Non-symbolic stimuli (i.e., dot arrays) are commonly used to study numerical cognition. However, in addition to numerosity, non-symbolic stimuli entail continuous magnitudes (e.g., total surface area, convex-hull, etc.) that correlate with numerosity. Several methods for controlling for continuous magnitudes have been suggested, all with the same underlying rationale: disassociating numerosity from continuous magnitudes. However, the different continuous magnitudes do not fully correlate; therefore, it is impossible to disassociate them completely from numerosity. Moreover, relying on a specific continuous magnitude in order to create this disassociation may end up in increasing or decreasing numerosity saliency, pushing subjects to rely on it more or less, respectively. Here, we put forward a taxonomy depicting the relations between the different continuous magnitudes. We use this taxonomy to introduce a new method with a complimentary Matlab toolbox that allows disassociating numerosity from continuous magnitudes and equating the ratio of the continuous magnitudes to the ratio of the numerosity in order to balance the saliency of numerosity and continuous magnitudes. A dot array comparison experiment in the subitizing range showed the utility of this method. Equating different continuous magnitudes yielded different results. Importantly, equating the convex hull ratio to the numerical ratio resulted in similar interference of numerical and continuous magnitudes.
We define a one parameter family of positions of a convex body which interpolates between the John position and the Loewner position: for r > 0, we say that K is in maximal intersection position of radius r if Voln(K ∩ rB n 2 ) ≥ Voln(K ∩ rT B n 2 ) for all T ∈ SLn. We show that under mild conditions on K, each such position induces a corresponding isotropic measure on the sphere, which is simply a normalized Lebesgue measure on r −1 K ∩ S n−1 . In particular, for r M satisfying r n M κn = Voln(K), the maximal intersection position of radius r M is an M -position, so we get an M -position with an associated isotropic measure. Lastly, we give an interpretation of John's theorem on contact points as a limit case of the measures induced from the maximal intersection positions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.