Optimal number of faces for fast self-folding kirigami

There is an increasing body of research studying how to obtain 3D structures at the microscale from the spontaneous folding of planar templates, using thermal fluctuations as the driving force. Here, combining numerical simulations and analytical calculations, we show that the total folding time of a regular pyramid is a non-monotonic function of the number of faces (N), with a minimum for five faces. The motion of each face is consistent with a Brownian process and folding occurs through a sequence of irreversible binding events between faces. The first one is well-described by a first-passage process in 2D, with a characteristic time that decays with N. By contrast, the subsequent binding events are first-passage processes in 1D and the time of the last one grows logarithmically with N. It is the interplay between these two different sets of events that explains the non-monotonic behavior. Implications in the self-folding of more complex structures are discussed.