<p>This study investigates dynamic transitions within the Lenia framework, a continuous cellular automaton that enables complex, autonomous lifeforms. We introduce Dynamic Growth Mechanisms (DGM), a novel approach consisting of three extensions: Dual-phase Growth, Kernel Scaling, and Kernel Translation. These mechanisms aim to promote self-organization and metamorphosis-like transitions from simple initial configurations. Dual-phase Growth combines Original and Asymptotic Lenia’ s growth rules. Kernel Scaling treats the kernel as an evolving cell attribute. Crucially, Kernel Translation breaks rotational symmetry by applying an offset vector (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{{\delta }}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">δ</mi> </mrow> </math></EquationSource> </InlineEquation>) to the kernel. Simulations of Dual-phase Lenia and DGM Lenia reveal complex behaviors, including pair production, fusion, and transitions reminiscent of insect metamorphosis. These dynamic behaviors and lifeform transitions are further illustrated in a supplementary online video gallery available at <a href="https://y-kayama.github.io/dgm-lenia/">https://y-kayama.github.io/dgm-lenia/</a>. DGM Lenia exhibits a variety of stable lifeforms and their transitions, suggesting potential evolutionary pathways from simple to more complex structures. Significantly, our detailed analysis reveals that the offset vector (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varvec{{\delta }}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">δ</mi> </mrow> </math></EquationSource> </InlineEquation>) in Kernel Translation plays a critical role in shaping the parameter space for organism emergence and transitions. Specifically, appropriate <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varvec{{\delta }}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">δ</mi> </mrow> </math></EquationSource> </InlineEquation> values establish a distinct region between areas dominated by Original Lenia and Asymptotic Lenia growth rules, significantly enhancing the emergence of diverse lifeforms, particularly Optium/Dioptium-class organisms and their associated transitions. Our findings highlight the potential of dynamic transitions between lifeforms and underscore the crucial role of symmetry breaking for promoting biodiversity, providing new avenues for exploring evolutionary mechanisms and stability in artificial life.</p>

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Metamorphic transitions in Lenia through Dynamic Growth Mechanisms

  • Yoshihiko Kayama

摘要

This study investigates dynamic transitions within the Lenia framework, a continuous cellular automaton that enables complex, autonomous lifeforms. We introduce Dynamic Growth Mechanisms (DGM), a novel approach consisting of three extensions: Dual-phase Growth, Kernel Scaling, and Kernel Translation. These mechanisms aim to promote self-organization and metamorphosis-like transitions from simple initial configurations. Dual-phase Growth combines Original and Asymptotic Lenia’ s growth rules. Kernel Scaling treats the kernel as an evolving cell attribute. Crucially, Kernel Translation breaks rotational symmetry by applying an offset vector ( \(\varvec{{\delta }}\) δ ) to the kernel. Simulations of Dual-phase Lenia and DGM Lenia reveal complex behaviors, including pair production, fusion, and transitions reminiscent of insect metamorphosis. These dynamic behaviors and lifeform transitions are further illustrated in a supplementary online video gallery available at https://y-kayama.github.io/dgm-lenia/. DGM Lenia exhibits a variety of stable lifeforms and their transitions, suggesting potential evolutionary pathways from simple to more complex structures. Significantly, our detailed analysis reveals that the offset vector ( \(\varvec{{\delta }}\) δ ) in Kernel Translation plays a critical role in shaping the parameter space for organism emergence and transitions. Specifically, appropriate \(\varvec{{\delta }}\) δ values establish a distinct region between areas dominated by Original Lenia and Asymptotic Lenia growth rules, significantly enhancing the emergence of diverse lifeforms, particularly Optium/Dioptium-class organisms and their associated transitions. Our findings highlight the potential of dynamic transitions between lifeforms and underscore the crucial role of symmetry breaking for promoting biodiversity, providing new avenues for exploring evolutionary mechanisms and stability in artificial life.