Plato's Cave: Non-Collinear Illusory Grouping

## Non-Collinear Illusory Grouping

Kanizsa [] shows that when curves defined by lines of dots are given excessive curvature, the percept of a smooth colinear grouping breaks into a percept of straight line segments joined by sharp vertices located at the dots, as shown below, where circles of dots of increasing curvature begin to appear as polygons. A similar effect is observed when a line of dots exceeds a critical curvature. These phenomena indicate that illusory boundary formation can occur in two modes, either completing smooth "colinear" curves, or through sharp vertices. The gradual transition between these two modes however suggests that a similar mechanism underlies both types of boundary completion.

Besides producing a sharp kink in a linear boundary, illusory boundary completion can also occur through vertices defined by multiple orientations in a large range of combinations, as shown below. Patterns A, B, C, and D shows arrangements of dots that produce illusory boundaries through vertices defined by one, two, three, and four dots respectively. A distance dependent relationship is seen in the grouping patterns of these dots, where the pattern defined by the nearest neighboring dots determines the percept of the vertex, and masks the groupings defined by more remote dots. For example, in B the horizontal grouping is masked by the stronger vertical grouping because the vertical distances are smaller than the horizontal ones. If alternate rows of B were removed, the horizontal grouping would emerge, showing that the horizontally adjacent dots are sufficiently close to generate a horizontal grouping, but that that grouping is masked by the nearer vertical grouping. Similarly, in C there are three nearest neighbors for each dot, which defines a three-way vertex passing through every dot in this pattern. If rows 2, 3, 6, 7, and 10 were removed from this pattern, a strong vertical grouping would immediately appear. Similarly, notice how pattern D promotes a four-way grouping at each dot, suppressing an alternative diagonal grouping between the dots.

These very specific properties observed in illusory vertex phenomena offer a means to test any proposed model of illusory vertex completion.