Finding the optimal nets for self-folding Kirigami

TL;DR

This paper introduces a deterministic method to identify optimal nets for self-folding Kirigami shells by mapping the problem onto maximum leaf spanning trees in a graph, enabling better design of larger structures.

Contribution

The authors develop a novel graph-theoretic approach to find optimal nets, improving upon previous random search methods and allowing for larger, more complex shell designs.

Findings

Method guarantees optimal nets with maximum vertex connections.

Enables design of larger, more complex self-folding shells.

Provides a complete catalog of maximum leaf spanning trees.

Abstract

Three-dimensional shells can be synthesized from the spontaneous self-folding of two-dimensional templates of interconnected panels, called nets. However, some nets are more likely to self-fold into the desired shell under random movements. The optimal nets are the ones that maximize the number of vertex connections, i.e., vertices that have only two of its faces cut away from each other in the net. Previous methods for finding such nets are based on random search and thus do not guarantee the optimal solution. Here, we propose a deterministic procedure. We map the connectivity of the shell into a shell graph, where the nodes and links of the graph represent the vertices and edges of the shell, respectively. Identifying the nets that maximize the number of vertex connections corresponds to finding the set of maximum leaf spanning trees of the shell graph. This method allows not only to design the self-assembly of much larger shell structures but also to apply additional design criteria, as a complete catalog of the maximum leaf spanning trees is obtained.