
3.0
3.67
2.67
0.0
0.0
4.0
3.33
3.0
1.0
1.0
0.0
0.0
0.0
0.0
3.0
3.0
3.0
4.0
0.0
3.0
3.0
0.0
4.0
3.0
3.0
3.0
0.0
0.0
0.0
0.0
1.0
1.0
3.0
3.33
4.0
0.0
0.0
4.0
4.0
0.0
0.0
4.0
4.0
0.0
0.0
3.5
4.0
3.5
4.0
0.0
3.0
0.0
0.0
3.0
0.0
4.0
3.5
4.0
3.5
0.0
0.0
4.0
4.0
0.0
0.0
4.0
4.0
0.0
0.0
4.0
4.0
0.0
0.0
4.0
4.0
0.0
0.0
3.5
4.0
3.5
4.0
0.0
3.0
0.0
0.0
3.0
0.0
4.0
3.5
4.0
3.5
0.0
0.0
4.0
4.0
0.0
0.0
4.0
4.0
0.0
0.0
4.0
3.33
3.0
1.0
1.0
0.0
0.0
0.0
0.0
3.0
3.0
3.0
4.0
0.0
3.0
3.0
0.0
4.0
3.0
3.0
3.0
0.0
0.0
0.0
0.0
1.0
1.0
3.0
3.33
4.0
0.0
0.0
2.67
3.67
3.0
2.0
0.0
0.0
3.0
3.0
0.0
1.0
2.0
3.5
0.0
4.0
0.0
0.0
4.0
0.0
3.5
2.0
1.0
0.0
3.0
3.0
0.0
0.0
2.0
3.0
3.67
2.67
0.0
0.0
4.0
4.0
3.0
3.0
4.0
2.67
0.0
2.0
2.33
2.0
1.0
0.0
3.5
3.0
4.0
4.0
3.0
3.5
0.0
1.0
2.0
2.33
2.0
0.0
2.67
4.0
3.0
3.0
4.0
4.0
0.0
0.0
3.5
4.0
3.0
0.0
3.0
4.0
3.5
2.0
3.0
3.0
0.0
1.0
2.0
3.0
3.5
3.5
3.0
2.0
1.0
0.0
3.0
3.0
2.0
3.5
4.0
3.0
0.0
3.0
4.0
3.5
0.0
0.0
0.0
3.0
4.0
4.0
0.0
0.0
3.0
3.0
2.0
0.0
3.0
2.0
1.0
3.0
4.0
4.0
3.0
1.0
2.0
3.0
0.0
2.0
3.0
3.0
0.0
0.0
4.0
4.0
3.0
0.0
0.0
0.0
3.0
0.0
3.5
4.0
0.0
3.0
3.0
3.0
0.0
2.0
3.0
2.33
0.0
0.0
3.5
3.5
0.0
0.0
2.33
3.0
2.0
0.0
3.0
3.0
3.0
0.0
4.0
3.5
0.0
3.0
0.0
0.0
3.5
4.0
3.0
0.0
3.0
3.0
3.0
0.0
3.0
3.0
2.0
2.0
3.0
0.0
0.0
0.0
0.0
3.0
2.0
2.0
3.0
3.0
0.0
3.0
3.0
3.0
0.0
3.0
4.0
3.5
0.0
0.0
4.0
3.67
0.0
3.0
3.0
3.0
0.0
3.0
3.0
3.0
3.5
0.0
3.0
0.0
0.0
0.0
0.0
3.0
0.0
3.5
3.0
3.0
3.0
0.0
3.0
3.0
3.0
0.0
3.67
4.0
0.0
0.0
4.0
4.0
0.0
3.0
3.0
0.0
3.0
3.0
3.0
0.0
4.0
2.67
0.0
0.0
4.0
4.0
0.0
0.0
2.67
4.0
0.0
3.0
3.0
3.0
0.0
3.0
3.0
0.0
4.0
4.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
3.0
3.0
3.0
0.0
0.0
3.0
4.0
0.0
1.0
4.0
4.0
1.0
0.0
4.0
3.0
0.0
0.0
3.0
3.0
3.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
3.0
3.0
0.0
0.0
0.0
3.0
3.0
0.0
4.0
4.0
0.0
3.0
2.0
1.0
0.0
0.0
1.0
2.0
3.0
0.0
4.0
4.0
0.0
3.0
3.0
0.0
0.0
0.0
3.0
3.0
0.0
0.0
3.0
3.0
0.0
0.0
0.0
0.0
0.0
3.0
3.5
4.0
3.0
3.0
3.0
3.0
0.0
0.0
3.0
3.0
3.0
3.0
4.0
3.5
3.0
0.0
0.0
0.0
0.0
0.0
3.0
3.0
0.0
0.0
4.0
4.0
3.0
3.0
0.0
4.0
3.67
4.0
0.0
3.0
4.0
4.0
3.67
3.33
4.0
4.0
3.33
3.67
4.0
4.0
3.0
0.0
4.0
3.67
4.0
0.0
3.0
3.0
4.0
4.0
0.0
0.0
4.0
4.0
3.0
3.0
0.0
4.0
4.0
3.5
3.0
0.0
3.5
4.0
2.67
4.0
4.0
4.0
4.0
2.67
4.0
3.5
0.0
3.0
3.5
4.0
4.0
0.0
3.0
3.0
4.0
4.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
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0.0
1 1a 2 2a 3 4 4a 5a 6 7 7a 8 8a 9 9a 10 10a
1 1a
0 944
1 8
2 16
3 56
4 0
5 0
6 0
7 0
8 0
992
8
16
8
0
0
0
0
0
984
16
8
16
0
0
0
0
0
51
0 944
1 8
2 16
3 56
4 0
5 0
6 0
7 0
992
8
16
8
0
0
0
0
984
16
8
16
0
0
0
0
992
0
32
0
0
0
0
0
976
0
48
0
0
0
0
0
936
0
32
16
24
16
0
0
976
16
32
0
0
0
0
0
1008
0
0
16
0
0
0
0
1008
16
0
0
0
0
0
0
1000
16
8
0
0
0
0
0
984
16
8
16
0
0
0
0
984
16
8
16
0
0
0
0
960
0
0
56
8
0
0
0
920
16
40
40
8
0
0
0
960
0
0
56
8
0
0
0
976
0
48
0
0
0
0
0
784
0
0
216
24
0
0
0
09_FAMILY ANALYSIS_DIAMOND FAMILY
52
53
09_FAMILY ANALYSIS_DIAMOND FAMILY Live/Dead Success
54
Live/Dead Fail
55
09_FAMILY ANALYSIS_DIAMOND FAMILY
32 40
Proximity Success 40
76
76 48 16
16
20
32
48
40
Proximity Graph Fail
48
76
48 48
48
20
16
20
48
16
32 16
32 40 40 76
48
76
Proximity Graph Success 16
48
16
72
16
48 48
48 48
72 72
56
72
72
Proximity Fail
57
09_FAMILY ANALYSIS_DIAMOND FAMILY Neighbors Success
58
Neighbors Fail
59
09_FAMILY ANALYSIS_CIRCLE FAMILY
60
61
09_FAMILY ANALYSIS_CIRCLE FAMILY Live/Dead Success
62
Live/Dead Fail
63
09_FAMILY ANALYSIS_CIRCLE FAMILY Proximity Success
Proximity Graph Fail
Proximity Graph Success
64
Proximity Fail
65
09_FAMILY ANALYSIS_CIRCLE FAMILY Neighbors Success
66
Neighbors Fail
67
10_FAMILY ANALYSIS_CONCLUSION In order to assess the models they are separated into two populations (successful and failure) and their input images are then diagrammed according to the selected criteria (proximity, life/death ratio, neighborhood).
68
69
11_FAIL FORMATIONS
Fail Patterns 70
These fail states are extracted to review their commonalities and distinctions. When used as initial start states they result in inactivity - they do not communicate with their neighbors and produce stability.
Fail base formations
71
11_FAIL FORMATIONS
72
73
11_FAIL FORMATIONS
74
75
11_FAIL FORMATIONS
76
77
12_GENERATING SUCCESS
78
79
12_GENERATING SUCCESS
80
81
12_GENERATING SUCCESS
examples of curing per layer for input image 3 curing occured in layers - 16, 20, 22, 27, 43, 48, 49, 69, 75, 76, 77, 83, 88, 90, 92, 96, 101, 103, 113, 114, 116, 121, 124, 126, 129, 136, 141, 145
82
input image
fail formaaon to search for
curing = turn on neighbors of a fail formaaon cured cells
uncured fail formaaon arising in the model
failed model
model before the ďŹ rst layer with curing
cured cells of fail formaaons
cured model
83