![]() We comment on common themes with other self-assembly models at the end of this article. We also find that shortest paths between 2D nets and the 3D polyhedron in a discrete model of the configuration space of foldings are a useful idealization of experimentally observed folding pathways. Our main finding is that compactness is a simple and effective design principle to maximize the yield of self-folding of polyhedra. A combinatorial explosion in the set of nets makes this study challenging. We study how the choice of initial net determines the folding pathway and yield for these polyhedra. In our experiments, the nets are realized as patterned panels of side length 300 μm, connected by solder hinges. The polyhedron is constructed by folding the net at the edges according to prescribed rules. In geometry, a net is an unfolding of the polyhedron that consists of a single, simply connected, nonoverlapping polygon made up of faces of the polyhedron attached at edges ( 17, chap. 21). We present an experimental and theoretical study of surface-tension driven self-folding of the dodecahedron, icosahedron, and truncated octahedron starting from a two-dimensional template called a net. Our focus in this work is on the role of discrete geometry in self-assembly. Several experiments, in combination with a growing body of theory, point the way to a future of algorithmic design of biomimetic devices and materials of increasing complexity ( 10 – 16). ![]() 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. 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. 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). Building such geometric models is, of course, part of a long tradition in biochemistry. 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). Two such abstractions are the Caspar–Klug (CK) theory of viral structure ( 2) and hydrophobic-polar (HP) lattice models for protein folding ( 3). 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. 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. Conversely, part of the promise of synthetic self-assembly has been that it may yield essential insights into the formation of biological structure. Our increased understanding of biological systems has inspired several synthetic methods of self-assembly ( 1). The image of truncated octahedra tessellating space is published under a Creative Commons Attribution-Share Alike 3.0 Unported license by Wikipedia user AndrewKepert.Nature uses hierarchical assembly to construct essential biomolecules such as proteins and nucleic acids and biological containers such as viral capsids. ![]() It was once conjectured that a structure with the same combinatorial properties as the truncated octahedral tessellation (or bitruncated cubic honeycomb) represented the ideal foam of equal-sized bubbles-i.e., it partitions equal volumes of space with the least surface area-but this distinction ultimately went to the uglier and weirder-looking Weaire-Phelan structure.ĭue both to its foam-like tessellatory efficiency and its permutohedral nature, the truncated octahedron can be considered a three dimensional analogue of the hexagon, though the cuboctahedron also has some claim to advancing the hexagonal legacy in 3-space. It is also, along with the lowly cube, one of only two space-filling uniform polyhedra, and is one of only five regular-faced convex polyhedra able to do so-along with the aforeementioned cube, the triangular and hexagonal prisms, and the gyrobifastigium, whatever the crap that is. Its vertices represent every combination of the coordinates 1, 2, 3, and 4 in 4-space, in same way that a hexagon can be embedded in 3-space with vertices at every permution of 1, 2, and 3 (i.e., bisecting a cube spanning coordinates 1, 1, 1 to 3, 3, 3). The truncated octahedron is the fourth order permutohedron, and forms constituent cells in higher-order permutohedra.
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