Randomness is perceived in two different extremes, in macroscopic homogeneity and local heterogeneity, but apparently far away from order. Here, the authors show that a periodic order can spontaneously arise from an ensemble of binary random copolymer sequences to induce recurrent folding of a self-assembled bilayer structure in water. Randomness is perceived in two different extremes, in macroscopic homogeneity and local heterogeneity, but apparently far away from order. Here we show that a periodic order spontaneously arises from a binary random copolymer when self-assembly occurs in an ensemble containing > 10(15) possible chain sequences. A Bernoullian distribution of hydrophilic and hydrophobic side chains grafted onto a linear backbone was constructed by random copolymerization. When the polymer chains associate in water, a sequence matching problem occurs because of the drastic heterogeneity in sequence: this is believed to generate local curvature mismatches which deviate from the ensemble-averaged interfacial curvature. Periodic folding of the self-assembled bilayer stabilizes the curvature instability as recurring hinges. Reminiscent of chain-folded lamellae found in polymer crystallization, this new liquid crystalline mesophase, characterized as bilayer-folded lamellae, manifests itself as an anisotropically alignable birefringent hydrogel with structural hierarchy across multiple length scales.