Fig. 1: Examples of trapezoid-based origami folded from cardstock.

(i) quasi-cylindrical ground states, (ii) rigid folding cylindrical isometry, (iii) non-rigid shear isometry, and (iv) diagramatic representation of compatibility conditions with line styles signifying the coupling between amplitudes on the adjoined vertices and triangles indicating periodic directions (see Supplementary Note 4 for more details). A Cylindrical geometry from Arc-Miura. B Extension of the pattern in panel A exhibiting a locked configuration. C Archimedean spiral from a graded Arc-Miura pattern. D Lemniscate of Bernoulli from a graded Arc-Miura with a parallelogram interface.