Extended Data Fig. 1: Overlap-dependent network aggregation.

a, A schematic of two ribbons with nominal separation \(\sqrt{{c}^{-1}}\) and length \(L\) defines the measures governing \({\mathscr{O}}\). b, Phase space of \({\mathscr{O}}\) for the range of ribbon separations, \(\sqrt{{c}^{-1}}\), and reaches, \(L/2\), investigated through the Monte Carlo model is presented. The heat map represents \({\mathscr{O}}{\mathscr{\in }}\left[\mathrm{1,1.12}\right]\) such that the regime for which \({\mathscr{O}} > 1.12\) is depicted in yellow, while the regime for which \({\mathscr{O}} < 1\) (wherein percolation is impossible) is depicted in blue. The red line represents the phase boundary to the left of which percolation occurred at least 50% (\(P\ge 0.5\)) of the time when visually determined over \(n=10\) simulations for each parameter combination. c-e, Monte Carlo networks and snapshots of the real system composed of LCE ribbons with an offset angle of 10° at 25 °C (left) and after heating to 175 °C (right) are shown for c, \(L=6\) mm, \(\sqrt{{c}^{-1}}\approx 1.6\) mm (\({\mathscr{O}} > 1.12\)); d, \(L=6\) mm, \(\sqrt{{c}^{-1}}\approx 3\) mm (\({\mathscr{O}} < 1.12\)); and e, \(L=8\) mm, \(\sqrt{{c}^{-1}}\approx 3\) mm (\({\mathscr{O}} > 1.12\)). f,e, Monte Carlo networks and snapshots of the real system composed of bilayer hydrogel ribbons at 25 °C (left) and after heating to 37 °C (right) are shown for f, \(L=5\) mm, \(\sqrt{{c}^{-1}}\approx 1.5\) mm (\({\mathscr{O}}{\mathscr{ < }}1.12\)); and g, \(L=10\) mm, \(\sqrt{{c}^{-1}}\approx 2\) mm (\({\mathscr{O}} > 1.12\)). Unlike the reach of LCE ribbons, the reach of hydrogel ribbons decreases by approximately 50% as a function of temperature due to deswelling (Supplementary Fig. 5).