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Many compliant shell mechanisms are periodically corrugated or creased. Being thin, their preferred deformation modes are inextensional, i.e., isometric. Here, we report on a recent characterization of the isometric deformations of periodic surfaces. In a way reminiscent of Gauss theorem, the result builds a constraint that relates the ways in which the periodic surface stretches, effectively but isometrically, to the ways in which it bends and twists. Several examples and use cases are presented. 

Abstract Human carbonic anhydrase 1 (CA1) has been suggested as a biomarker for identification of several diseases including cancers, pancreatitis, diabetes and Sjogren's syndrome. However, the lack of a rapid, cheap, accurate and easy‐to‐use quantification technique has prevented widespread utilization of CA1 for practical clinical applications. To this end, we present a label‐free electronic biosensor for detection of CA1 utilizing highly sensitive graphene field effect transistors (G‐FETs) as a transducer and specific RNA aptamers as a probe. The binding of CA1 with aptamers resulted in a positive shift in Dirac voltageV_Dof the G‐FETs, the magnitude of which depended on target concentration. These aptameric G‐FET biosensors showed the binding affinity (K_D) of ~2.3 ng/ml (70 pM), which is four orders lower than that reported using a gel shift assay. This lower value ofK_Denabled us to achieve a detection range (10 pg/ml –100 ng/ml) which is well in line with the clinically relevant range. These highly sensitive devices allowed us to further prove their clinical relevance by successfully detecting the presence of CA1 in human saliva samples. Utilization of this label‐free biosensor could facilitate the early‐stage identification of various diseases associated with changes in concentration of CAs.