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Entropy and energy influence polygonal nets folding into Platonic solids

Entropy and energy influence polygonal nets folding into Platonic solids

What makes an object successful at folding? Protein scientists study how an object transforms between 2D surfaces, and tridimensional objects by using universal nets, that provide a balance between entropy loss and potential energy gain. This also explains why some of their geometrical attributes (such as compactness) represent a good predictor for the folding preference of a given shape. Researchers from the University of Michigan study the thermodynamic foldability of 2D nets for all five Platonic solids.

Our goal is to understand how topology affects yield in the stochastic folding of 3D objects. The advantage is threefold. First, by using a collection of sheets folding into the same target shape, we isolate the geometric attributes responsible for high-yield folding. Second, the model allows exhaustive computation of the pathways followed by the nets during folding, elucidating how some nets achieve high yield. Third, by studying increasingly more complex objects—from tetrahedra to icosahedra—we can use the folding mechanisms quantified in the simplest objects to predict, and potentially validate, their occurrence in the more complex shapes.

Albrecht Dürer, a 16th-century Dutch artist, studied whether 2D cuts of nonoverlapping, edge-joined polygons might be folded into Platonic and Archimedean polyhedra. Dürer cuts were eventually referred to as “nets,” but for a long time, their popularity was confined to the area of mathematics. Self-folding origami is a contemporary innovation that brings a modern twist to the ancient art of paper folding. Self-folding takes Dürer’s thoughts to the forefront of many study disciplines, from medicine to robotics, by offering a technique for generating complex 3D geometries from low-dimensional objects without the need for manipulation of the constituent pieces.

Several recent efforts have used physical forces such as light, pH, capillary forces, cellular traction, and thermal expansion to accomplish controlled folding of 3D structures. Other studies have looked at the link between the geometric properties of the object being folded and its proclivity for effective folding. The effect of different cut patterns on the material’s stress-strain behavior has been elucidated in the macroscopic folding of kirigami sheets—origami-like structures containing cuts and creases—and the “inverse design problem” of finding cuts leading to the folding of a specific target structure.

Workflow for polyhedra folding study.

Despite being the most basic and symmetric 3D polytopes, the Platonic forms family suffices to show the fast expansion in design space as shapes get more complex: A tetrahedron has two net representations, cubes and octahedra have 11 nets, while dodecahedra and icosahedra have 43,380 unique edge unfoldings apiece.

To identify the nets able to fold reliably into their polyhedron of origin we performed hundreds of cooling simulations for each net, using both a fast and a slow cooling protocol. The two distinct nets for the tetrahedron, hereafter referred to as the triangular net and the linear net, showed remarkably different folding propensities for the fast cooling protocol.

Effect of net topology on the folding probability for the simplest Platonic solids. Folding yield, defined by the fraction of simulations that reached the folded state, was calculated for two cooling rates. Nets for each polyhedron are ordered from highest to lowest yield at low cooling rate.

Nets that fold the most reliably are the most compact and have the most leaves on their cutting graph. If a net has a large number of leaves (the vertices with degree one on the cutting tree) and a small diameter, it is considered to be more compact (the longest shortest path between any two faces on the face graph). Most notably, the folding probability may be decreased from 99 percent to 17 percent in nets that differ only by the position of a single face.

What causes one shape to fold nearly perfectly every time while a slightly different one fails to do so almost as frequently? And why do net “leafiness” and “compactness” correlate with folding yield?

All nets follow a folding pathway that achieves a narrow balance between reduction of degrees of freedom and gain of potential energy. In practice, high-temperature folding occurs locally, so that the system seeks to optimize its conformational entropy at each step of the process. Furthermore, as the number of faces rises, the folding tendency of a net diminishes. For example, while the 4-sided tetrahedron folds nearly flawlessly, the 20-sided icosahedron cannot fold.

Our results can be used to guide the stochastic folding of nanoscale objects into drug-delivery devices and thermally folded robots.

Universal folding pathways of polyhedron nets, Paul M. Dodd, Pablo F. Damasceno, and Sharon C. Glotzer

Published: July 2018, Proceedings of the National Academy of Sciences
DOI: 10.1073/pnas.1722681115

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