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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...

## Contexts in source publication

**Context 1**

... 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. ...

Machine-matter, of which mechanical metamaterials and meta-devices are important sub-categories, is emerging as a major paradigm for designing advanced functional materials. Various exciting applications of these concepts have been recently demonstrated, ranging from exotic mechanical properties to device-like and adaptive functionalities. The vast majority of the studies published to date have, however, focused on the quasi-static behavior of such devices, neglecting their rich dynamic behavior. Recently, we proposed a new class of strain rate-dependent mechanical metamaterials that are made from bi-beams (i.e., viscoelastic bilayer beams). The buckling direction of such bi-beams can be controlled with the applied strain rate. The proposed approach, however, suffers from a major limitation: 3D printing of such bi-beams with such a 'strong' differential strain rate-dependent response is very challenging. Here, we propose an alternative approach that only requires a 'weak' differential response and a rationally designed geometric artifact to control the buckling direction of bi-beams. We present an analytical model that describes the landscape of all possible combinations of geometric designs and hyperelastic as well as viscoelastic properties that lead to the desired strain rate-dependent switching of the buckling direction. We also demonstrate how multi- and single-material 3D printing techniques can be used to fabricate the proposed bi-beams with microscale and submicron resolutions. More importantly, we show how the requirement for a weak differential response eliminates the need for multi-material 3D printing, as the change in the laser processing parameters is sufficient to achieve effective differential responses. Finally, we use the same 3D printing techniques to produce strain rate-dependent gripper mechanisms as showcases of potential applications.

Flexible tactile sensors can be used as the human–machine interactions for tactile information. Integration of flexible tactile sensor onto complex curved surfaces usually generates high assembly strains or stresses, thus reducing the accuracy of tactile sensing. This paper presents a novel flexible tactile sensor using a devisable kirigami structural design for conformal integration of curved surfaces with low assembly strain. The target sphere object's surface is divided into several fan‐shaped pieces and then flattened into 2D patterns for the structural design and arrangement of tactile sensors. Each piece of tactile sensor has three sensing units including one triangular‐shaped and two quadrilateral‐shaped sensing units. The fabricated tactile sensor with tri‐unit and quad‐unit is characterized with the sensitivity of 0.0112 and 0.0081 kPa−1 in sensing range of 0–80 kPa, respectively. Numerical simulation results show the tactile sensor has generally low assembly strain (<1.0%) when integrated onto a tennis ball, and experimental results demonstrate that the relative changes of initial resistance and calculated sensitivity are less than 4.0% before and after integration. The contact force sensing in tennis ball games with different actions demonstrates the potential of using the designed tactile sensor for further athletic training and other applications. A flexible and conformal tactile sensor is developed to be integrated onto spherical surface for contact force sensing. Conformal structure contains contour shape designed by kirigami and patterned units to reduce the assembly strains and stress of tactile sensor. Applications of tennis ball motion detection demonstrate the feasibility of the tactile sensor for sports training and interactions between humans and sports equipment.