Neighborhood Topologies

Cellular Automata Tutor: Mustafa El Sayed Ilya Pereyaslavtsev Pavlina Vardoulaki Cosku ร inkiliรง Ahmed Shokir

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Table of content: - CA explained - Code - 2D Expermentation - 3D Stacking - Selective stacking - Probability map - Physical models

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STAY

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true --->> true 1 --->> 1

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Grid based system

Neighborhood system

Rule driven simulation

The grid logic is inextricably linked with 2d(3d) array grid, it stores and calculates information and it is also a method of computing voxels in space.

Array based systems allows us to find neighbors more easily than system without array hierarchy.

Every frame in the simulation allows the system to check the array and depending on the previous state make intelligent decisions based on its number of neighbors.

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The code infrastructure explores the basic CA system through logics like image base where the cellular automata follows a static image imprint. Age based where the Voxels have a set age limit after that they are eliminated. And Rule shifting where the Rules shift states between different rules accord to time limit.

Grid

Neighbourhood Time

Grid

Rules

Manual input

Image input

State

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0 1

Age

Age limit

Time

State

0

Image based

Neighbourhood Population Rules

Stacking

Age based

Selective stacking

Rule shifting

Rule shifting Age limit

Stacking

Selective stacking Display

Manual input

2D Cellular automata // Code

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The voxels rapidly change state from 1 to 0 to create the effect of “on and off�.

pulsation

The voxels are in a stable state, They are consistent and not changing position or state.

stable In a glider state the voxels are Change position in pair or triplet.

Transfer from one grid point to another in a Organized fashion.

gliders

In this condition the Voxels grow and constantly Change their boundary .

growing 6

2D Cellular automata patterns

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Growth_Rule13_ref3

Growth_Rule15_ref3

Growth_Rule22_ref7

Elimination_Rule3_ref1

Growth_Rule17_ref2

Growth_Rule24_ref2

Elimination_Rule12_ref1

Pulsation_Rule2d_ref1

Elimination_ Rule19_ref5

Pulsation_Rule2d_ref3 8

Growth_Rule18_ref1

Growth_Rule21_ref3

Growth_Rule21_ref2

Growth_Rule29_ref3

Growth_Rule29_ref2

Elimination_Rule20_ref5

Elimination_Rule37_ref1

Elimination_Rule28_ref3

Pulsation_Rule4_ref4

Pulsation_Rule32_ref1

2D Cellular automata patterns

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Max Living - 532 if (mystate== 1 && neighbour < 3) state=0; else if (mystate==1 && neighbour <= 5 && neighbour >=3 ) state=1; else if (mystate==1 &&neighbour > 6) state=0; else if (mystate==0 && neighbour== 3) state=1;

The voxels consistently grow until they reach the edge of the grid, Pulsation happens right after rapidly expanding until they reach saturation.

2D cellular automata patterns // Growth // Rule13_ref3

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Max Living - 476 if (mystate== 1 && neighbour < 2) state=0; //any conditions else if (mystate==1 && neighbour < 4 && neighbour>=2 ) state=1; else if (mystate==0 &&neighbour >= 3 && neighbour <7) state=1; else if (mystate==0 && neighbour > 7) state=1; else if (mystate==1 && neighbour == 9) state=1;

The voxels consistently grow from the coners of the grid until they connect, Pulsation happens with more voids in between the grid spaces.

2D cellular automata patterns // Growth // Rule21_ref4

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Living - 378 if (mystate== 1 && neighbour < 2) state=0; else if (mystate==1 && neighbour < 4 && neighbour>=2 ) state=1; else if (mystate==0 &&neighbour >= 3 && neighbour <5) state=1; else if (mystate==1 && neighbour > 5) state=0;

The voxels grow from the edges of the image creating different pulsation effects, in this model pulsation happens as the voxels grow.

2D cellular automata patterns // Pulsation // Rule 2D _ ref3

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Max Living - 354 if (mystate== 1 && neighbour < 2) state=0; else if (mystate==1 && neighbour < 4 && neighbour>=2 ) state=1; else if (mystate==0 &&neighbour >= 3 && neighbour <5) state=1; else if (mystate==1 && neighbour > 5) state=0;

The voxels grow from the corners with unpredicted behaviour the voxels tend to connect on more broader scope.

2D cellular automata patterns // Pulsation // Rule 4_ref4

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Max Living - 208 if (mystate== 1 && neighbour < 2) state=0; //any conditions else if (mystate==1 && neighbour < 3 && neighbour>=1 ) state=1; else if (mystate==0 &&neighbour >= 3 && neighbour <4) state=1; else if (mystate==1 && neighbour > 4) state=0;

The voxels are switching off and creating different patterns as the grid is they end up all eliminated.

2D cellular automata patterns // Elimination // Rule3_ref1

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Max Living - 152 if (mystate== 1 && neighbour < 3) state=0; //any conditions else if (mystate==1 && neighbour <= 4 && neighbour >=3 ) state=1; else if (mystate==1 &&neighbour > 4) state=0; else if (mystate==0 && neighbour== 3) state=1;

2D cellular automata patterns // Elimination // Rule12_ref1

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After learning the behaviours of cellular automata in two- dimension, we started to apply these patterns in three dimensional space to see the effects and how can we develop it more.

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3D cellular automata // Stacking

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Rule1_ref1

Rule2d_ref1

Rule2d_ref3

Rule3_ref1

Rule15_ref3

Rule17_ref2

Rule18_ref1

Rule20_ref5

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Rule3_ref3

Rule4_ref2

Rule4_ref4

Rule13_ref3

Rule21_ref3

Rule21_ref4

Rule24_ref2

Rule28_ref3

3D cellular automata // Stacking

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Rule1_ref1

Rule4_ref4

Rule13_ref3

if (mystate== 1 && neighbour < 2) state=0; else if (mystate==1 && neighbour < 4 && neighbour>=1 ) state=1; else if (mystate==0 &&neighbour >= 3 && neighbour <4) state=1; else if (mystate==1 && neighbour > 5) state=0;

if (mystate== 1 && neighbour < 2) state=0; else if (mystate==1 && neighbour <= 3 && neighbour>=2 ) state=1; else if (mystate==0 &&neighbour >= 3 && neighbour <=4) state=1; else if (mystate==1 && neighbour > 5) state=0;

if (mystate== 1 && neighbour < 3) state=0; else if (mystate==1 && neighbour <= 5 && neighbour >=3 ) state=1; else if (mystate==1 &&neighbour > 6) state=0; else if (mystate==0 && neighbour== 3) state=1;

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Rule15_ref3

Rule21_ref4

Rule24_ref2

if (mystate== 1 && neighbour < 2 && neighbour > 2) state=0; else if (mystate==1 && neighbour <= 4 && neighbour >=2 ) state=1; else if (mystate==1 &&neighbour > 4) state=0; else if (mystate==0 && neighbour== 3) state=1; else if (mystate==0 && neighbour== 8) state=1;

if (mystate== 1 && neighbour < 2) state=0; else if (mystate==1 && neighbour < 4 && neighbour>=2 ) state=1; else if (mystate==0 &&neighbour >= 3 && neighbour <7) state=1; else if (mystate==0 && neighbour > 7) state=1; else if (mystate==1 && neighbour == 9) state=1;

if (mystate== 1 && neighbour < 2 || neighbour ==8) state=0; else if (mystate==0 && neighbour < 4 && neighbour>=2 ) state=1; else if (mystate==0 &&neighbour >= 3 && neighbour <6) state=1; else if (mystate==0 && neighbour > 6) state=1; else if (mystate==1 && neighbour == 9) state=1;

Cellular automata // Stacking

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Cellular automata // Selective stacking In this stage we decided to apply Age limit, rule shifting and multiple states to gain more control over the system.

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Simple selective stacking Fractal surfaces

Fractal connections

Age rule shifting

Fractal trees

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Fractal trees

Manual input

Fractal trees

Cellular automata // Selective stacking

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Cellular automata // Simple selective stacking // Fractal surfaces

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Cellular automata // Simple selective stacking // Fractal surfaces

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Cellular automata // Simple selective stacking // Fractal surfaces

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Cellular automata // Simple selective stacking _Fractal connections

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Cellular automata // Simple selective stacking _Manual imput

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Cellular automata // Selective stacking Age rule shifting_Fractal connections

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Cellular automata // Selective stacking Rule shifting according to living voxels_Fractal trees

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Probability

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Grid

Neighbourhood Addition Ca system

Grid

External ruletion Ca system

manual input

Image input

Time

Rules

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Age

Time

State

probability map

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State Image based Neighbours Rules

Stacking

Age based

Rule shifting

selective stacking

Age Limit Simple stacking Selective stacking

Display

Manual input

probability

<

<

< 39

rules

growing pulsation Through exploration of diffirent rules behevioral patterns, various methouds of initial state inputs and few controls logics we came up with directions we interested in.

Ameba rule which keeps solid core and tends to grow throug pulsation prudeces outstanding pattern in a stacked model

elimination gliders blinking

input

control

all on/off image manual random resolution

rule shifting age limit population selective STACKING

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Ameba rule discovery

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Age limit

Surface forming by cells with less than 8 neighbors

8 8 8 8 8

8 8 8 8

8 8 8 8

8 8 8 8

8 8 8 8

8 8 8 8

Age limit was introduced as atempt to destibilize contsantly growing rools. CA cells witch stay without changes during certan simulatuon cycles eliminates.

Surface area forms by rules which could keep solid core. Due to this only cells having less than 8 neighbors stacks.

8 8 8

N

The second algotithm to form surface from solid-core rules. This method allows only recently borned cells be stacked.

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N N N

Surface forming by new cells Both algorithms were combined to produce surface in finals models

N N

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N N

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This rule tends to grow and fills all grid size given. As an istrument to regulate growing areas of this rule a age limit elimitation was used. But amount of control it gives was not enought.

Control through age limit

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data

representation

NULL NULL NULL NULL NULL NULL NULL NULL NULL NULL

2d data array

born rules

NULL NULL NULL NULL NULL

100% B

0% S

75 B

25 S

50 B

50 S

25 B

75 S

0%B

100% S

NULL NULL NULL NULL NULL NULL NULL NULL NULL NULL

filling each frame 0.0 - 1.0 0% - 100%

cleaning each frame val * 0.8 val < 0.3 = 0

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Probability map was introduced as more intellegent way to limit growing area of rule inside of the array grid. Probability map - 2d array of data filled with numbers maped to 0.0-1.0 range. Each frame the map could be filled by external logic. In our case the simplest point movement inside of the grid was took as the filling logic.

ca rule

stay rules

CA rules were divided into two parts to be affected by probabiblty map. The first part, calling borning applies only on hight probability areas. Chance that this part would be applied vatiates from 0 to 100% each frame. If the borning part of rule didnâ&#x20AC;&#x2122;t apply the state part of rule applies.

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Ipos

speed I = +2

power I = 1

speed I = +2

power I = 2

power J = 1

speed J = 1

Ipos

Jpos

speed J = 1

speed J = 0

Jpos

Through these five logics we were able to control the probability map and how it can create different effects using cellular automata. Position start, speed of movement, power effect on the array grid and how it reacts when it reachs the boundry of the grid.

Probability map logic

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bouncing

bouncing full

These different effects where tested with careful documentation and rigor.

going through

pulsation

pulsation 2 direction

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pulsation

1 point rotation

2 point rotation 47

masses 1

masses 2

masses 3 48

Length verses number of Edges (cantilevers)

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texture 1

texture 2

texture 3 52

texture verses Richness

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folds 1

folds 2

folds 3 56

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Surface area vs most likable

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Physical models

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Physical models

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