Voxel Neighbourhood Stability Study

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.

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CONTENTS Research parameters

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Initial experimentation

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Iteration 1

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Iteration 2

56

Iteration 3

68

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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

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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.

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10X10

10X1010X10

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10X10

10X1010X10

10X10

10X10

10X1010X10

10X10

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20x2020x20

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20x2020x20

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20x2020x20

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30x3030x30

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30x30

30x30 30x30

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30x30

30x30 30x30

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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 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.

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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

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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

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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

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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

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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.

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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

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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.

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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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10X10

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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%.

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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.

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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.

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© AADRL 2017