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The presence of incomplete cuts in a thin planar sheet can dramatically alter its mechanical and geometrical response to loading, as the cuts allow the sheet to deform strongly in the third dimension, most beautifully demonstrated in kirigami art-forms. We use numerical experiments to characterize the geometric mechanics of kirigamized sheets as a function of the number, size and orientation of cuts. We show that the geometry of mechanically loaded sheets can be approximated as a composition of simple developable units: flats, cylinders, cones and compressed Elasticae. This geometric construction yields scaling laws for the mechanical response of the sheet in both the weak and strongly deformed limit. In the ultimately stretched limit, this further leads to a theorem on the nature and form of geodesics in an arbitrary kirigami pattern, consistent with observations and simulations. Finally, we show that by varying the shape and size of the geodesic in a kirigamized sheet, we can control the deployment trajectory of the sheet, and thence its functional properties as an exemplar of a tunable structure that can serve as a robotic gripper, a soft light window or the basis for a physically unclonable device. Overall our study of disordered kirigami sets the stage for controlling the shape and shielding the stresses in thin sheets using cuts. 

Porcine hearts (N = 14) underwent ex vivo diffusion tensor imaging (DTI) at 3T. DTI analysis showed regional differences in helix angle (HA) range. The HA range in the posterior free wall was significantly greater than that of the anterior free wall (p = 0.02), the lateral free wall (p < 0.001) and the septum (p = 0.008). The best-fit transmural HA function also varied by region, with eight regions best described by an arctan function, seven by an arcsine function, and a single region by a linear function. Tractography analysis was performed, and the length that the tracts spanned within the epicardial, midwall, and endocardial segments was measured. A high number of tracts span the epicardial and mid-wall thirds, with fewer tracts spanning the mid-wall and endocardial thirds. Connectivity analysis of the number of tracts connecting different ventricular regions showed a high prevalence of oblique tracts that may be critical for long-range connectivity. 

Porcine hearts (N = 14) underwent ex vivo diffusion tensor imaging (DTI) at 3T. DTI analysis showed regional differences in helix angle (HA) range. The HA range in the posterior free wall was significantly greater than that of the anterior free wall (p = 0.02), the lateral free wall (p < 0.001) and the septum (p = 0.008). The best-fit transmural HA function also varied by region, with eight regions best described by an arctan function, seven by an arcsine function, and a single region by a linear function. Tractography analysis was performed, and the length that the tracts spanned within the epicardial, midwall, and endocardial segments was measured. A high number of tracts span the epicardial and mid-wall thirds, with fewer tracts spanning the mid-wall and endocardial thirds. Connectivity analysis of the number of tracts connecting different ventricular regions showed a high prevalence of oblique tracts that may be critical for long-range connectivity.