Cellular Automata Simulator Form
Example Data Table
This example uses rule 90, width 15, 6 generations, wrap boundary, and a single active starting cell.
| Generation | State | Active Cells | Density |
|---|---|---|---|
| 0 | 000000010000000 | 1 | 6.67% |
| 1 | 000000101000000 | 2 | 13.33% |
| 2 | 000001000100000 | 2 | 13.33% |
| 3 | 000010101010000 | 4 | 26.67% |
| 4 | 000100000001000 | 2 | 13.33% |
| 5 | 001010000010100 | 4 | 26.67% |
Formula Used
The simulator uses an elementary one-dimensional binary cellular automaton. Each next cell state depends on the current left, center, and right cells.
Neighborhood encoding: p = 4L + 2C + R
Update rule: s(t+1, i) = bit(rule, p)
Here, L, C, and R are binary values from the current generation.
The encoded neighborhood p ranges from 0 to 7, and the chosen rule decides whether the next cell becomes 0 or 1.
Additional metrics are calculated from the simulated rows, including total active cells, average density, repeated states, stability detection, and average cell transitions between consecutive generations.
How to Use This Calculator
- Enter a rule number between 0 and 255.
- Choose the width and number of generations to simulate.
- Select a boundary method for edge handling.
- Pick the starting pattern type.
- Provide optional random density, block size, or custom seed values.
- Press Run Simulation to generate the grid and summary table.
- Review density, stability, repeat behavior, and active cell trends.
- Download the CSV or PDF file for reporting or reuse.
Frequently Asked Questions
1. What does the rule number control?
The rule number defines how each three-cell neighborhood transforms into the next state. Different rule values create very different behaviors, from static rows to chaotic and repeating patterns.
2. What is the difference between wrap and fixed boundaries?
Wrap boundaries connect the left and right edges, making the row circular. Fixed boundaries treat missing neighbors outside the grid as zeros, which usually changes edge growth behavior.
3. Why do random runs change between submissions?
Random mode generates a fresh starting row each time the form is submitted. That means the same rule can produce different visual outcomes because the initial pattern changed.
4. What does average density mean?
Average density is the mean proportion of active cells across all simulated generations. It helps compare sparse rules, dense rules, and patterns that shift their activity over time.
5. How is stability detected?
A stable generation is reported when a computed next row exactly matches the current row. This indicates that the automaton has stopped changing under the selected rule and boundary conditions.
6. What counts as a repeated state?
A repeated state appears when a generation matches any earlier row. This can reveal cycles, oscillations, or returns to previous structures, depending on the rule and starting conditions.
7. Can I use a custom seed shorter than the grid width?
Yes. The custom binary seed can be shorter than the width. It will be placed either at the center or left edge, while the remaining cells stay inactive.
8. When should I use CSV instead of PDF?
Use CSV when you want the full generation table for spreadsheets or analysis. Use PDF when you need a concise, readable summary for sharing, printing, or documentation.