VOXEL NEIGHBOURHOODS BHAVATARINI KUMARAVEL LARA NIOVI VARTZIOTIS IRFAN BHAKRANI XIUJING WANG
DW 102C / VOXEL NEIGHBOURHOODS LED BY
MOSTAFA EL SAYED OCTAVIAN GHEORGHIU
TEAM
BHAVATARINI KUMARAVEL LARA NIOVI VARTZIOTIS IRFAN BHAKRANI XIUJING WANG
WORKSHOP BRIEF The workshop explores the generative potential of self-regulating neighbourhoods of voxels that interact through simple local rule sets and result in complex organisations across large populations. Experimenting through explicit models of interactions and observable patterns of agency the workshop explores the capacity for these systems to evolve structural elements with the capacity to self-structure.
RESEARCH BRIEF The research seeks to evaluate a voxel neighbourhood system through stability. The inherent stability of the system is analysed and strategies to enhance it are evolved. The local rules and the initializing seed of the cellular automata voxel neighbourhood are rated in terms of their stability quotients across iterations and the best rules and seeds deduced out of them are employed in an attempt to building systems of increased heights, employing the stabilizing strategies discovered. 3
4
CONTENTS Research parameters
6
Initial experimentation
18
Iteration 1
24
Iteration 2
56
Iteration 3
68
5
RESEARCH PARAMETERS The system of study, cellular automata, the rules to be used, and the strategies sought to be employed in the reserahc are briefly discussed. 6
FIG 1. CELLULAR AUTOMATON
FIG 2. A VOXEL
CELLULAR AUTOMATA Cellular automata is a grid of cells that exist in one of a finite number of states, where a new generation is born by the application of a fixed rule on the grid, and the states re-determined. A voxel is a unit value on a regular three dimensional grid.
ALIVE (1)
DEAD (0)
FIG 3. VOXEL - TWO STATES 7
DIES
DIES
DIES
LONELINESS
Loneliness neliness Loneliness
DIES
DIES
DIES
OVERCROWDING
Overcrowding Overcrowding Overcrowding FOR ALIVE VOXELS (Environment range)
FIG 4. CONWAY‘S GAME OF LIFE - RULES
GAME OF LIFE The Game of life is a one model of cellular automata devised by British mathematician John Conway in 1970. The voxels in the model exist in either of two states – life or death. The state of a voxel is determined by the states of its eight neighbouring cells in two dimensions, in terms of ‘population’. An alive voxel dies if it is surrounded by less than two voxels, due to ‘loneliness’. Also, if an alive voxel is surrounded by more than three alive neighbours, it dies of ‘overcrowding’. Hence for an alive voxel to stay alive, it must have two to three alive neighbours around. This is termed as the ‘Environment range‘. Similarly for a dead voxel to come alive, it needs to have three neighbours around it. This is called the ‘Fertility range’. Thus, a ruleset is a combination of four integers, the first two of which define the limit of the environment range and the later two of the fertility range. 8
STAYS ALIVE
STAYS ALIVE
IDEAL NEIGHBOURHOOD
STAYS ALIVE
COMES ALIVE
IDEAL NEIGHBOURHOOD
Ideal neighbourhood Ideal neighbourhood Ideal neighbourhood Ideal neighbourhood FOR DEAD VOXELS (Fertility range)
SEED The game of life is essentially a zero-player game, where the ruleset acts on an initializing pattern called the seed.
3D GAME The game can be extended in three dimensions, by stacking the generations of voxels formed after every single application of the ruleset, one over the other. This adds the attribute of age to the voxels, where voxels that stay alive for more generations get ‘older’ and voxels that are newly made alive are ‘younger’.
FIG 5. SEED - EXAMPLES
GENERATION 7 GENERATION 7 GENERATION 6 GENERATION 6
GENERATION 5 GENERATION 5 GENERATION 4 GENERATION 4
GENERATION 3 GENERATION 3 GENERATION 2 GENERATION 2 GENERATION 1 GENERATION 1
ELEVATION ELEVATION
FIG 6. GAME OF LIFE IN 3D 9
OLD VOXEL (RED)
YOUNG VOXEL (BLACK)
FIG 7. SEED GROWTH IN 3D - TOP VIEW 10
SEED_LN 50X50 (2311) A0 D 61440 T 61440 0%
SEED_LN 50X50 (2312) A0 D 61440 T 61440 0%
SEED_LN 50X50 (2313) A 18212 D 43228 T 61440 29.64 %
SEED_LN 50X50 (2314) A 18448 D 42992 T 61440 30.02 %
SEED_LN 50X50 (2315) A 18716 D 42724 T 61440 30.46 %
SEED_LN 50X50 (2316) A 19624 D 41816 T 61440 31.94 %
SEED_LN 50X50 (2312) A 19984 D 41456 T 61440 32.52 %
SEED_LN 50X50 (1333) A 14832 D 46608 T 61440 24.14 %
SEED_LN 50X50 (2333) A 13652 D 47788 T 61440 22.22 %
SEED_LN 50X50 (3333) A 8416 D 53024 T 61440 13.69 %
SEED_LN 50X50 (3433) A 8416 D 53024 T 61440 13.69 %
SEED_LN 50X50 (3533) A 22204 D 39236 T 61446 36.13 %
SEED_LN 50X50 (3633) A 24224 D 37215 T 61440 39.42 %
SEED_LN 50X50 (2335) A 15668 D 45772 T 61440 25.50 %
SEED_LN 50X50 (2336) A 16334 D 45096 T 61440 26.58 %
SEED_LN 50X50 (2332) A 16940 D 44550 T 61440 27.57 %
SEED STUDIES FIG 8. SEED - LN 50X50 LN 50x50
SEED_LN 50X50 (3733) A 25848 D 35592 T 61440 42.07 %
SEED_LN 50X50 (2334) A 14264 D 47176 T 61440 23.21 %
A - ALIVE D - DEAD T - TOTAL POPULATION 11
SEED_X 50X50 (2311) A 4164 D 57276 T 61440 6.77 %
SEED_X 50X50 (2312) A 4688 D 56752 T 61440 7.63 %
SEED_X 50X50 (2313) A 15072 D 46368 T 61440 24.53 %
SEED_X 50X50 (2314) A 15688 D 45752 T 61440 25.53 %
SEED_X 50X50 (2315) A 16144 D 45296 T 61440 26.27 %
SEED_X 50X50 (2316) A 18524 D 42916 T 61440 30.14 %
SEED_X 50X50 (23120) A 18944 D 42496 T 61440 30.83 %
SEED_X 50X50 (1333) A 10308 D 51132 T 61440 16.77 %
SEED_X 50X50 (2333) A 8244 D 53196 T 61440 13.41 %
SEED_X 50X50 (3333) A 3816 D 57624 T 61440 6.21 %
SEED_X 50X50 (3433) A 3816 D 57624 T 61440 6.21 %
SEED_X 50X50 (3533) A 15664 D 45776 T 61446 25.49 %
SEED_X 50X50 (3633) A 19680 D 41760 T 61440 5.91 %
SEED_X 50X50 (3733) A 21972 D 39468 T 61440 35.76 %
SEED_X 50X50 (2334) A 10764 D 50576 T 61440 17.51 %
SEED_X 50X50 (2335) A 11696 D 49744 T 61440 19.03 %
SEED_X 50X50 (2336) A 14552 D 46888 T 61440 23.68 %
SEED_X 50X50 (23320) A 15350 D 46140 T 61440 24.90 %
SEED STUDIES FIG 9. SEED - X 50X50 X 50x50 12
A - ALIVE D - DEAD T - TOTAL POPULATION
SEED_5 50X50 (2311) A 4972 D 56468 T61440 8.09%
SEED_5 50X50 (2312) A 5448 D 55992 T 61440 8.87%
SEED_5 50X50 (2313) A 16956 D 44484 T 61440 27.60%
SEED_5 50X50 (2314) A 18152 D 43288 T 61440 29.54%
SEED_5 50X50 (2315) A 18684 D 42756 T 61440 30.41%
SEED_5 50X50 (2316) A 21692 D 39748 T 61440 35.31%
SEED_5 50X50 (23120) A 22396 D 39044 T 61440 36.45%
SEED_5 50X50 (1333) A 10384 D 51056 T 61440 16.90 %
SEED_5 50X50 (2333) A 8200 D 53240 T 61440 13.35 %
SEED_5 50X50 (3333) A 2416 D 59024 T 61440 3.93 %
SEED_5 50X50 (3433) A 2416 D 59024 T 61440 3.93 %
SEED_5 50X50 (3533) A 15796 D 45644 T 61440 25.71 %
SEED_5 50X50 (3633) A 22776 D 38664 T 61440 37.07 %
SEED_5 50X50 (2335) A 12260 D 49180 T 61440 19.95%
SEED_5 50X50 (2336) A 14352 D 47088 T 61440 23.36%
SEED_5 50X50 (23320) A 15872 D 45568 T 61440 25.83%
SEED STUDIES FIG 10. SEED - 5 50X50 5 50x50
SEED_5 50X50 (3733) A 24796 D 36644 T 61440 40.35 %
SEED_5 50X50 (2334) A 8200 D 53240 T 61440 18.36%
A - ALIVE D - DEAD T - TOTAL POPULATION 13
Percentage Alive
Gravity Engine
Percentage density Stability tolerance Percentage stable
FINALIZING RULES + SEED
FINDING STABILITY
EVALUATING STABILITY
FIG 11. RESEARCH METHODOLOGY 14
Aggregation Mass differentiation
CREATING STABILITY
VOXEL
MASS
GRAVITY
FIG 12. MASS + GRAVITY
FINDING STABILITY
FIG 13. UNITY - GRAVITY ENGINE
The voxel system is created using the Unity Game Engine scripted through C-Sharp programming language. Using the Gravity Engine by Unity, the system is tested of its stability by assigning masses to individual voxels. 15
INITIAL POSITION (x1, y1, z1)
DISPLACEMENT Compared with Stability Tolerance
FINAL POSITION (x2, y2, z2)
FIG 14. DISPLACEMENT ON GRAVITY
MEASURING STABILITY A voxel is considered stable if its displacement after the application of the Gravity Engine does not exceed a constant called the Stability tolerance. Thus, voxels are classified as stable and unstable and the counts are measured as percentages of the total alive population for further analysis. 16
DISPLACEMENT >= STABILITY TOLERANCE
< STABILITY TOLERANCE
UNSTABLE VOXEL
STABLE VOXEL
UNSTABLE VOXEL COUNT / ALIVE COUNT
STABLE VOXEL COUNT / ALIVE COUNT
PERCENTAGE UNSTABLE
PERCENTAGE STABLE
FIG 15. CALCULATING STABILITY FIGURES
2D DENSITY 8 NEIGHBOURS
3D DENSITY 26 NEIGHBOURS
AVERAGE 2D DENSITY
AVERAGE 3D DENSITY
FIG 16. NEIGHBOURHOOD DENSITY
RELATING DENSITY The relationship between the stability of the system and its voxel density is speculated. Hence, the average two-dimensional and three-dimensional density of the system are computed and employed in analysis to check relatable figures.
17
25x25
25x25
25x25
50x50
50x50
50x50
100x100
100x100
100x100
SEED_LN
SEED_X
SEED_5
FIG 17. SEEDS FOR INITIAL EXPERIMENTATIONRULESET – RULE(2333) HEIGHT – 15,30
INITIAL EXPERIMENTATION The initial set of test records are performed on three seeds operating at three different resolutions ranging from 25 x 25 to 100 x 100. The rule applied in Conway’s original rule of 2333 and two heights – 15 and 30 – are tried. For every seed index and ruleset combination, the total number of voxels, the number of voxels alive, the number dead, the average 3D density, the average 2D density, the stability tolerance and the percentage of stability after turning on the gravity engine are recorded. 18
AGE MODEL
UNDER GRAVITY
3D DENSITY MODEL
STABILITY CHECK
2D DENSITY MODEL
STABLE VOXELS
FIG 18. SEED X 50X50 UNDER RULE(2333)
Ruleset EnvironmentLower Limit Environment Upper limit Fertility Lower Limit Fertility Upper Limit Width Length Height Population Alive Dead Percentage Alive Pecentage Dead Stability tolerance Stable Unstable Percentage Stable Percentage Unstable Average 3D density Average 2D density
2333 2 3 3 3 50 50 15 65536 8244 57292 12.58% 87.42% 1 269 7975 3.26% 96.74% 38% 51%
Ruleset EnvironmentLower Limit Environment Upper limit Fertility Lower Limit Fertility Upper Limit Width Length Height Population Alive Dead Percentage Alive Pecentage Dead Stability tolerance Stable Unstable Percentage Stable Percentage Unstable Average 3D density Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
2333 2 3 3 3 50 50 15 65536 8244 57292 12.58% 87.42% 1 269 7975 3.26% 96.74% 38% 51%
SEED_X 50x50
Stable Voxel Color Unstable Voxel Color
19
Percentage Voxel Age Color
AGE MODEL
UNDER GRAVITY
3D DENSITY MODEL
STABILITY CHECK
2D DENSITY MODEL
STABLE VOXELS
FIG 19. SEED 5 100X100 UNDER RULE(2333)
Ruleset Ruleset EnvironmentLower EnvironmentLower Limit Environment Environment Upper Upper limit Fertility Fertility Lower Limit Lower Limit Fertility Fertility Upper Upper Limit Limit Width Width Length Length Height Height Population Population Alive Alive Dead Dead Percentage Alive Percentage Alive Pecentage Dead Pecentage Dead Stability tolerance Stability tolerance Stable Stable Unstable Unstable Percentage Stable Percentage Stable Percentage Unstable Percentage Unstable Average 3D density Average 3D density Average 2D density Average 2D density 20
Percentage Voxel Age Color
2333 2333 22 33 33 33 50 100 50 100 15 30 65536 507904 8244 39842 57292 468062 12.58% 7.84% 87.42% 92.16% 1 269 270 7975 39572 3.26% 0.68% 96.74% 99.32% 37% 38% 44% 51%
Ruleset EnvironmentLower Limit Environment Upper limit Fertility Lower Limit Fertility Upper Limit Width Length Height Population Alive Dead Percentage Alive Pecentage Dead Stability tolerance Stable Unstable Percentage Stable Percentage Unstable Average 3D density Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color Stable Voxel Color Unstable Voxel Color
2333 2 3 3 3 50 50 15 65536 8244 57292 12.58% 87.42% 1 269 7975 3.26% 96.74% 38% 51%
SEED_5 100x100
AGE MODEL
UNDER GRAVITY
3D DENSITY MODEL
STABILITY CHECK
2D DENSITY MODEL
STABLE VOXELS
FIG 20. SEED LN 25X25 UNDER RULE(2333)
Ruleset EnvironmentLower Limit Environment Upper limit Fertility Lower Limit Fertility Upper Limit Width Length Height Population Alive Dead Percentage Alive Pecentage Dead Stability tolerance Stable Unstable Percentage Stable Percentage Unstable Average 3D density Average 2D density
2333 2333 2 2 3 3 3 50 25 50 25 15 65536 16384 8244 3072 57292 13312 12.58% 18.75% 87.42% 81.25% 1 269 249 7975 2823 3.26% 8.11% 96.74% 91.89% 38% 51% 57%
Ruleset EnvironmentLower Limit Environment Upper limit Fertility Lower Limit Fertility Upper Limit Width Length Height Population Alive Dead Percentage Alive Pecentage Dead Stability tolerance Stable Unstable Percentage Stable Percentage Unstable Average 3D density Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
2333 2 3 3 3 50 50 15 65536 8244 57292 12.58% 87.42% 1 269 7975 3.26% 96.74% 38% 51%
SEED_LN 25x25
Stable Voxel Color Unstable Voxel Color
21
Percentage Voxel Age Color
CREATING STABILITY Due to the poor stability observed in a system of fragmented voxels, it is essential to include parameters of aggregation within the system. In this case parameters are set to initiate face to face linkages in between voxels to create connected clusters on respective axes to improve the global stability of the system.
AGGREGATION The first parameter of aggregation is created by creating linkages in between voxels that are unsupported from beneath, this forms a system of local cantilevers where unsupported fragments are connected to a supported whole whereas if there is no possibility of a face-to-face connection, the voxel dies. Also, the voxels supported from boneath form connections with the voxels beneath them resulting in vertical aggregation aiding in stability.
CANTILEVERED VOXEL
No alive neighbour below it
NON-CANTILEVERED VOXEL Has alive neighbour below it
CANTILEVERED VOXEL WITHOUT FACE-TO-FACE CONNECTION KILLED CANTILEVERED VOXEL WITH FACE-TO-FACE CONNECTION AGGREGATED
NON-CANTILEVERED VOXEL VERTICALLY AGGREGATED
FIG 21. AGGREGATION STRATEGIES 22
CANTILEVERED VOXEL WITHOUT FACE-TO-FACE CONNECTION KILLED
MODEL
ON GRAVITY
AFTER GRAVITY
BEFORE AGGREGATION TOTAL - 48 STABLE - 32 (67%)
MODEL
AGGREGATED MODEL
AFTER GRAVITY
AFTER AGGREGATION TOTAL - 48 KILLED - 4 STABLE - 44 (92%)
FIG 22. EFFECTS OF AGGREGATION ON STABILITY 23
ITERATION 1 The strategies of aggregation discussed are put to test with four seeds in three simple resolutions of 10x10, 20x20 and 30x30, for a height of 8 layers, across a vast set of rulesets. For every seed and ruleset combination, the total population, the number alive, the average 3D density and the stability percentage are recorded before aggregation. After performing aggregation, the number of cantilver voxels found, the number fixed , the number killed and the subsequent percentage of stability are recorded. 24
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10X1010X10
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10X10
10X1010X10
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10X10
10X1010X10
10X10
10
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20x2020x20
20x20
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20x2020x20
20x20
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20x2020x20
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20
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30x3030x30
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30
SEED_X
SEED_X
_LINE SEED_LINE SEED_LINE SEED_X
SEED_X SEED_POINTSEED_POINTSEED_POINTSEED_POINT SEED_POINT X SEED_POINT X SEED_P
RULESET â&#x20AC;&#x201C; 1122, 1177, 1211, 1233, 1234, 1235, 1236, 1237, 1522, 1724, 2211, 2333, 2433, 2533, 2644, 3344, 3355, 3411, 5511, 7711
FIG 23. SEEDS AND RULES FOR ITERATION 1
FIG 24. SEED BASED GRAPHS FOR ITERATION 1 - TEMPLATE
SEED BASED GRAPHING The data recorded are represented in graphs that are seed based and resolution specific; rules are taken along the X Axis and the percentage scale is laid out along the Y Axis. The plot area has four lines, each of which represent the percentage of life, the percentage density, the percentage of stability before aggregation and the percentage of stability after aggregation respectively. The graphs help to visulize the relationship between the four parameters across various ruleset and seed combinatons. 25
POINT
LINE
PERCENTAGE ALIVE
AVERAGE DENSITY
PERCENTAGE STABLE BEFORE AGGREGATION
FIG 25. SEED BASED GRAPHS FOR ITERATION 1 - 10X10 SEEDS 26
PERCENTAGE STABLE AFTER AGGREGATION
X
POINT X
PERCENTAGE ALIVE
AVERAGE DENSITY
PERCENTAGE STABLE BEFORE AGGREGATION
PERCENTAGE STABLE AFTER AGGREGATION
27
POINT
LINE
PERCENTAGE ALIVE
AVERAGE DENSITY
PERCENTAGE STABLE BEFORE AGGREGATION
FIG 26. SEED BASED GRAPHS FOR ITERATION 1 - 20X20 SEEDS 28
PERCENTAGE STABLE AFTER AGGREGATION
X
POINT X
PERCENTAGE ALIVE
AVERAGE DENSITY
PERCENTAGE STABLE BEFORE AGGREGATION
PERCENTAGE STABLE AFTER AGGREGATION
29
POINT
LINE
PERCENTAGE ALIVE
AVERAGE DENSITY
PERCENTAGE STABLE BEFORE AGGREGATION
FIG 27. SEED BASED GRAPHS FOR ITERATION 1 - 30X30 SEEDS 30
PERCENTAGE STABLE AFTER AGGREGATION
X
POINT X
PERCENTAGE ALIVE
AVERAGE DENSITY
PERCENTAGE STABLE BEFORE AGGREGATION
PERCENTAGE STABLE AFTER AGGREGATION
31
POINT GROWTH POINT GROWTH
ARCHING ARCHING
UNIFORM GROWTH UNIFORM GROWTH
INVERTED GROWTH INVERTED GROWTH
FIG 28. GROWTH PATTERNS IDENTIFIED IN ITERATION 1 32
BEFORE AGGREGATION (31% STABILITY)
AFTER AGGREGATION (77% STABILITY)
FIG 29. POINT GROWTH IN POINT 20X20 UNDER RULE(1236)
POINT GROWTH NON-UNIFORM GROWTH PATTERN MAY RESULT IN STUNTED OR SPARSE GROWTH
Point growth shows a non-uniform STABILIZATION STRATEGY growth pattern which may result in stunCANTILEVER AGGREGATION ted or sparce growth in some conditions. VERTICAL AGGREGATION
Ruleset EnvironmentLower Limit Environment Upper limit Ruleset Fertility Lower Limit EnvironmentLower Limit Fertility Upper Limit Environment Upper limit Width Fertility Lower Limit Length Fertility Upper Limit Height Width Population Length Alive Height Dead Population Percentage Alive Alive Pecentage Dead Dead Stability tolerance Percentage Alive Stable Pecentage Dead Unstable Stability tolerance Percentage Stable Stable Percentage Unstable Unstable Average 3D density Percentage Stable Average 2D density Percentage Unstable Average 3D density Average 2D density Percentage Voxel Age Color
2333 2 3 2333 3 2 3 3 50 3 50 3 15 50 65536 50 8244 15 57292 65536 12.58% 8244 87.42% 57292 1 12.58% 269 87.42% 7975 1 POINT 20X20 – RULE(1236) 3.26% 269 96.74% 8 HEIGHT 7975 38% TOTAL POPULATION – 2048 3.26% 51% 96.74% ALIVE – 708 (34%) 38% AVERAGE DENSITY – 40% 51% PERCENTAGE STABLE BEFORE AGGREGATION – 31%
PERCENTAGE STABLE AFTER AGGREGATION – 77%
Sparse Color - Dense Color Percentage Voxel Age Color
AGE: YOUNG - OLD
Stable Voxel Color Sparse Color - Dense Color Unstable Voxel Color
STABILITY: STABLE - UNSTABLE
Stable Voxel Color Unstable Voxel Color
33
BEFORE AGGREGATION (82% STABILITY)
AFTER AGGREGATION (84% STABILITY)
FIG 30. ARCHING IN POINT 30X30 UNDER RULE(1724)
ARCHING ARCHING BEHAVIOUR AT THE PERIPHERAL FACES ASTABILIZATION STRATEGY growth pattern that exhibits on its CANTILEVER AGGREGATION peripheral faces, staggered stacking of VERTICAL AGGREGATION cantilevered voxels one above the other is termed as arching. 34
Ruleset EnvironmentLower Limit Environment Upper limit Ruleset Fertility Lower Limit EnvironmentLower Limit Fertility Upper Limit Environment Upper limit Width Fertility Lower Limit Length Fertility Upper Limit Height Width Population Length Alive Height Dead Population Percentage Alive Alive Pecentage Dead Dead Stability tolerance Percentage Alive Stable Pecentage Dead Unstable Stability tolerance Percentage Stable Stable Percentage Unstable Unstable Average 3D density Percentage Stable Average 2D density Percentage Unstable Average 3D density Average 2D density Percentage Voxel Age Color
2333 2 3 2333 3 2 3 3 50 3 50 3 15 50 65536 50 8244 15 57292 65536 12.58% 8244 87.42% 57292 1 12.58% 269 87.42% 7975 1 POINT 30X30 – RULE(1724) 3.26% 269 96.74% HEIGHT 7975 8 38% POPULATION – 8192 TOTAL 3.26% 51% 96.74% ALIVE – 4872 (59%) 38% AVERAGE DENSITY – 58% 51% PERCENTAGE STABLE BEFORE AGGREGATION – 82%
PERCENTAGE STABLE AFTER AGGREGATION – 84%
Sparse Color - Dense Color Percentage Voxel Age Color
AGE: YOUNG - OLD
Stable Voxel Color Sparse Color - Dense Color Unstable Voxel Color
STABILITY: STABLE - UNSTABLE
Stable Voxel Color Unstable Voxel Color
BEFORE AGGREGATION (100% STABILITY)
AFTER AGGREGATION (100% STABILITY)
FIG 31. UNIFORM GROWTH IN X 20X20 UNDER RULE(1522)
UNIFORM GROWTH UNIFORM GROWTH WITH VOXELS AGING UNIFORMLY STABLE AT LOW HEIGHTS Uniform growth shows columner growth PRONE TO BUCKLING AT INCREASED HEIGHTS patterns which are stable at lower layers STABILIZATION STRATEGY but fail at higher layers due to the action ofVERTICAL AGGREGATION buckling.
Ruleset EnvironmentLower Limit Environment Upper limit Ruleset Fertility Lower Limit EnvironmentLower Limit Fertility Upper Limit Environment Upper limit Width Fertility Lower Limit Length Fertility Upper Limit Height Width Population Length Alive Height Dead Population Percentage Alive Alive Pecentage Dead Dead Stability tolerance Percentage Alive Stable Pecentage Dead Unstable Stability tolerance Percentage Stable Stable Percentage Unstable Unstable Average 3D density Percentage Stable Average 2D density Percentage Unstable Average 3D density Average 2D density Percentage Voxel Age Color
2333 2 3 2333 3 2 3 3 50 3 50 3 15 50 65536 50 8244 15 57292 65536 12.58% 8244 87.42% 57292 1 12.58% 269 87.42% 7975 1 X 20X20 – RULE(1522) 3.26% 269 96.74% HEIGHT 7975 8 38% POPULATION – 2048 TOTAL 3.26% 51% 96.74% ALIVE – 480 (23%) 38% AVERAGE DENSITY – 39% 51% PERCENTAGE STABLE BEFORE AGGREGATION – 100%
PERCENTAGE STABLE AFTER AGGREGATION – 100%
Sparse Color - Dense Color Percentage Voxel Age Color
AGE: YOUNG - OLD
Stable Voxel Color Sparse Color - Dense Color Unstable Voxel Color
STABILITY: STABLE - UNSTABLE
Stable Voxel Color Unstable Voxel Color
35
BEFORE AGGREGATION 34% STABILITY)
AFTER AGGREGATION (36% STABILITY)
FIG 32. UNIFORM GROWTH IN X 20X20 UNDER RULE(7711)
INVERTED GROWTH WITH INCREASING GENERATIONS VOXELS INCREASING IN NUMBER OCCURS IN RULESETS WHERE FERTLILITY RANGE EXCEEDS This growth pattern shows an increasing ENVIRONMENT RANGE number of voxels with increase in geneSTABILIZATION STRATEGY rations . This pattern can be distinctly INVARIABLY FAILS WITH ALL STRATEGIES observed in rulesets where the fertility range exceeds the environment range. This system invariably fails for all applied strategies due insufficient voxels at its base.
36
Ruleset EnvironmentLower Limit Environment Upper limit Ruleset Fertility Lower Limit EnvironmentLower Limit Fertility Upper Limit Environment Upper limit Width Fertility Lower Limit Length Fertility Upper Limit Height Width Population Length Alive Height Dead Population Percentage Alive Alive Pecentage Dead Dead Stability tolerance Percentage Alive Stable Pecentage Dead Unstable Stability tolerance Percentage Stable Stable Percentage Unstable Unstable Average 3D density Percentage Stable Average 2D density Percentage Unstable Average 3D density Average 2D density Percentage Voxel Age Color
2333 2 3 2333 3 2 3 3 50 3 50 3 15 50 65536 50 8244 15 57292 65536 12.58% 8244 87.42% 57292 1 12.58% 269 87.42% 7975 POINT X 20X20 – RULE(7711) 1 3.26% 269 HEIGHT 96.74% 8 7975 38% POPULATION – 2048 TOTAL 3.26% 51%– 480 (23%) ALIVE 96.74% AVERAGE DENSITY – 22% 38% 51% PERCENTAGE STABLE BEFORE AGGREGATION – 34%
PERCENTAGE STABLE AFTER AGGREGATION – 36%
Sparse Color - Dense Color Percentage Voxel Age Color
AGE: YOUNG - OLD
Stable Voxel Color Sparse Color - Dense Color Unstable Voxel Color
STABILITY: STABLE - UNSTABLE
Stable Voxel Color Unstable Voxel Color
DATA CONSOLIDATION The seeds in 10x10 resolutions didn‘t offer much growth and hence are eliminated. Also rulesets like Rule(3344), that have no alive population are eliminated. Hence, rulesets and seeds with active growths are consolidated and put for a clarified data graphing.
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SEED_X
SEED_X
LINE SEED_LINE SEED_LINE SEED_X
SEED_X SEED_POINTSEED_POINTSEED_POINTSEED_POINT SEED_POINT X SEED_POINT X SEED_P
RULESET – 1211, 1233, 1234, 1235, 1236, 1522, 1724, 2211, 2333, 2433, 2533, 2644, 3411, 5511, 7711
FIG 33. CONSOLIDATED SEEDS AND RULESETS FOR ITERATION 1 37
PERCENTAGE STABILITY WITH AGGREGATION
PERCENTAGE STABLE
PERCENTAGE STABILITY WITHOUT AGGREGATION AVERAGE 3D DENSITY
SEEDS
FIG 34. RULE BASED GRAPHS FOR ITERATION 1 - TEMPLATE
RULE BASED GRAPHING The consolidated data are then plotted in graphs that are rule-based with the seeds along X-axis and the percentage scale set along Y-axis. The plot are has three lines representing the average 3D density, the percentage stability without aggregation and the percentage stability after aggregation. The relationship betweent the three parameters for each ruleset seed combination are studied and the combinations are chosen for further iterations. 38
RULE(2533)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 35. RULE BASED GRAPHS FOR ITERATION 1 - RULE(2533)
Stable Voxel Color Unstable Voxel Color
39
RULE(1522)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 36. RULE BASED GRAPHS FOR ITERATION 1 - RULE(1522) 40
Stable Voxel Color Unstable Voxel Color
RULE(1211)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 37. RULE BASED GRAPHS FOR ITERATION 1 - RULE(1211)
Stable Voxel Color Unstable Voxel Color
41
RULE(1233)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 38. RULE BASED GRAPHS FOR ITERATION 1 - RULE(1233) 42
Stable Voxel Color Unstable Voxel Color
RULE(1724)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 39. RULE BASED GRAPHS FOR ITERATION 1 - RULE(1724)
Stable Voxel Color Unstable Voxel Color
43
RULE(1522) RULE(2644)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 40. RULE BASED GRAPHS FOR ITERATION 1 - RULE(2644) 44
Stable Voxel Color Unstable Voxel Color
RULE(1211) RULE(1234)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 41. RULE BASED GRAPHS FOR ITERATION 1 - RULE(1234)
Stable Voxel Color Unstable Voxel Color
45
RULE(2211)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 42. RULE BASED GRAPHS FOR ITERATION 1 - RULE(2211) 46
Stable Voxel Color Unstable Voxel Color
RULE(3411)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 43. RULE BASED GRAPHS FOR ITERATION 1 - RULE(3411)
Stable Voxel Color Unstable Voxel Color
47
RULE(1235)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 44. RULE BASED GRAPHS FOR ITERATION 1 - RULE(1235) 48
Stable Voxel Color Unstable Voxel Color
RULE(2333)
AVERAGE 3D DENSITY
PERCENTAGE STABILITY BEFORE AGGREGATION
2333 Ruleset 2 EnvironmentLower Limit 3 Environment Upper limit 3 Fertility Lower Limit 3 Fertility Upper Limit 50 Width 50 Length 15 Height 65536 Population 8244 Alive 57292 Dead 12.58% Percentage Alive 87.42% Pecentage Dead 1 Stability tolerance 269 Stable 7975 Unstable PERCENTAGE STABILITY AFTER AGGREGATION 3.26% Percentage Stable 96.74% Percentage Unstable 38% Average 3D density 51% Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 45. RULE BASED GRAPHS FOR ITERATION 1 - RULE(2333)
Stable Voxel Color Unstable Voxel Color
49
RULE(1235) RULE(5511)
Ruleset EnvironmentLower Limit Environment Upper limit Fertility Lower Limit Fertility Upper Limit Width Length Height Population Alive Dead Percentage Alive Pecentage Dead Stability tolerance Stable Unstable Percentage Stable Percentage Unstable Average 3D density Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 46. RULE BASED GRAPHS FOR ITERATION 1 - RULE(5511) 50
Stable Voxel Color Unstable Voxel Color
2333 2 3 3 3 50 50 15 65536 8244 57292 12.58% 87.42% 1 269 7975 3.26% 96.74% 38% 51%
RULE(2333) RULE(1236)
Ruleset EnvironmentLower Limit Environment Upper limit Fertility Lower Limit Fertility Upper Limit Width Length Height Population Alive Dead Percentage Alive Pecentage Dead Stability tolerance Stable Unstable Percentage Stable Percentage Unstable Average 3D density Average 2D density
2333 2 3 3 3 50 50 15 65536 8244 57292 12.58% 87.42% 1 269 7975 3.26% 96.74% 38% 51%
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 47. RULE BASED GRAPHS FOR ITERATION 1 - RULE(1236)
Stable Voxel Color Unstable Voxel Color
51
RULE(2433)
Ruleset EnvironmentLower Limit Environment Upper limit Fertility Lower Limit Fertility Upper Limit Width Length Height Population Alive Dead Percentage Alive Pecentage Dead Stability tolerance Stable Unstable Percentage Stable Percentage Unstable Average 3D density Average 2D density
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 48. RULE BASED GRAPHS FOR ITERATION 1 - RULE(2433) 52
Stable Voxel Color Unstable Voxel Color
2333 2 3 3 3 50 50 15 65536 8244 57292 12.58% 87.42% 1 269 7975 3.26% 96.74% 38% 51%
RULE(7711)
Ruleset EnvironmentLower Limit Environment Upper limit Fertility Lower Limit Fertility Upper Limit Width Length Height Population Alive Dead Percentage Alive Pecentage Dead Stability tolerance Stable Unstable Percentage Stable Percentage Unstable Average 3D density Average 2D density
2333 2 3 3 3 50 50 15 65536 8244 57292 12.58% 87.42% 1 269 7975 3.26% 96.74% 38% 51%
Percentage Voxel Age Color Sparse Color - Dense Color
FIG 49. RULE BASED GRAPHS FOR ITERATION 1 - RULE(7711)
Stable Voxel Color Unstable Voxel Color
53
RULE(1211)
RULE(1233)
RULE(12
RULE(1236)
RULE(1522)
RULE(17
RULE(1235) RULE(1233)
RULE(12
RULE(1234) RULE(1211)
RULE(2333)
RULE(2433)
RULE(25
RULE(1724) RULE(1236) RULE(1234) RULE(1211)
RULE(2211) RULE(1522) RULE(1235) RULE(1233)
RULE(17
RULE(3411)
RULE(5511)
RULE(77
RULE(2533) RULE(2333) RULE(1724) RULE(1236)
RULE(2644) RULE(2433) RULE(2211) RULE(1522)
RULE(25 RULE(17
RULE(12
CHOSEN RULESET – SEED COMBINATIONS ELIMINATED RULESETS RULE(7711) RULE(3411) RULE(2533) RULE(2333)
RULE(5511) RULE(2644) RULE(2433)
RULE(77
RULE(25
FIG 50. CHOOSING RULES AND SEEDS FOR FURTHER ITERATIONS 54
CHOSEN RULESET – SEED COMBINATIONS
RULE(1724)
RULE(2211)
RULE(1236) RULE(1234)
RULE(1522) RULE(1235)
RULE(2533)
RULE(2644)
RULE(2333) RULE(1724)
RULE(2433) RULE(2211)
RULE(17
RULE(25
CHOSEN RULESET – SEED COMBINATIONS ELIMINATED RULESETS RULE(7711) RULE(3411) RULE(2533)
RULE(5511) RULE(2644)
RULE(77
CHOSEN RULESET – SEED COMBINATIONS ELIMINATED RULESETS RULE(7711)
CHOSEN SAMPLES Through the analyzed graphs an inference is drawn upon the performance of voxel systems through multiple rule sets and seeds. The combinations showing consistently low figures of stability through various tests of aggregation are eliminated (Rule 1211, 2211, 3411, 5511, 7711). The preferred combinations selected to carry out further tests show high figures of stability and are highly receptive to aggregation, although combinations that are absolutely stable are eliminated as well. 55
ITERATION 2 The chosen ruleset seed combinations from iteration 1 are posed with additional strategies of aggregation and the results are graphed and studied 56
20x20 20x20
20x20 20x20
20x20 20x20
20x20 20x20
20x20 20x20
20x20 20x20
SEED_LINE SEED_X HEIGHT – HEIGHT –88
SEED_LINE SEED_X HEIGHT – HEIGHT –88
SEED_POINT SEED_X HEIGHT – HEIGHT –88
SEED_POINT SEED_X HEIGHT –88 HEIGHT –
SEED_POINT X SEED_POINT HEIGHT – HEIGHT –88
SEED_POINT X SEED_POINT HEIGHT –88 HEIGHT –
RULE(1234) RULE(1522) RULE(1235) RULE(1724) RULE(1236) RULE(1724) RULE(2433)
RULE(1522) RULE(1234) RULE(1724) RULE(1235) RULE(1236) RULE(1724) RULE(2433)
RULE(1234) RULE(1234) RULE(1235) RULE(1235) RULE(1236) RULE(1236) RULE(1724) RULE(1522) RULE(2433) RULE(1724) RULE(2533) RULE(2644)
RULE(1234) RULE(1234) RULE(1235) RULE(1235) RULE(1236) RULE(1236) RULE(1724) RULE(1522) RULE(2433) RULE(1724) RULE(2533) RULE(2644)
RULE(1234) RULE(1233) RULE(1235) RULE(1522) RULE(1236) RULE(1724) RULE(1522) RULE(2433) RULE(1724) RULE(2533) RULE(2533) RULE(2644)
RULE(1233) RULE(1234) RULE(1522) RULE(1235) RULE(1724) RULE(1236) RULE(2433) RULE(1522) RULE(2533) RULE(1724) RULE(2533) RULE(2644)
FIG 51. SEEDS AND RULES FOR ITERATION 2 57
MODEL WITH HEIGHT EXCEEDING BASE DIMENSION MODEL WITH HEIGHT GREATER THAN BASE DIMENSIONS
VERTICAL AGGREGATION VERTICAL AGGREGATION
VERTICAL AGGREGATION HORIZONTAL AGGREGATION
FIG 52. HORIZONTAL AGGREGATION VS VERTICAL AGGREGATION 58
NO AGGREGATION
VERTICAL AGGREGATION
HORIZONTAL AGGREGATION
FIG 53. HORIZONTAL AGGREGATION VS VERTICAL AGGREGATION UNDER GRAVITY
HORIZONTAL AGGREGATION Horizontal aggregation involves voxels in one layer getting connected to one another through face-to-face connections. They have a wider load-bearing area and hence work better in models with heights exceeding the base dimensions.
VERTICAL AGGREGATION Vertical aggregation involves voxels getting connected to the voxels beneath them through face-to-face aggregation. The aggregation leads to stranding with low load-bearing area and hence, buckling occurs at increasing heights. 59
OLD VOXEL LOWER MASS
YOUNG VOXEL HIGHER MASS AGE BASED MASSING AGE-BASED MASSING
CANTILEVER VOXEL LOWER MASS
NON-CANTILEVER VOXEL LOWER MASS CANTILEVER MASSING CANTILEVER MASSING
TOP LAYER LOWER MASS
BOTTOM LAYER HIGHER MASS LAYER BASED MASSING LAYER-BASED MASSING
FIG 54. DIFFERENTIAL MASSING STRATEGIES 60
BEFORE AGGREGATION
CANTILVER AGGREGATION + VERTICAL AGGREGATION
PERCENTAGE STABLE
HORIZONTAL AGGREGATION
LAYER-BASED DIFFERENTIAL MASSING
HORIZONTAL AGGREGATION + LAYER-BASED DIFFERENTIAL MASSING
FIG 55. PARAMETERS RECORDED AT FIVE STAGES OF EXPERIMENT
FIG 56. DIFFERENTIAL MASSING STRATEGIES 61
FIG 57. RULE(1234) ON POINT 20X20 (INCONSISTENT GROWTH)
FIG 58. RULE(1236) ON POINT 20X20 (CONSISTENT NON-UNIFORM GROWTH)
FIG 59. RULE(1235) ON POINT 20X20 (CONSISTENT NON-UNIFORM GROWTH)
FIG 60. RULE(1522) ON POINT 20X20 (COLUMNAR GROWTH) A - AGE MODEL G - ONLY GRAVITY C+V - CANTILEVER + VERTICAL AGGREGATION H - HORIZONTAL AGGREGATION H+L - HORIZONTAL AGGREGATION + LAYER-BASED MASSING L - LAYER-BASED MASSING
62
FIG 61. RULE(1724) ON POINT 20X20 (COLUMNAR GROWTH)
FIG 62. RULE(2644) ON POINT 20X20 (COLUMNAR GROWTH)
FIG 63. RULE(2533) ON POINT 20X20 (CONSISTENT GROWTH)
FIG 64. RULE(1234) ON X 20X20 (COLUMNAR GROWTH) A - AGE MODEL G - ONLY GRAVITY C+V - CANTILEVER + VERTICAL AGGREGATION H - HORIZONTAL AGGREGATION H+L - HORIZONTAL AGGREGATION + LAYER-BASED MASSING L - LAYER-BASED MASSING
63
FIG 65. RULE(1235) ON X 20X20 (COLUMNAR GROWTH)
FIG 66. RULE(1724) ON X20X20 (COLUMNAR GROWTH)
FIG 67. RULE(1236) ON X 20X20 (COLUMNAR GROWTH)
FIG 68. RULE(2433) ON X 20X20 (CONSISTENT NON-UNIFORM GROWTH) A - AGE MODEL G - ONLY GRAVITY C+V - CANTILEVER + VERTICAL AGGREGATION H - HORIZONTAL AGGREGATION H+L - HORIZONTAL AGGREGATION + LAYER-BASED MASSING L - LAYER-BASED MASSING
64
FIG 69. RULE(1233) ON POINT X 20X20 (COLUMNAR GROWTH)
FIG 70. RULE(1724) ON POINT X 20X20 (COLUMNAR GROWTH)
FIG 71. RULE(1522) ON POINT X 20X20 (COLUMNAR GROWTH)
FIG 72. RULE(2533) ON POINT X 20X20 (COLUMNAR GROWTH) A - AGE MODEL G - ONLY GRAVITY C+V - CANTILEVER + VERTICAL AGGREGATION H - HORIZONTAL AGGREGATION H+L - HORIZONTAL AGGREGATION + LAYER-BASED MASSING L - LAYER-BASED MASSING
65
FIG 73. RULE(2644) ON POINT X 20X20 (COLUMNAR GROWTH)
FIG 74. RULE(1724) ON LINE 20X20 (COLUMNAR GROWTH)
FIG 75. RULE(1522) ON LINE 20X20 (INSUFFICIENT GROWTH) A - AGE MODEL G - ONLY GRAVITY C+V - CANTILEVER + VERTICAL AGGREGATION H - HORIZONTAL AGGREGATION H+L - HORIZONTAL AGGREGATION + LAYER-BASED MASSING L - LAYER-BASED MASSING
CHOSEN SAMPLES Analyzing the selected seed and rule combinations through specific tests of aggregation yields three distinct form and growth observations with a significant impact to its stability counts. A selection process is further carried out where new peaks and lows are observed and unresponsive systems are eliminated. Systems showing insufficient growth patters are eliminated due to low figures of development. These systems show inadequate voxels for carrying out stability tests. Seeds developing in a columner fashion show uniform vertical growth with high stability counts, although these are eliminated from further consideration due to structural monotony and uninteresting growth patterns. The seeds exhibit a development pattern with consistent layer growth and adjescent layery patterns dissimilar to one and other, this growth system is carried forward for further examination due to its organic growth pattern. 66
INSUFFICIENT GROWTH
COLUMNAR GROWTH
CONSISTENT NON-UNIFORM GROWTH
FIG 76. ITERATION 2 - GROWTH PATTERNS IDENTIFIED 67
ITERATION 3 The four ruleset seed combinations with consistent non-uniform growth patterns are tested for their stability at increased heights, such as 100 layers. Extensive aggregation strategies and their combinations are put into aid for stabilization, and their subsequent effects and behaviour are studied. 68
20x20
20x20
SEED_X HEIGHT – 100
SEED_POINT HEIGHT – 100
RULE(2433)
RULE(1235) RULE(1236) RULE(2533)
FIG 77. SEEDS AND RULES FOR ITERATION 3 69
+
WITHOUT AGGREGATION
CANTILEVER + VERTICAL AGGREGATION
HORIZONTAL AGGREGATION
The neighbourhood system is observed to be the most unstable without the addition of external parameters. The process of applying gravity to a non aggregated system enables us to analyze points of failure and create sytems to attain stability.
The addition of this parameter helps to attain higher stability counts due to the face to face linkages of voxels on the verical axis, at the same time, cantilevering voxels that cannot form a link with a voxel underneath. However this system under performs when tested with greater number of layers due to the action of buckling leading it to a point of failure.
The parameter of horizontal aggregation significantly increases the stability of the system since it forms face to face linkages on the horizontal axis which aids in distributing the axial forces acting on the system over a larger surface area hence negating the action of buckling. But the system doesnâ&#x20AC;&#x2122;t attain absolute stability since voxels that cannot form a link collapse under gravity.
+
+
LAYER-BASED DIFFERENTIAL MASSING
LAYER-BASED DIFFERENTIAL MASSING + HORIZONTAL AGGREGATION
HORIZONTAL + CANTILEVER AGGREGATION
Adding layer based differential massing to a system reduces the collateral damage caused by unstables voxels on the stable ones when they collapse under gravity. This parameter does not create a significant amount of stability by itself although it acts as a catalyst when combined with another parameter.
A combination of layer based differential massing with horizontal aggregation creates a significant improvement in the stability counts of the system since the unstable voxels do not displace the aggregated ones when acted upon by gravity, at the same time it improves the stability of the upper layers by forming a stronger foundation due to a higher mass at the base. This set of parameters achieves a high amount of stability and perfect stability in some cases.
Highest amount of stability is achieved through a comninaton of these parameters since horizontal aggregation creates a higher global stability in the system by distributing the forces; on the other hand cantilevering helps the unsupported voxels to attach themselves to the aggregated ones. Through this strategy we achieve minimum points of failure and maximum amounts of totally stable systems.
70
POINT 20X20 – RULE(2533)
X 20X20 – RULE(2433)
HEIGHT - 100
HEIGHT - 100
ITERATION 3
ITERATION 3
FIG 78. COLUMNAR GROWTH PATTERN GROWTH PATTERNS
GROWTH PATTERNS
These samples are seen to develop columnar growth pattern at heights of 100 layers. Hence, they are eliminated for stability studies. 71
WITHOUT AGGREGATION (1.59% STABLE)
LAYER-BASED DIFFERENTIAL MASSING (2.00% STABLE)
CANTILEVER + VERTICAL AGGREGATION (99.75% STABLE)
HORIZONTAL AGGREGATION (87.09% STABLE)
HORIZONTAL AGGREGATION + LAYER-BASED DIFFERENTIAL MASSING (88.63% STABLE)
HORIZONTAL + CANTILEVER AGGREGATION (100.00% STABLE)
FIG 79. POINT 20X20 UNDER RULE(1235) - ITERATION 3 72
99,75%
100%
Point 20X20 Rule(1235)
87,09%
90%
100,00%
88,63%
80% 70% 60% 50% 40% 30% 20% 10% 2,00%
1,59% 0%
G
V+C
H
H+L
L
H+C
FIG 80. POINT 20X20 UNDER RULE(1235) - GRAPH
POINT 20x20 Rule(1235): Alive: 5440, Dead: 20160, Rule(1235): Alive: 5440, Dead: 20160, Total Population: 25600 Total Population: 25600 Average 3D density: 37% Average 3D density: 37% WITHOUT AGGREGATION WITHOUT AGGREGATION Stability : 1.59% Stability : 1.59% CANTILEVER + VERTICAL AGGREGATION CANTILEVER AGGREGATION Stability : 99.75% Stability : 99.75% HORIZONTAL AGGREGATION HORIZONTAL AGGREGATION Stability : 87.09% Stability : 87.09% LAYER BASED DIFFERENTIAL MASS LAYER BASED DIFFERENTIAL MASS Stability : 2.00% Stability : 2.00% HORIZONTAL AGGREGATION + LAYER-BASED DIFF.MASS HORIZONTAL AGGREGATION + LAYER-BASED DIFF.MASS Stability : 88.63% Stability : 88.63% HORIZONTAL + CANTILEVER AGGREGATION HORIZONTAL + CANTILEVER AGGREGATION Stability : 100.00% Stability : 100.00%
STABILITY ANALYSIS The model depicts high stability values under cantilver aggregation (99.75%) and horizontal aggregation (87.09%). Hence, while combining these two strategies, the model is seen to exhibit 100% stability. 73
WITHOUT AGGREGATION (2.70% STABLE)
LAYER-BASED DIFFERENTIAL MASSING (2.65% STABLE)
CANTILEVER + VERTICAL AGGREGATION (100.00% STABLE)
HORIZONTAL AGGREGATION (90.44% STABLE)
HORIZONTAL AGGREGATION + LAYER-BASED DIFFERENTIAL MASSING (90.54% STABLE)
HORIZONTAL + CANTILEVER AGGREGATION (100.00% STABLE)
FIG 81. POINT 20X20 UNDER RULE(1236) - ITERATION 3 74
100,00%
100%
Point 20X20 Rule(1236) 90,44%
90%
100%
90,54%
80% 70% 60% 50% 40% 30% 20% 10% 0%
2,70% G
2,65% V+C
H
H+L
L
H+C
FIG 82. POINT 20X20 UNDER RULE(1236) - GRAPH
POINT 20x20 Rule(1236): Alive: 3988, Dead: 21612, Rule(1236): Alive: 3988, Dead: 21612, Total Population: 25600 (57 LAYERS) Total Population: 25600 (57 LAYERS) Average 3D density: 40% Average 3D density: 40% WITHOUT AGGREGATION WITHOUT AGGREGATION Stability : 2.70% Stability : 2.70% CANTILEVER + VERTICAL AGGREGATION CANTILEVER AGGREGATION Stability : 100.00% Stability : 100.00% HORIZONTAL AGGREGATION HORIZONTAL AGGREGATION Stability : 90.44% Stability : 90.44% LAYER BASED DIFFERENTIAL MASS LAYER BASED DIFFERENTIAL MASS Stability : 2.65% Stability : 2.65% HORIZONTAL AGGREGATION + LAYER-BASED DIFF.MASS HORIZONTAL AGGREGATION + LAYER-BASED DIFF.MASS Stability : 90.54% Stability : 90.54% HORIZONTAL + CANTILEVER AGGREGATION HORIZONTAL + CANTILEVER AGGREGATION Stability : 100.00% Stability : 100.00%
STABILITY ANALYSIS The model stunts at 57 layers, nevertheless has an interesting growth pattern. It achieves 100% stability under cantilever aggregation and a combination of horizontal and cantilever aggregation strategies. 75
INCREASING RESOLUTION Following the success of the Point seed at 20x20 resolution under rules 1235 and 1236, the resolution is increased to 30x30 and acted upon by the same rules and strategies, to check their stability quotient. 76
20x20
30x30
SEED_POINT HEIGHT – 100
SEED_POINT HEIGHT – 100
RULE(1235) RULE(1236)
RULE(1235) RULE(1236)
FIG 83. INCREASING RESOLUTION OF POINT SEED 77
WITHOUT AGGREGATION (3.30% STABLE)
LAYER-BASED DIFFERENTIAL MASSING (3.58% STABLE)
CANTILEVER + VERTICAL AGGREGATION (90.32% STABLE)
HORIZONTAL AGGREGATION (89.38% STABLE)
HORIZONTAL AGGREGATION + LAYER-BASED DIFFERENTIAL MASSING (90.08% STABLE)
HORIZONTAL + CANTILEVER AGGREGATION (88.09% STABLE)
FIG 84. POINT 30X30 UNDER RULE(1235) - ITERATION 3 78
Point 30x30 Rule(1235) 100% 90,32%
90%
89,38%
90,08%
88,09%
80% 70% 60% 50% 40% 30% 20% 10%
3,58%
3,30%
0% G
V+C
H
H+L
L
H+C
FIG 85. POINT 30X30 UNDER RULE(1235) - GRAPH
POINT 30x30 Rule(1235): Alive: 12764, Dead: 89636, Rule(1235): Alive: 12764, Dead: 89636, Total Population: 102400 Total Population: 102400 Average 3D density: 40% Average 3D density: 40% WITHOUT AGGREGATION WITHOUT AGGREGATION Stability : 3.30% Stability : 3.30% CANTILEVER AGGREGATION CANTILEVER + VERTICAL AGGREGATION Stability : 90.32% Stability : 90.32% HORIZONTAL AGGREGATION HORIZONTAL AGGREGATION Stability : 89.38% Stability : 89.38% LAYER BASED DIFFERENTIAL MASS LAYER BASED DIFFERENTIAL MASS Stability : 3.58% Stability : 3.58% HORIZONTAL AGGREGATION + LAYER-BASED DIFF.MASS HORIZONTAL AGGREGATION + LAYER-BASED DIFF.MASS Stability : 90.08% Stability : 90.08% HORIZONTAL + CANTILEVER AGGREGATION HORIZONTAL + CANTILEVER AGGREGATION Stability : 88.09% Stability : 88.09%
STABILITY ANALYSIS The model exhibits thin columnar growth towards the higher layers. The cantiver and vertical aggregation strategy offers the highest stability figur of 90.32% with only the linear columnar elements buckling and failing. The horizontal aggregation strategy coupled with the layer-based differential massing strategy also offers a commendable stability value of 90.08%. 79
WITHOUT AGGREGATION (0.88% STABLE)
CANTILEVER + VERTICAL AGGREGATION (98.02% STABLE)
HORIZONTAL AGGREGATION (93.28% STABLE)
LAYER-BASED DIFFERENTIAL MASSING (1.34% STABLE)
HORIZONTAL AGGREGATION + LAYER-BASED DIFFERENTIAL MASSING (94.21% STABLE)
HORIZONTAL + CANTILEVER AGGREGATION (100.00% STABLE)
FIG 86. POINT 30X30 UNDER RULE(1236) - ITERATION 3 80
Point 30x30 Rule(1236) 98,02%
100%
93,28%
100%
94,21%
90% 80% 70% 60% 50% 40% 30% 20% 10%
1,34%
0,88% 0% G
V+C
H
H+L
L
H+C
FIG 87. POINT 30X30 UNDER RULE(1236) - GRAPH
POINT 30x30 Rule(1236): Alive: 32592, Dead: 69808, Rule(1236): Alive: 32592, Dead: 69808, Total Population: 102400 Total Population: 102400 Average 3D density: 41% Average 3D density: 41% WITHOUT AGGREGATION WITHOUT AGGREGATION Stability : 0.88% Stability : 0.88% CANTILEVER AGGREGATION CANTILEVER + VERTICAL AGGREGATION Stability : 98.02% Stability : 98.02% HORIZONTAL AGGREGATION HORIZONTAL AGGREGATION Stability : 93.28% Stability : 93.28% LAYER BASED DIFFERENTIAL MASS LAYER BASED DIFFERENTIAL MASS Stability : 1.34% Stability : 1.34% HORIZONTAL AGGREGATION + LAYER-BASED DIFF.MASS HORIZONTAL AGGREGATION + LAYER-BASED DIFF.MASS Stability : 94.21% Stability : 94.21% HORIZONTAL + CANTILEVER AGGREGATION HORIZONTAL + CANTILEVER AGGREGATION Stability : 100.00% Stability : 100.00%
STABILITY ANALYSIS The model exhibits a very consisten and healthy population which offers high stability values under the cantilever and vertical aggregation strategies (98.02%) and the horizontal aggregation and layer-based massing strategies (94.21%). When the horizontal and cantilever aggregation strategies are carried out together, the model achieves 100% stability. 81
POINT 20x20 – R(1235)
POINT 20x20 – R(1236)
100% STABLE UNDER HORIZONTAL + CANTILEVER AGGREGATION
100% STABLE UNDER HORIZONTAL + CANTILEVER AGGREGATION
FIG 88. STABILITY CREATED 82
POINT 3 POINT 30
90.3 UNDER VERT AGG
1236) – R(1236)
POINT 30x30 – POINT 30x30 – R(1235) R(1235) POINT 20x20 – R(1235)
TABLE E TAL + ZONTAL + GGREGATION GATION
90.39% STABLE 90.39% STABLE 100% STABLE UNDER VERTICAL + CANTILEVER UNDER VERTICAL + CANTILEVER UNDER HORIZONTAL + AGGREGATION AGGREGATION CANTILEVER AGGREGATION
POINT 30x30 – POINT 30x30 – R(1236) R(1236) POINT 20x20 – R(1236) 100% STABLE UNDER HORIZONTAL + CANTILEVER AGGREGATION
POINT 30
90.3 UNDER VERT AGG
Through this research and its multiple iterations, the rules and seeds in the cellular automaton Game of Life that have the inherent ability to produce sufficiently stable systems and are also positively receptive to further stabiliazation strategies are discovered. Also, the potential of various strategies in improving stability are analyzed and through amalgamations of Rules, seeds and strategies, stable systems are created and resolved. 83
© AADRL 2017