Extended Data Fig. 8: Pattern recognition capacity.

To analyze the pattern-recognition capabilities of the designed tile set, the map-training algorithm (see Supplementary Information section 2.4) was run for increasingly larger sets of random images. a-c, Example images mapped to concentration patterns for sets with 1, 12, and 18 trained images per shape, with the intended target shape for each image indicated. Following the same procedure as used for the experimental system, with the same weighting of locations, 30 × 30 images with 10 possible grayscale values and matching histograms were mapped exponentially to tile concentrations in the 917 tile system; however, all images were generated randomly. Training was done using only the Window Nucleation Model with a window size k of either 2, 4, or 6, with a limit of 400,000 steps (Supplementary Information section 2.5). For each number of images per shape considered, ten repetitions of training (starting from random assignments) were performed (to account for variability of the training algorithm) for each of three different sets of images (to account for variability in sets of images). d, As the number of images in the set increases, the selectivity of nucleation using the trained map decreases. For larger k, the pixel-tile map can exploit higher-order correlations and can thus accommodate more images. For each fully-trained system, nucleation rates were calculated using the Stochastic Greedy Model, described in Supplementary Information section 2.2, at G_se = 5.4, which roughly corresponds to a temperature of 48.6 °C, and with concentrations comparable to the experimental system. Selectivity was calculated as the nucleation rate of the target shape for each image divided by the total nucleation rate of all three shapes for that image, averaged over all images in the system, and over all 30 systems (10 repetitions for each of 3 sets of images) for each point, with 90% confidence intervals shown. Star shows selectivity calculated from nucleation model results for the experimentally-implemented system. Alternatively, dashed lines show results (at G_se = 5.5) for maps constructed by a simpler training method that assigns the highest w² previously-unassigned pixels in each training image to a unique w × w region in the target shape, detailed in Supplementary Information section 2.6. These maps have at least as much capacity as the model-trained maps within the time constraints of these tests, suggesting a robustness to training method. e, As the number of images increases, pattern recognition must increasingly rely on patterns of concentrations of shared tiles, rather than choosing a pixel-to-tile map that places high-concentration pixels on tiles unique to the target shape. Histograms show average concentrations of tiles in different shapes or combinations of shape (including the average across tile categories) for images in training cases a–c, and the experimental system. The change can also be seen in the concentration maps of a–c, with the sharp checkerboard of high concentration tiles in target shapes in a becoming less apparent in b and c.