The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers

Abstract

During language processing, humans form complex embedded representations from sequential inputs. Here, we ask whether a "geometrical language" with recursive embedding also underlies the human ability to encode sequences of spatial locations. We introduce a novel paradigm in which subjects are exposed to a sequence of spatial locations on an octagon, and are asked to predict future locations. The sequences vary in complexity according to a well-defined language comprising elementary primitives and recursive rules. A detailed analysis of error patterns indicates that primitives of symmetry and rotation are spontaneously detected and used by adults, preschoolers, and adult members of an indigene group in the Amazon, the Munduruku, who have a restricted numerical and geometrical lexicon and limited access to schooling. Furthermore, subjects readily combine these geometrical primitives into hierarchically organized expressions. By evaluating a large set of such combinations, we obtained a first view of the language needed to account for the representation of visuospatial sequences in humans, and conclude that they encode visuospatial sequences by minimizing the complexity of the structured expressions that capture them.

Fig 1. Paradigm.

(A) Basic geometrical rules used to create sequences: rotations (+1, +2, -1, -2), axial symmetries (H: horizontal, V: vertical, A,B: oblique) and rotational symmetry (P). From one location of the octagon, each of the 7 others can be reached by the application of one or more primitives. (B) Screen shot from experiment 1. The orange dot appears at successive locations on the octagon, and subjects are asked to predict the next location. (C) Examples of sequences presented to French adults (blue), kids and Munduruku adults (yellow), or both (green).

Fig 2. Performance of adult participants in experiment 1.

Top panels show the evolution of error rate across successive steps (data points 3–16 in adults) for each regular sequence (error bars = 1 SEM). The gray curve in the background shows the error rate for irregular sequences, which serve as a baseline. Bottom panels show the percentage of responses at a given location for each data point. White dots indicate the correct location. Vertical dashed lines mark the transition between the two 8-item subsequences that constitute the full 16-item sequences.

Fig 3. Complexity predicts error rates.

For each sequence, the y axis represents the mean error rate, and the x axis the sequence complexity, as measured by minimal description length. Panels show data from French adults (top, experiment 1), preschool children (middle, pooling over experiments 2 and 3), and Munduruku teenagers and adults (bottom, experiment 4). For each group, a regression line is also plotted and the Spearman’s correlation coefficient is displayed. In French children and Munduruku adults, the “4diagonals” and “2crosses” are clear outliers—as explained in the main text, the regression can be improved by assuming that their “language of thought” does not include rotational symmetry P.

Fig 4. Performance of preschool children in experiment 2.

Same format as Fig 2. In children, only data points 6 to 8 and 12 to 16 were collected. Vertical dashed lines indicate the transition between the first and the second presentations of the 8-item sequences.

Fig 5. Performance of preschool children in experiment 3.

Same format as Fig 4.

Fig 6. Performance of Munduruku participants in experiment 4.

Same format as Fig 4.

Fig 7. Model fits to subjects’ data.

Comparisons of the correct rates exhibited in completing regular and “irregular” sequences by French adults (top), preschool children (middle) and Munduruku teenagers and adults (bottom) with the performance of our model in its full version (for French adults—top), then in a noisy version (for children—middle), and finally in a version that includes a reduced instruction set (for children—middle; and Mundurukus—bottom).