In previousposts, I have examined how variations in cellular automaton rule application methods produce distinct changes in resulting pattern formations. One such exploration demonstrated the effects of applying rules to a circular grid topology, where the cellular automaton grid wraps around such that after processing the final cell in a row, the algorithm returns to the initial cell of that same row.
Building upon this foundation, another fundamental question emerges regarding the directional methodology of rule application. In computational practice, particularly in data processing applications, 2D matrix values are conventionally read from left to right, progressing systematically from the leftmost element to the rightmost element in each row.
What occurs when this conventional approach is reversed, specifically when values are processed from right to left?
Having previously investigated bottom-to-top rule application in cellular automaton grids, the logical next step involves examining right-to-left processing patterns.
This post explores the application of cellular automaton rules in a right-to-left direction while maintaining the original rule specifications unchanged. The analysis will encompass both grid topologies:
1. Circular Grid Configuration: Where boundaries wrap around to create continuous connectivity 2. Closed-Ended Grid Configuration: Where boundaries remain fixed without wrapping
This post aims to determine whether directional processing affects pattern emergence and evolution in cellular automata systems, while preserving the fundamental rule structures that govern cellular state transitions.
Closed-Ended Grid Configuration Analysis
The closed-ended grid configuration has been previously demonstrated to generate subtly different patterns compared to circular grid configurations. This section examines the application of a specific cellular automaton rule processed in both left-to-right and right-to-left directions. The analysis reveals that the resulting patterns exhibit mirror symmetry, consistent with the directional reversal applied.
Rule 2
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Evolution of Elementary Cellular Automaton Rule 2, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 2, starting from a single active cell at the top, with rule applied from right to left.
Rule 28
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Evolution of Elementary Cellular Automaton Rule 28, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 28, starting from a single active cell at the top, with rule applied from right to left.
Rule 45
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Evolution of Elementary Cellular Automaton Rule 45, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 45, starting from a single active cell at the top, with rule applied from right to left.
Rule 60
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Evolution of Elementary Cellular Automaton Rule 60, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 60, starting from a single active cell at the top, with rule applied from right to left.
Rule 145
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Evolution of Elementary Cellular Automaton Rule 145, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 145, starting from a single active cell at the top, with rule applied from right to left.
Rule 149
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Evolution of Elementary Cellular Automaton Rule 149, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 149, starting from a single active cell at the top, with rule applied from right to left.
Rule 177
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Evolution of Elementary Cellular Automaton Rule 177, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 177, starting from a single active cell at the top, with rule applied from right to left.
Rule 193
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Evolution of Elementary Cellular Automaton Rule 193, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 193, starting from a single active cell at the top, with rule applied from right to left.
Rule 195
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Evolution of Elementary Cellular Automaton Rule 195, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 195, starting from a single active cell at the top, with rule applied from right to left.
Rule 225
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Evolution of Elementary Cellular Automaton Rule 225, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 225, starting from a single active cell at the top, with rule applied from right to left.
Rule 230
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Evolution of Elementary Cellular Automaton Rule 230, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 230, starting from a single active cell at the top, with rule applied from right to left.
Rule 238
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Evolution of Elementary Cellular Automaton Rule 238, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 238, starting from a single active cell at the top, with rule applied from right to left.
Comparative Analysis: Circular vs. Closed-Ended Configurations
A systematic comparison between results obtained from circular grid configurations and closed-ended grid configurations demonstrates notable but subtle differences in pattern formation. Despite these variations in boundary conditions, both configurations produce mirror images when processing direction is reversed, though with some exceptions (for example, rule 177).
Rule 215
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Evolution of Elementary Cellular Automaton Rule 215, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 215, starting from a single active cell at the top, with rule applied from right to left.
Rule 149
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Evolution of Elementary Cellular Automaton Rule 149, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 149, starting from a single active cell at the top, with rule applied from right to left.
Rule 177
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Evolution of Elementary Cellular Automaton Rule 177, starting from a single active cell at the top, with rule applied from left to right.
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Evolution of Elementary Cellular Automaton Rule 177, starting from a single active cell at the top, with rule applied from right to left.
Directionally Invariant Patterns
The article concludes with an examination of patterns that remain unchanged regardless of left-to-right or right-to-left rule application. These directionally invariant formations represent a unique subset of cellular automaton behaviors that maintain structural consistency independent of processing direction.
Rule 18
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Evolution of Elementary Cellular Automaton Rule 18, starting from a single active cell at the top, with rule applied from left to right (or from right to left).
Rule 105
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Evolution of Elementary Cellular Automaton Rule 105, starting from a single active cell at the top, with rule applied from left to right (or from right to left).
Rule 150
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Evolution of Elementary Cellular Automaton Rule 150, starting from a single active cell at the top, with rule applied from left to right (or from right to left).
Rule 182
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Evolution of Elementary Cellular Automaton Rule 182, starting from a single active cell at the top, with rule applied from left to right (or from right to left).
Substantial opportunities remain for further exploration, particularly regarding the integration of probabilistic elements into directional processing variations. The interaction between processing direction and stochastic rule application represents a promising avenue for future cellular automaton articles.
