One Dimensional Cellular Automata

Elementary Cellular Automata

Elementary Cellular Automata (ECA) have a radius of one and can be in only two possible states. Here we show a couple of examples:

Rule 18

using CellularAutomata, Plots

states = 2
radius = 1
generations = 50
ncells = 111
starting_val = zeros(Bool, ncells)
starting_val[Int(floor(ncells/2)+1)] = 1

rule = 18

ca = CellularAutomaton(DCA(rule), starting_val, generations)

heatmap(ca.evolution,
    yflip=true,
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false)

Rule 30

states = 2
radius = 1
generations = 50
ncells = 111
starting_val = zeros(Bool, ncells)
starting_val[Int(floor(ncells/2)+1)] = 1

rule = 30

ca = CellularAutomaton(DCA(rule), starting_val, generations)

heatmap(ca.evolution,
    yflip=true,
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false)

Multiple States Cellular Automata

General Cellular Automata have the same rule of ECA but they can have a radius larger than unity and/or a number of states greater than two. Here are provided examples for every possible permutation, starting with a Cellular Automaton with 3 states.

Rule 7110222193934

using CellularAutomata, Plots

states = 3
radius = 1
generations = 50
ncells = 111
starting_val = zeros(ncells)
starting_val[Int(floor(ncells/2)+1)] = 2

rule = 7110222193934

ca = CellularAutomaton(DCA(rule,states=states,radius=radius),
                       starting_val, generations)

heatmap(ca.evolution,
    yflip=true,
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false,
    size=(ncells*10, generations*10))

Larger Radius Cellular Automata

The following examples shows a Cellular Automaton with radius=2, with two only possible states:

Rule 1388968789

using CellularAutomata, Plots

states = 2
radius = 2
generations = 30
ncells = 111
starting_val = zeros(ncells)
starting_val[Int(floor(ncells/2)+1)] = 1

rule = 1388968789

ca = CellularAutomaton(DCA(rule,states=states,radius=radius),
                           starting_val, generations)

heatmap(ca.evolution,
    yflip=true,
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false,
    size=(ncells*10, generations*10))

And finally, three states with a radius equal to two:

Rule 914752986721674989234787899872473589234512347899

states = 3
radius = 2
generations = 30
ncells = 111
starting_val = zeros(ncells)
starting_val[Int(floor(ncells/2)+1)] = 2

rule = 914752986721674989234787899872473589234512347899

ca = CellularAutomaton(DCA(rule,states=states,radius=radius),
                       starting_val, generations)

heatmap(ca.evolution,
    yflip=true,
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false,
    size=(ncells*10, generations*10))

It is also possible to specify asymmetric neighborhoods, giving a tuple to the kwarg detailing the number of neighbors to considerate at the left and right of the cell: Rule 1235

states = 2
radius = (2,1)
generations = 30
ncells = 111
starting_val = zeros(ncells)
starting_val[Int(floor(ncells/2)+1)] = 1

rule = 1235

ca = CellularAutomaton(DCA(rule,states=states,radius=radius),
                       starting_val, generations)

heatmap(ca.evolution,
    yflip=true,
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false,
    size=(ncells*10, generations*10))

Totalistic Cellular Automata

Totalistic Cellular Automata takes the sum of the neighborhood to calculate the value of the next step.

Rule 1635

using CellularAutomata, Plots
states = 3
radius = 1
generations = 50
ncells = 111
starting_val = zeros(Integer, ncells)
starting_val[Int(floor(ncells/2)+1)] = 1

rule = 1635

ca = CellularAutomaton(DCA(rule, states=states),
                       starting_val, generations)

heatmap(ca.evolution,
    yflip=true,
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false)

Rule 107398

states = 4
radius = 1
generations = 50
ncells = 111
starting_val = zeros(Integer, ncells)
starting_val[Int(floor(ncells/2)+1)] = 1

rule = 107398

ca = CellularAutomaton(DCA(rule, states=states),
                       starting_val, generations)

heatmap(ca.evolution,
    yflip=true,
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false)

Here are some results for a bigger radius, using a radius of 2 as an example.

Rule 53

states = 2
radius = 2
generations = 50
ncells = 111
starting_val = zeros(Integer, ncells)
starting_val[Int(floor(ncells/2)+1)] = 1

rule = 53

ca = CellularAutomaton(DCA(rule, radius=radius), 
                           starting_val, generations)

heatmap(ca.evolution, 
    yflip=true, 
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false)

Continuous Cellular Automata

Continuous Cellular Automata work in the same way as the totalistic but with real values. The examples are taken from the already mentioned book NKS.

Rule 0.025

using CellularAutomata, Plots

generations = 50
ncells = 111
starting_val = zeros(Float64, ncells)
starting_val[Int(floor(ncells/2)+1)] = 1.0

rule = 0.025

ca = CellularAutomaton(CCA(rule), starting_val, generations)

heatmap(ca.evolution,
    yflip=true,
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false)

Rule 0.2

radius = 1
generations = 50
ncells = 111
starting_val = zeros(Float64, ncells)
starting_val[Int(floor(ncells/2)+1)] = 1.0

rule = 0.2

ca = CellularAutomaton(CCA(rule, radius=radius),
                       starting_val, generations)

heatmap(ca.evolution,
    yflip=true,
    c=cgrad([:white, :black]),
    legend = :none,
    axis=false,
    ticks=false)