Complex Polygon Research Articles

Pleasing and interesting visual complexity: support for Berlyne.

Berlyne has stated that there are two kinds of exploratory behavior, specific and diversive, and has cited data relating to “pleasingness” and “interestingness” of complex visual patterns in support of his position. In order to test Berlyne's views, 28 Ss rank ordered nine polygons, of varying degrees of complexity, in terms of “pleasingness” and “interestingness.” Highly complex polygons were found to be more interesting, in support of Berlyne. No significant differences emerged for the “pleasingness” category. Birth order data showed a nonsignificant tendency for first borns to choose less complex polygons as more pleasing than higher complexity shapes.

Perceived creativity, set, and preference for simple or complex shapes.

82 female student nurses were asked to choose their three most preferred polygons from 12 presented to them. Half the Ss were instructed that the top row, which contained the simplest shapes, are preferred by creative people. The other half of the Ss were instructed that the bottom row, which contained the complex shapes, are preferred by creative people. Each S also stated whether or not she perceived of herself as creative. Their sets significantly affected choices, with Ss choosing the simple or complex polygons in accord with the kind they were told creative Ss choose. Perceived creativity tended to be related to choice of complex polygons, but this trend did not reach statistical significance.

AESTHETIC PREFERENCES OF SCHIZOPHRENICS.

Twenty schizophrenic males and 20 male college students expressed their aesthetic preferences for polygons and poems. The schizophrenics preferred the less complex polygons and the less novel poem, in contrast to the choices made by the normal Ss. There was no consistency between degree of novelty and complexity for the schizophrenics, but there was for the college students. The results are consistent with earlier findings that aesthetic preferences can differentiate two diverse groups, and with the theoretical work of McReynolds and Kubie.

The symmetry groups of the regular complex polygons

The symmetry groups of the regular complex polygons

The Groups of Regular Complex Polygons

The two-dimensional unitary space, U2, is a complex vector space of points (x, y) = (x1 + ix2, y1 + iy2), for which the distance between (x, y) and (x', y') is defined by . A unitary transformation is a linear transformation which preserves distance. A line is the set of points (x, y) satisfying some complex equation ax + by = c. A unitary transformation is a (unitary) reflection if it is of finite period n > 1 and leaves a line pointwise invariant. Thus à unitary matrix represents a reflection if its two characteristic roots are 1 and a complex nth root (n > 1) of 1.

A Regular Quaternion Polygon

Repeated application of the unitary reflectionsto the point (1,0) and line x + iy = 1 yields 24 points and 24 lines. These are the vertices and edges of the regular complex polygon 4 {3} 4 whose group has the abstract definition R4 = I, RSR = SRS [l]. The purpose of this note is to introduce the notion of regular quaternion polygon and give an example analogous to the preceding one.