The 1D elementary CA model code is in the block below. It returns a matrix of cell states, with each system state a row in the matrix.

The key function is one_d_ca(rule, n, density, t), which returns a t\(\times\)n matrix, with initial density of 1’s given by the density parameter, and the update rule being Wolfram code rule.

Code

library(dplyr)# initialise random state, specified size and densityinit_state <-function(n =50, density =0.5) {as.numeric(runif(n) < density)}# pre- and post-fix the last and first elements of the statewrap_state <-function(x) {c(tail(x, 1), x, x[1])}# binary vector to decimal integer (most significant bits first)as_decimal <-function(b) {sum((2^ ((length(b) -1):0)) * b)}# convert decimal integer to vector of bits (most significant first)as_binary <-function(n, n_bits =8) { result <-c()for (i in1:n_bits) { result <-c(n %%2, result) n <- n %/%2 } result}rule_description <-function(rule_num) {paste(rule_num, paste(as_binary(rule_num), collapse =""), sep =": ")}# takes a matrix of CA states (each row a timestep) and appends# a new row for the next system stategeneration <-function(states, rule) { state <-tail(states, 1) %>%c() w_state <-wrap_state(state) result <-c()for (i in1:length(state)) { nbhd <- w_state[seq(i, i +2)] result <-c(result, rule[as_decimal(nbhd) +1]) }matrix(c(states %>%t() %>%c(), result), nc =ncol(states), byrow =TRUE)}# generate a matrix of cell states where each row is the system stateone_d_ca <-function(rule =30, n =64, density =0.5, t =192) { s <-init_state(n, density) ss <-matrix(s, nr =1)for (i in2:t) { ss <-generation(ss, rev(as_binary(rule))) } ss}

Making plots

Figure 8.2

The matrix format is convenient for computation, and fine for plotting a single model result but that has to be converted to a dataframe to be plotted using ggplot2. So here is the pure R version of Figure 8.2, but with time running from top to bottom, more cells, and more time.

Code

library(ggplot2)library(reshape2)one_d_ca(rule =110, n =192, density =0.5, t =384) %>% reshape2::melt(c("t", "x"), value.name ="state") %>%mutate(state =as.logical(state)) %>%ggplot() +geom_raster(aes(x = x, y =-t, fill = state)) +scale_fill_brewer(palette ="Paired") +coord_equal() +theme_void()

Figure 8.1

And here are the results for all Wolfram’s elementary 1D CA.

Code

# convenience function to return a dataframe instead of a matrixget_df <-function(rule, n =64, density =0.5, t =192) {one_d_ca(rule = rule, n = n, density = density, t = t) %>% reshape2::melt(c("t", "x"), value.name ="state") %>%mutate(state =as.logical(state),rule = rule)}# The 32 rules that match Wolfram's selection criteriarules <-c( 0, 4, 18, 22, 32, 36, 50, 54,72, 76, 90, 94, 104, 108, 122, 126,128, 132, 146, 150, 160, 164, 178, 182,200, 204, 218, 222, 232, 236, 250, 254)df <-get_df(rules[1], t =128)for (rule in rules[2:32]) { df <-bind_rows(df, get_df(rule, t =128))}# give the rules a better labeldf$rule <-factor(df$rule, levels = rules,labels =lapply(rules, rule_description))ggplot(df) +geom_raster(aes(x = t, y = x, fill = state)) +scale_fill_brewer(palette ="Paired") +coord_equal() +guides(fill ="none") +facet_wrap( ~ rule, ncol =4) +theme_void()

Code

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