Fig. 2

Radially symmetric columns assembled from triangular plates. a Schematic and time-lapse optical microscopy images of the projections of the same column, overlaid with contour lines color-coded according to the local curvature (the inverse of the locally fitted circle radius R)⁴⁰. The color bar represents curvature values. The first-layer plate’s orientation θ is defined as the angle of one long side of the plate relative to the horizontal axis. b The distribution of local curvature of the projection contours tracked at 0, 0.2, and 0.5 s in a. c Orientations of two plates (one in gray, the other in black) in a pair of nearest neighbor columns inside the lattice over time, and time-lapse optical microscopy images with the plate orientations labeled in yellow. The measurement errors are ±5°. d Time-lapse optical microscopy images (top) and corresponding Voronoi cell representations (bottom) of the hexagonal lattice, showing the annealing of imperfectly coordinated sites. The arrows in the top panel are color-coded by the velocity magnitude |v_j| of columns calculated over a period of 0.7 s (12 frames). The color of each Voronoi cell denotes the number of nearest neighbors Z_j per column j. e Radial distribution function of the lattice in experiment (green curve) and that of an ideal hexagonal lattice (gray lines). The inset defines the center-to-center distance r between columns. A D value of 2.36 µm is used in the plot. f The local order–local density (\(|{\psi _{6j}}|,\rho _j\)) histogram based on single column tracking of the hexagonal lattice (32 frames in Supplementary Movie 3). Here \(|\psi _{6j}| = |\frac{1}{{Z_j}}\mathop {\sum }_{k = 1}^{Z_j} \exp ( {6i{\mathrm{\beta }}_{jk}})|\), where the summation goes over all the nearest neighbors of column j, and β_jk is the angle between the bond linking column j and its kth neighbor and an arbitrary reference axis^46,47. Ionic strength: 0.5 mM. Scale bars: 1 µm in a; 2 µm in c, d