Shaping the way from the unknown to the known: The role of convex hull shape in numerical comparisons

Abstract

Various studies have shown that numerical processing is modulated by non-numerical physical properties. One such physical property is the convex hull -the smallest convex polygon surrounding all items in an array. The convex hull is usually discussed only in terms of its area. However, our group has shown that observers use the convex hull shape, as defined according to the number of vertices of the convex hull, to make numerical estimations (Katzin et al., 2020). Yet, it is still unknown if and how the convex hull shape affects comparison tasks, and how it interacts with its counterpart, convex hull area. Here we re-examine the data collected by Katzin and colleagues (2019). Using image processing, we extracted the information on the convex hull shape and showed that the shape affects latency and accuracy of numerical comparisons. We found that both the convex hull shape and other physical properties (i.e., convex hull area, average diameter, density, total circumference, and total surface area) have distinct effects on performance. Finally, the convex hull shape effect was found in counting and estimation ranges, however its effect decreased with numerosities above the counting range. Our results indicate that the interplay between convex hull shape and other physical properties, including convex hull area and numerosity, plays an important role in numerical decisions. We suggest that the convex hull shape should be controlled for when designing non-symbolic numerical tasks.

A Methodological Framework for Stimuli Control: Insights from Numerical Cognition (published)

“…Another example comes from our own experience. We lately suggested that the shape of the convex hull is important for numerical judgments (Katzin et al, 2019;Shilat et al, 2021). Previous studies have not recorded data on the shape of the convex hull and have not provided their stimuli.…”

“…Accordingly, intrinsic properties describe information extracted from individual array items. In contrast, extrinsic properties describe information extracted from the array as a whole (Shilat et al, 2021). In case of disagreement between annotators, extrinsic properties were considered relational, and intrinsic as non-relational (Lewis, 1983;Marshall & Weatherson, 2023).…”

“…New ideas could be tested by reanalyzing previous studies when both the stimuli images and their corresponding responses are provided. For example, Shilat et al, (2021) reanalyzed images generated by Salti et al's (2017) GM (Salti et al, 2017).…”

“…Stimuli should be paired with behavioral and neural response data (where applicable) for each trial. This way, new ideas could be tested by reanalyzing previous studies (e.g., Shilat et al, 2021).…”

The Interaction between Congruency and Numerical Ratio Effects in the Nonsymbolic Comparison Test

“…First, it has been shown that participants are able to process several visual parameters. In particular, scholars have manipulated several visual features and identified partial (in)congruency and full (in)congruency conditions [30,36]. In partial (in)congruency conditions, arrays were (in)congruent in one of the controlled visual features (e.g., cumulative area), while in fully (in)congruency conditions, arrays were (in)congruent on all controlled visual properties [30].…”