Extended Data Fig. 9: Admissible sheet shapes for parallel scars in the ruled-surface model.

a, State diagram of double-scarred shapes. The rescaled scar radius R^* as function of rescaled pitch T^* and groove angle β is shown (colour bar). Contours of constant R^* are shown for clarity (dashed grey lines). b, Zoom-in of experimentally relevant portion of the state diagram. Real sheets with parallel scars exhibit \(R* \lesssim \frac{1}{2}\). c, Examples of double-scarred shapes throughout the phase diagram, corresponding to coloured dots in b; scar lines (blue/green) and grooves (grey) are indicated. Constant W_f = 1 and scar arc length of π/40W_f between adjacent grooves were chosen for visualization. Coils (left), cylinders (bottom left), cones (bottom) and helices can be distinguished. Point T^* = β = 0 produces cylindrical surfaces with arbitrary value of R^* (purple star, dashed box); there are no other surfaces with β = 0. Solid (dashed) black boxes indicate qualitative experimentally observed shapes for equal (opposite)-parity scars.