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# Tiling method. Steps in obtaining a correct tiling pattern of pantographs: a dual graph of the net, where the vertices are the faces of the triangulated surface, b directed dual graph with correct orientation, c correct tiling pattern, d correct tiling pattern with θ ¼ 20

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Dilational structures can change in size without changing their shape. Current dilational designs are only suitable for specific shapes or curvatures and often require parts of the structure to move perpendicular to the dilational surface, thereby occupying part of the enclosed volume. Here, we present a general method for creating dilational struc...

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... first construct the dual graph to the triangulated surface. In this graph, there is one node for each triangular face and two nodes are connected if the two corresponding faces share an edge. Such a graph is shown in Fig. 6a for the octahedron. Note that, as was mentioned in Section "Triangulation", at most two faces share an edge since the original surface is a 2-manifold. The dual graph to an octahedron is shown in Fig. 6a. For closed surfaces, this creates a simple, connected, 3-regular graph. We assign a direction to each of the edges in the graph to ...
Context 2
... there is one node for each triangular face and two nodes are connected if the two corresponding faces share an edge. Such a graph is shown in Fig. 6a for the octahedron. Note that, as was mentioned in Section "Triangulation", at most two faces share an edge since the original surface is a 2-manifold. The dual graph to an octahedron is shown in Fig. 6a. For closed surfaces, this creates a simple, connected, 3-regular graph. We assign a direction to each of the edges in the graph to represent the motions of the pantograph mechanisms placed on the triangulated surface. A directed dual graph for an octahedron is shown in Fig. 6b. Since each edge in the dual graph can only have a single ...
Context 3
... surface is a 2-manifold. The dual graph to an octahedron is shown in Fig. 6a. For closed surfaces, this creates a simple, connected, 3-regular graph. We assign a direction to each of the edges in the graph to represent the motions of the pantograph mechanisms placed on the triangulated surface. A directed dual graph for an octahedron is shown in Fig. 6b. Since each edge in the dual graph can only have a single direction, the sides of the triangles are enforced to move along with each ...

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... On the one side of the spectrum, mechanical metamaterials [1][2][3] are rationally designed to exhibit exotic material-like properties, such as negative effective values of the Poisson's ratio [4][5][6] , thermal expansion [7][8][9] , and stiffness 10,11 . On the other side of the spectrum, however, one finds such concepts as mechanical logic gates 12,13 , adaptive-stiffness mechanisms 14,15 , and shape-shifting designs [16][17][18][19] , which exhibit device-like functionalities. As opposed to traditional machines, which are usually highly dynamic, the vast majority of the mechanical metamaterials designed to date work quasi-statically. ...
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