Mechanics of turning a page

Abstract

A mundane activity of turning a page arises not only in reading, but also in production of paper and textile. Here we analyze the shape and stability of a thin sheet of paper, bound to a book in one side and compressed from the other side, to gain mechanical understanding of the page turning. We start with describing the shape of a sheet that is buckled between the bound end and gripping fingers, using the Euler elastica. Then we predict the fate of the bent page when the external load is removed, to determine the condition for the page to turn to the other side. Upon corroborating our theory with experiments of various geometry and materials of thin sheets, we suggest the shortest path of fingers to turn pages for both lazy readers and efficient manufacturing robots.