Abstract

A piece of paper coiled up in a tube hugs the wall in order to minimize its elastic energy. However, a moment's observation shows that the paper fails to make complete contact near its ends. Here, we show that the detached region in the inner part of the coil touches down the tube wall in an angle that is independent of the system size, thickness and stiffness of the surface. This angle is universal for any coiled sheet whose unstressed state is initially flat. Our results show how the shape and stress distribution for this detached region define the geometry and mechanical equilibrium for the rest of the sheet. In particular, we find scaling relations to describe the structure of the outer part of the coil, in contact with the tube wall, as a function of the number of times the sheet is coiled, N. © 2008 The Royal Society.