Cellular Automaton

Abstract

We have an arrangement of atoms/cells. Each cell belongs to one in a finite set of states. With the progression of a unit time cells change their state in a parallel fashion determined by the rules of the system.

Depending on the initial state and thr rules of the system we get a variety of behaviours. Some systems lead to homogenous states, some exhibit periodic oscillations between states, some are chaotic non-deterministic behaviours, ,.etc . The study of these discrete and abstract computational systems is called cellular automaton. is a great resource to understand and explore these systems.

Game of Life

Game of Life is a popular cellular automaton devised by British mathematician John Conway. It consists of a two dimensional grid of cells with a start state with each cell being either dead or live at any point in time.

The rules of the system are as follows :-

• A cell with less than two neighbours dies due to underpopulation.
• A cell with two or three neighbours lives.
• A cell with greater than three neighbours dies by overpopulation.
• A dead cell with three live neighbours becomes live.

We will implement this game of life in minecraft to generate some interesting patterns.

using PiCraft

First let's write our function which will transform from one state to another.

function nextState(A::Array{Bool, 2})
    B = similar(A)
    dims = size(A)
    for i = 1:dims[1], j = 1:dims[2]
        liveNeighbours = 0
        for p = -1:1, q = -1:1
            if checkbounds(Bool, A, i + p, j + q) && (p == 0 && q == 0)
                liveNeighbours += 1
            end
        end
        if A[i, j] == false && liveNeighbours == 3
            B[i, j] = true
        elseif liveNeighbours < 2 || liveNeighbours > 3
            B[i, j] = false
        end
    end
    return B
end

Now to render an arbitrary 2-D board in Minecraft Let live cells be white wool and dead be black wool

function displayState(p::Tuple{Int, Int, Int}, A::Array{Bool, 2})
    dims = size(A)
    for i = 1:dims[1], j = 1:dims[2]
        if A[i, j]
            setBlock(p .+ (i, j, 0), Block(35))
        else
            setBlock(p .+ (i, j, 0), Block(35, 15))
        end
    end
end

p = getTile() .+ (3, 4, 3)
A = rand(Bool, 10, 10)

for i = 1:20
    displayState(A)
    A = nextState(A)
    sleep(0.5)
end

A variety of interesting patterns can be found It is encouraged that you experiment with various patterns.

3-D implementation

If we want to consider a 3-D implementation of the Game of Life then we must realise that in 3-D a cell will have 26 Neighbours (3^3 - 1).

We generally express a 3-D system as a Life wxyz system. The rules of such a system are :

• The number of live neighbours must be between w and x to sustain life.
• The number of live neighbours must be between y and z to bring a dead back to life.

Conway's Game of Life is hence a Life 2333 system. Since a large number of combinations are possible they have been studied extensively for their properties. Most systems quickly collapse or explode. Life 4555 and 5766 are the only valid systems which exhibit life like properties.

Let's implement Life 4555. For that we will rewrite the nextState and displayState funcitons.

function nextState(A::Array{Bool, 3})
    B = similar(A)
    dims = size(A)
    for i = 1:dims[1], j = 1:dims[2], k = 1:dims[3]
        liveNeighbours = 0
        for p = -1:1, q = -1:1, r = -1:1
            if checkbounds(Bool, A, i + p, j + q, k + r) && (p == 0 && q == 0 && r == 0)
                liveNeighbours += 1
            end
        end
        if A[i, j, k] == false && liveNeighbours == 5
            B[i, j, k] = true
        elseif liveNeighbours < 4 || liveNeighbours > 5
            B[i, j, k] = false
        end
    end
    return B
end

function displayState(p::Tuple{Int, Int, Int}, A::Array{Bool, 3})
    dims = size(A)
    for i = 1:dims[1], j = 1:dims[2], k = 1:dims[3]
        if A[i, j, k]
            setBlock(p .+ (i, j, k), Block(35))
        else
            setBlock(p .+ (i, j, k), Block(35, 15))
        end
    end
end

p = getTile() .+ (3, 4, 3)
A = rand(Bool, 7, 7, 7)

for i = 1:20
    displayState(A)
    A = nextState(A)
    sleep(0.5)
end

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