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<p>The phenomenon of self-organization, where structures spontaneously form without external instruction, is ubiquitous in nature and holds immense potential for engineering fields. However, elucidating the self-organizing mechanisms behind the complex, hierarchical structures seen in biological organisms remains a critical challenge. This study addresses this by proposing a hierarchical extension of the Ishida model, a two-dimensional cellular automaton (CA), to generate complex forms through simple, localized interactions between neighboring elements. The model employs a multi-level hierarchical structure where transition rules are applied sequentially to each layer, often with a time lag, to promote stable pattern formation and prevent unintended side effects. By adjusting a small set of core parameters (notably <i>w</i><sub><i>1</i></sub> and <i>w</i><sub><i>2</i></sub>), the model successfully generated a variety of static and dynamic hierarchical patterns. Specifically, it first produced a double-ring structure, which was then internally segmented into patterns with two, four, or six divisions. Furthermore, using an asymmetric initial configuration, the model generated dynamic patterns that maintained their shape while in motion, as well as forms with controlled, elongating protrusions. These results demonstrate that simple, iterative operations can create diverse, complex morphologies, offering a valuable framework for synthetic morphology and potential applications in fields like modular swarm robotics.</p>

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Cellular automata model that generates hierarchical forms by simple operations between neighboring elements

  • 期刊论文

  • 发表日期：

    2025年8月12日

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  • Takeshi Ishida

摘要

The phenomenon of self-organization, where structures spontaneously form without external instruction, is ubiquitous in nature and holds immense potential for engineering fields. However, elucidating the self-organizing mechanisms behind the complex, hierarchical structures seen in biological organisms remains a critical challenge. This study addresses this by proposing a hierarchical extension of the Ishida model, a two-dimensional cellular automaton (CA), to generate complex forms through simple, localized interactions between neighboring elements. The model employs a multi-level hierarchical structure where transition rules are applied sequentially to each layer, often with a time lag, to promote stable pattern formation and prevent unintended side effects. By adjusting a small set of core parameters (notably w1 and w2), the model successfully generated a variety of static and dynamic hierarchical patterns. Specifically, it first produced a double-ring structure, which was then internally segmented into patterns with two, four, or six divisions. Furthermore, using an asymmetric initial configuration, the model generated dynamic patterns that maintained their shape while in motion, as well as forms with controlled, elongating protrusions. These results demonstrate that simple, iterative operations can create diverse, complex morphologies, offering a valuable framework for synthetic morphology and potential applications in fields like modular swarm robotics.