Sensitivity to geometric shape regularity in humans and baboons: A putative signature of human singularity

Abstract

Among primates, humans are special in their ability to create and manipulate highly elaborate structures of language, mathematics, and music. Here we show that this sensitivity to abstract structure is already present in a much simpler domain: the visual perception of regular geometric shapes such as squares, rectangles, and parallelograms. We asked human subjects to detect an intruder shape among six quadrilaterals. Although the intruder was always defined by an identical amount of displacement of a single vertex, the results revealed a geometric regularity effect: detection was considerably easier when either the base shape or the intruder was a regular figure comprising right angles, parallelism, or symmetry rather than a more irregular shape. This effect was replicated in several tasks and in all human populations tested, including uneducated Himba adults and French kindergartners. Baboons, however, showed no such geometric regularity effect, even after extensive training. Baboon behavior was captured by convolutional neural networks (CNNs), but neither CNNs nor a variational autoencoder captured the human geometric regularity effect. However, a symbolic model, based on exact properties of Euclidean geometry, closely fitted human behavior. Our results indicate that the human propensity for symbolic abstraction permeates even elementary shape perception. They suggest a putative signature of human singularity and provide a challenge for nonsymbolic models of human shape perception.

Keywords: comparative cognition; developmental psychology; geometry; human singularity; neural network modeling.

Fig. 1.

Geometric regularity effect in humans. (A) Stimuli. We selected 11 quadrilaterals, here ordered according to their number of geometrical regularities (parallelism, equal sides, equal angles, or right angles). For each quadrilateral, four deviants were generated by moving the bottom right corner by a fixed distance, thus shortening, lengthening, or rotating the bottom side. (B) Examples of intruder-task displays. (Left) Circular display used in experiment 1. Participants had to tap the intruder. (Center) Rectangular display used in experiment 2 and subsequently. In the canonical presentation, five shapes exemplified a fixed quadrilateral, with variations in size and orientation, and the remaining shape was a deviant. In the swapped presentation, those two shapes were swapped. In either case, participants had to tap the intruder. (Right) Sequential presentation, unfolding from top to bottom and from left to right over a span of 1.8 s. Participants had to answer “correct” for properly placed dots (in green) and “incorrect” for deviant dots (example in red). (C) Geometric regularity effect in experiment 1. The error rate varied massively with shape regularity in French adults. Shapes are ordered by performance, and each is labeled with a color that is consistent across graphs, including A. Error bars represent the SE pooled over all participants; in this figure, it is smaller than dot size. (D–H) Replications of the geometric regularity effect with swapped vs. canonical trials in French adults (D), subjective judgments of shape complexity on a scale of 1 to 100 (E), sequential presentation of the four corners (F), French kindergartners (G), and uneducated Himba adults from rural Namibia (H).

Fig. 2.

Visual search paradigm. (A) Examples of visual search displays. In the visual search task, 6, 12, or 24 shapes were randomly positioned inside a circle, and the participant had to decide whether all the shapes were identical, irrespective of rotation and scaling, or whether there was one that differed from the others. They gave their binary present/absent response by pressing one of two possible keys on the keyboard. (B) Error rates in the visual search task. Errors rates increased with both the number of shapes and their complexity (geometric irregularity). The latter effect closely correlated with the average error rate in the intruder task. (C) Search times. (Left) Slope of the visual search as a function of the number of displayed items, the presence or absence of an outlier, and the shape. (Right) Correlation between the slope of the visual search on present trials and the error rates of the intruder task (experiment 2).

Fig. 3.

The geometric regularity effect is absent in baboons. (A) Training procedure. Each animal was trained for thousands of trials on the intruder task, first with a small number of fixed images (three; training stage 1), then with a larger number of images (up to six; training stage 5) and with variations in size and orientation. Mastery of the task was verified through two generalization tests using novel images. Each baboon moved from one stage to the next only when the error rate fell below 20%. (B) Summary of baboon training performance (first and last blocks of 88 trials each). Each color represents one baboon. Most animals attained the criterion on the 10 pairs of shapes used for training (Top) and successfully generalized to 10 new pairs of shapes (Bottom Left) and to three pairs of easily distinguishable polygons (Bottom Right; chance: 83.3% errors with six shapes). (C) performance in the geometric intruder task. (Left) Average performance for each geometric shape at three stages: the first 33 test blocks, the middle 33 test blocks, and the last 33 test blocks. Each block contained 88 trials, and baboons took at most 99 blocks. (Right) correlation between the average error rate in baboons and in French adults taking the same test (experiment 2).

Fig. 4.

A double dissociation in geometric shape perception. (A) Symbolic model. Each shape is coded by a vector of discrete geometric properties (equal angles, parallel sides, equal lengths, and right angles; each relationship is assumed to be detected with a tolerance of 12.5%). The distance between the standard and outlier vectors is then used as a predictor of the ease of intruder detection. (B) Neural network model (modified from ref. , with permission from the authors). CORnet, a model of the ventral visual pathway for image recognition, is used to encode each of the six shapes of a given trial by an activation vector in inferotemporal cortex (IT). The shape whose vector is the most distant (L₂-norm) from the average of the five others is taken as the network’s intruder response. The predicted error rate is obtained by averaging across hundreds of trials. (C) Simple correlation matrix across shapes between the performance of individual baboons (names in capitals; Top), the predictions of the two models (Middle), and various human groups (Bottom). Color indicates the correlation coefficient, r. (D) Standardized regression weights (β) in a multiple regression of the data from various human and nonhuman primate groups across 44 data points (11 shapes × 4 outlier types) using the symbolic and neural network models as predictors. Stars indicate significance level (●P < 0.05; *P < 0.01; **P < 0.001; ***P < 0.0001).