Extended Data Fig. 10: Twisting transition in a simple elastic model.

a, Left: sheets are modelled as ruled surfaces. The scar line forms a helical directrix (orange line) with radius of curvature R_scar and pitch 2πT. Grooves (grey lines) are approximated as straight lines of length W_f at angle β to the scar, resulting in a cylindrical shape twisted over angle α. Right: elastic deformation modes of the sheet cost energy (see Eq. (19)) and are resisted with stiffness K_k (scar rolling), K_gs (scar torsion), K_β (wing angle changes), K_s (sheet splay), and K_r (sheet rolling). b, Energy-minimizing sheet shapes as function of W_f for given deformation resistance K_k, K_gs, K_β = 1, K_s, K_r = 0.1. Consistent with experiments (see Extended Data Fig. 7a), R_scar grows from its initial value R⁰ by amount ΔR. A twisting bifurcation in T and α is observed at critical \({W}_{f}^{\ast }\). Post-bifurcation values T^* and α^* at the end of the studied width range W_f = 150 mm are indicated. Sheet shapes shown left correspond to black circles in each graph. c, The overall sheet shape and existence of the twisting bifurcation, quantified by ΔR, \({W}_{f}^{* }\), T^*, and α^*, are robust against variations of deformation resistances over several orders of magnitude (legend).