Algorithmic design of self-folding polyhedra

Abstract

Self-assembly has emerged as a paradigm for highly parallel fabrication of complex three-dimensional structures. However, there are few principles that guide a priori design, yield, and defect tolerance of self-assembling structures. We examine with experiment and theory the geometric principles that underlie self-folding of submillimeter-scale higher polyhedra from two-dimensional nets. In particular, we computationally search for nets within a large set of possibilities and then test these nets experimentally. Our main findings are that (i) compactness is a simple and effective design principle for maximizing the yield of self-folding polyhedra; and (ii) shortest paths from 2D nets to 3D polyhedra in the configuration space are important for rationalizing experimentally observed folding pathways. Our work provides a model problem amenable to experimental and theoretical analysis of design principles and pathways in self-assembly.microfabrication | origami | programmable matter | viral capsid N ature uses hierarchical assembly to construct essential biomolecules such as proteins and nucleic acids and biological containers such as viral capsids. Our increased understanding of biological systems has inspired several synthetic methods of selfassembly (1). Conversely, part of the promise of synthetic selfassembly has been that it may yield essential insights into the formation of biological structure. In order to realize these ambitions, it is necessary to develop model experimental systems and theoretical analyses that make precise the analogies between natural and synthetic self-assembly. Abstraction of the essentials of complex biochemical processes is an important step in this process, and perhaps the simplest abstraction is of the geometric form of a biological structure. Two such abstractions are the CasparKlug (CK) theory of viral structure (2) and hydrophobic-polar (HP) lattice models for protein folding (3). The consequences of geometry alone can be striking in such models: The CK theory provides a valuable classification of virus shapes by T number, and much of the detailed architecture of compact proteins such as helices, and antiparallel and parallel sheets emerges from purely steric restrictions on long chain molecules (4). Building such geometric models is, of course, part of a long tradition in biochemistry. What is now striking is the ability to build basic geometric structures such as polyhedra in laboratory self-assembly experiments using molecules such as DNA (5-8) or 100-nm to 1-mm scale lithographically interconnected panels (9). In addition to the intellectual value of such experiments, many of the self-assembled structures realized cannot be fabricated by alternate methods, and they are of technological relevance in optics, electronics, and medicine. In order to translate these self-assembly processes from the laboratory to a manufacturing setting, there is a need to uncover rules that govern yield and defect tolerance. Several experiments, in combination with a growing body of theory, point ...

AMS Fall Sectional Sampler

“…The cube has only 11 nets, each of which may be tested in the lab. However, the dodecahedron has 43,380 nets, and, to my surprise and delight, simple heuristics along with our computations revealed the best nets in the lab [4]. Since then our work has evolved into a study of the pathways of self-assembly [3].…”

Origami-Inspired Printed Robots

“…Recent work has found origami folding to be particularly useful for fabrication and assembly at the microand nanoscales, where 3-D fabrication and direct manipulation are not practical. Demonstrated folded micro/nanostructures include various DNA shapes [21] and devices [22], molecular devices [23], supramolecular structures [24], micromirrors [25], electrochemical capacitors [26], structures with bidirectional curvature [27], folded graphene sheets [28], algorithmically designed self-folding polyhedra [29], and pop-up book inspired folded microdevices [30].…”

Three-dimensional polymeric microtiles for optically-tracked fluidic self-assembly

“…Three-and two-dimensional components are normally designed with polyhedral [18] and polygonal [13] symmetrical shapes, respectively. This makes the topology of the stochastically assembled structures more predictable and easier to control-particularly for populations with components of a unique type.…”