Goodness of compound patterns based on symmetry groups

Abstract

This study examined the ordinal agreement and order conservation hypotheses of Imai (1986), which predict that a pattern is considered good if it is invariant for more transformations. Undergraduates (N=144) made goodness ratings for 21-dot compound patterns. The patterns consisted of 8-dot and 13-dot figures with solid and/or open circles. Both the figures and the patterns were invariant under the transformations of rotation and reflection, forming cyclic (C₁, C₂, C₄) or dihedral (D₁, D₂, D₄) groups. The results showed no significant differences among the four combinations of solid and open circles. When the 13-dot C₁ figure was overlapped with the 8-dot figures, the C₁ compound patterns were rated the poorest. The goodness ratings of the C_n and D_n patterns were the increasing functions of the number of transformations, which supports the ordinal agreement hypothesis. When alternated with the 8-dot D₄ figure, the 8-dot C₁ figure superimposed with the 13-dot figures reduced the goodness ratings of the C₁ patterns, but the order of the ratings was conserved, which supports the order conservation hypothesis.