Intro to 1D Cellular Automata

Rules and Space-Time Diagrams

A 1D cellular automaton updates each cell using its local neighborhood. We will use elementary 3-cell rules, such as Rule 30.

Figure 1: Rule 30 space-time diagram.

Define the rule

A rule is an 8-bit binary table that maps each neighborhood to a new cell value.

rule_number = 30
rule_bin = np.array([int(x) for x in f"{rule_number:08b}"])

Apply the rule (template)

def apply_rule(state, rule_bin):
    new_state = np.zeros_like(state)
    for i in range(1, len(state) - 1):
        neighborhood = state[i - 1 : i + 2]
        index = 7 - int("".join(neighborhood.astype(str)), 2)
        new_state[i] = rule_bin[index]
    return new_state

Build a space-time diagram

iterations = 60
grid_size = 101

grid = np.zeros((iterations, grid_size), dtype=int)
grid[0, grid_size // 2] = 1

for t in range(1, iterations):
    grid[t] = apply_rule(grid[t - 1], rule_bin)

plt.imshow(grid, cmap="binary", interpolation="nearest", aspect="auto")
plt.show()

Interactive initial row

The script amlab/cellular_automata/cellular.py lets you click on the first row to toggle cells and recompute the automaton.

---
title: "Intro to 1D Cellular Automata"
subtitle: "Rules and Space-Time Diagrams"
format: html
---

A 1D cellular automaton updates each cell using its local neighborhood. We will use elementary 3-cell rules, such as Rule 30.

```{python}
#| label: fig-ca-rule30
#| fig-cap: 'Rule 30 space-time diagram.'
#| fig-width: 7
#| fig-height: 4
#| echo: false

import numpy as np
import matplotlib.pyplot as plt

from amlab.cellular_automata.cellular import apply_rule

rule_number = 30
rule_bin = np.array([int(x) for x in f"{rule_number:08b}"])

iterations = 60
grid_size = 101

grid = np.zeros((iterations, grid_size), dtype=int)
grid[0, grid_size // 2] = 1

for t in range(1, iterations):
    grid[t] = apply_rule(grid[t - 1], rule_bin)

fig, ax = plt.subplots(figsize=(7, 4))
ax.imshow(grid, cmap="binary", interpolation="nearest", aspect="auto")
ax.set_xlabel("Cell index")
ax.set_ylabel("Time")
plt.show()
plt.close()
```

## Define the rule

A rule is an 8-bit binary table that maps each neighborhood to a new cell value.

```python
rule_number = 30
rule_bin = np.array([int(x) for x in f"{rule_number:08b}"])
```

## Apply the rule (template)

```python
def apply_rule(state, rule_bin):
    new_state = np.zeros_like(state)
    for i in range(1, len(state) - 1):
        neighborhood = state[i - 1 : i + 2]
        index = 7 - int("".join(neighborhood.astype(str)), 2)
        new_state[i] = rule_bin[index]
    return new_state
```

## Build a space-time diagram

```python
iterations = 60
grid_size = 101

grid = np.zeros((iterations, grid_size), dtype=int)
grid[0, grid_size // 2] = 1

for t in range(1, iterations):
    grid[t] = apply_rule(grid[t - 1], rule_bin)

plt.imshow(grid, cmap="binary", interpolation="nearest", aspect="auto")
plt.show()
```

## Interactive initial row

The script [amlab/cellular_automata/cellular.py](../../amlab/cellular_automata/cellular.py) lets you click on the first row to toggle cells and recompute the automaton.