Beyond natural geometry: on the nature of proto-geometry

What curves are parallel? The core feature of preschoolers' intuitive parallel category

Existing evidence has revealed that humans can spontaneously categorize many geometric shapes without formal education. Children around 4 years could distinguish between intersecting lines and parallel lines. Three features can be used to identify parallel lines, namely "translational congruence," "never meet," and "constant distance." This study separated them by using pairs of curves that possess only one of these features. Two experiments across 2021-2023, respectively, compared the relative priority of "translational congruence" with "constant distance," and "never meet" with "constant distance" among 3- to 5-year-old Chinese preschoolers (Ntotal = 314, 48% female). The results showed that preschoolers consistently grouped "constant distance" curves with parallel lines. This suggests that the core feature of intuitive parallel category is "constant distance" at this age.

Graded human sensitivity to geometric and topological concepts

In a seminal study, Dehaene et al. (2006) found evidence that adults and children are sensitive to geometric and topological (GT) concepts using a novel odd-one-out task. However, performance on this task could reflect more general cognitive abilities than intuitive knowledge of GT concepts. Here, we developed a new 2-alternative forced choice (2-AFC) version of the original task where chance represents a higher bar to clear (50% vs. 16.67%) and where the role of general cognitive abilities is minimized. Replicating the original finding, American adult participants showed above-chance sensitivity to 41 of the 43 GT concepts tested. Moreover, their performance was not strongly driven by two general cognitive abilities, fluid intelligence and mental rotation, nor was it strongly associated with mathematical achievement as measured by ACT/SAT scores. The performance profile across the 43 concepts as measured by the new 2-AFC task was found to be highly correlated with the profiles as measured using the original odd-one-out task, as an analysis of data sets spanning populations and ages revealed. Most significantly, an aggregation of the 43 concepts into seven classes of GT concepts found evidence for graded sensitivity. Some classes, such as Euclidean geometry and Topology, were found to be more domain-specific: they "popped out" for participants and were judged very quickly and highly accurately. Others, notably Symmetry and Geometric transformations, were found to be more domain-general: better predicted by participants' general cognitive abilities and mathematical achievement. These results shed light on the graded nature of GT concepts in humans and challenge computational models that emphasize the role of induction.