Using quantitative phase microscopy, we have discovered a quadratic relationship between the radius R and the thickness t of helical ribbons that form spontaneously in multicomponent cholesterol-surfactant mixtures. These helical ribbons may serve as mesoscopic springs to measure or to exert forces on nanoscale biological objects. The spring constants of these helices depend on their submicroscopic thickness. The quadratic relationship (R proportional to t(2)) between radius and thickness is a consequence of the crystal structure of the ribbons and enables a determination of the spring constant of any of our helices solely in terms of its observable geometrical dimensions.