Helicoid to Spiral Ribbon Transition

We present a continuum description for the transition between the helicoid and spiral ribbon structures of chiral materials. At a critical value of the ratio between the bending and stretching moduli, the F\"oppl--von K\'arm\'an number, we encounter a continuous buckling transition from a straight helicoid to a spiral ribbon. Two of the three persistence lengths of the ribbon become very short at the transition point, indicating strong thermal shape fluctuations. The transition is discontinuous if the ribbon width is treated as a free thermodynamic variable.