[Group theoretical approach to the cognition of transformational structures and goodness of patterns]

In this article, we tried to reformulate the transformational structure theory of goodness judgment of patterns within the framework of the group theory. We found that reformulation based on the group concepts results in a more general theory, which we call the transformation group theory of goodness judgment of patterns. Main properties of the generalized theory are: 1. Each of the basic cognitive transformations is replaced by a transformation group. 2. They are the dihedral group consisting of mirror-image transformations and a cyclic transformation, the translation group, and the value-reversal transformation group. These groups are derivable independently of the given configurations. 3. The transformation group has the following properties: duality of invariance, non-divergency of transformations produced, and availability of indirect test of invariance. These properties are convenient for cognition of transformational structure of patterns in goodness judgment. 4. The intraconfigurational transformation structure of configurations can be defined by the transformation group so that order among goodness judgments can be predicted, in parallel with the existing transformational structure theory.