Self-assembled DC Resistive Circuits with Self-controlled Voltage-Based Growth

A new model for analog self-assembly is introduced, the circuit Tile Assembly model (cTAM), in which a supply voltage creates electric “glues” that attach small resistive circuits to a seed to form larger circuits. Component circuits can only attach to the growing circuit if the voltage across the output terminals of the partial assembly exceeds a given threshold. Thus, as the circuit grows, the supply voltage progressively dissipates until additional circuit components can no longer attach. Thus, the supply voltage acts as a finite resource that is used up as the circuit assembles, like nutrientfor bacterial colonies. Assemblies in the shape of resistive ladders and grids are analyzed. For ladder-like circuits, the size of the assembled circuits remain within the order of the logarithm of the ratio of the supply voltage to the threshold, and is inversely proportional to the golden ratio \(\phi \), a universal constant pervasive in architecture, engineering, and biology. For grids, empirical results are presented showing bounded growth and unique terminal assemblies. The model exhibits intriguing properties, such as self-controlled growth without glue or seed programming, and communication at a distance within the assembly without signaling programming. In addition, a generalization of the model is proposed in which construction is driven by energy minimization in response to boundary conditions on the perimeter of the assembly.