Intrinsic curvature determines the crinkled edges of "crenellated disks"

Elastic curvature constants determine many structural and functional properties of fluid membranes. Methods to measure the mean curvature modulus have proved to be robust. In contrast, Gaussian curvature is an intrinsic property of a surface. Thus, measuring the relevant modulus $\bar{k}$ in fluid membranes remains a challenging task. Inspired from colloidal "crenellated disks" observed in a model system composed of hard rods, we propose a concise relation between the two curvature moduli and the parameters associated with the free, crinkled edges. Our approach offers a straightforward way to determine $\bar{k}$ of these reconfigurable membranes, where various complex topologies can be nanosculpted. Further, we reveal the structure and stability of the "crenellated disks."