Tommaso Toffoli's 1977 technical report explores the mechanics of cellular automata, emphasizing their role as abstract dynamical systems and physical models. The study investigates their computational capabilities, particularly under reversibility, which preserves their ability to compute and construct. This reversibility allows for the representation of computing processes in a physical-like medium, aligning with fundamental physical principles. The report also discusses the physical implementability of cellular automata in small dimensions, addressing synchronization, initialization, and the potential for spontaneous organization in statistical assemblies. It highlights the relevance of interconnection patterns in sequential networks as computing resources. Toffoli acknowledges contributions from the Logic of Computers Group and funding sources. The report outlines the theoretical foundations of cellular automata, their connection to physical systems, and their potential applications in modeling complex behaviors. It also addresses challenges in implementing cellular automata, including synchronization and thermodynamic constraints. The study concludes that cellular automata, as models of physical computation, offer a bridge between mathematics and physics, with implications for understanding complex systems and computational processes.Tommaso Toffoli's 1977 technical report explores the mechanics of cellular automata, emphasizing their role as abstract dynamical systems and physical models. The study investigates their computational capabilities, particularly under reversibility, which preserves their ability to compute and construct. This reversibility allows for the representation of computing processes in a physical-like medium, aligning with fundamental physical principles. The report also discusses the physical implementability of cellular automata in small dimensions, addressing synchronization, initialization, and the potential for spontaneous organization in statistical assemblies. It highlights the relevance of interconnection patterns in sequential networks as computing resources. Toffoli acknowledges contributions from the Logic of Computers Group and funding sources. The report outlines the theoretical foundations of cellular automata, their connection to physical systems, and their potential applications in modeling complex behaviors. It also addresses challenges in implementing cellular automata, including synchronization and thermodynamic constraints. The study concludes that cellular automata, as models of physical computation, offer a bridge between mathematics and physics, with implications for understanding complex systems and computational processes.