Excerpt from
L.Kleine-Horst: Empiristic theory of visual gestalt perception. Hierarchy and interactions of visual functions. (ETVG), Part 6, I

The complexity of ico-structures

It is difficult to determine what is meant with "complexity"; this term has overtones of "quantity" ("more-ness"), "composedness", and "complicatedness". Common to these terms is, that, in a "complex" object, a number of parts or aspects (qualities) exist. We must, however, distinguish - as usual within the frame of ETVG - between phenomenal and functional complexity. The percept "moon in the sky" is phenomenologically the least complex ico-system; it consists of only one figure and one outfield, and the figure itself of one infield and one contour. Functionologically, this simple ico-system is more complex, as it is conditioned by a hierarchy of twelve (positive/ negative) functions. In the following, "complexity" is predominantly to be understood in the phenomenological sense.

"Moreness" of structures and qualities does not suffice to form an impression of a clear "complexity"; both differences of these structures and qualities and/or differences in their connections must be added. Complexity can express itself in respect to the entire figure/outfield system, the figure alone, the contour exclusively, the infield, or the outfield. In the following, examples of these possibilities will be presented. An approximately complete treatment of "complexity" cannot be carried out here, because I have yet to think about it systematically. A lot of theoretical work must be done in the future.

First one must distinguish between the order-level complexity and the (gestalt) factor-based complexity. Order-level complexity refers to the system "figure", or to the system "figure/outfield" (ico), as a whole, whereas factor-based complexity refers to the gestalt factors involved in these systems and provides degrees of complexity within a certain order-level.

1. The order-level complexity of figures, and the factor-based complexity

We shall deal with several examples of both order-level complexity and factor-based complexity. The order-level of a figure corresponds to the degree of forming "superfigures". The first-order level is occupied by a single figure, as there is no superfigure. Such a first-order figure can possess a number of contours, i.e. a certain shape or form, it can possess inner contours, i.e. a partitioned infield, and also - as in most cases - several outfields, without losing its property of being a first-order figure. Fig. 6-1 shows examples of first- to fourth-order figures. Let us first concern ourselves with the first-order figures. In figure A, a "(black) formless disk in a (white) surrounding" is shown, for whose appearance the factor Fl must be actualized only once (of course, in conjunction to the actualizing of its lower-level factors, on which it is grounded). Figure D needs the actualization of the factors up to and including the eighth factor level ("elongatedness"), whereas for the figure C the actualization of the hierarchy at least up to and including the ninth level is necessary, and for E and G (and for F, if F is perceived as a rectangle) up to the tenth level. All these figures are, in regard to the order-level complexity, first-order figures, although they are, in regard to the factor-based complexity, of different complexities. In the case of B, one can question whether it is a single figure with a certain shape, or whether it is two figures that are "open toward one another", as both of them are not fully closed. Indeed, when dealing with complexity, one must take "in-betweens" into consideration.

Every percept, that seems to be composed of several first-order figures, belongs to the second or a higher-order level, and is a "superfigure" in respect to the figures of which they are composed.  As with every first-order figure, every superfigure is, or can be, additionally formed by gestalt factors. These gestalt factors make the difference between the forms of the figures, which are, within the same order-level, derived from the same basic figure. This is already clear by comparing the first-order figures in Fig. 6-1, that are derived from the disk A.

If we view the second-order figures in Fig.6-1, the pure "twoness" (factor Q at value 2) of figures like H already provides the foundation of a second-order figure, independent of whether or not this twoness is experienced as "elongated" (factor E), and/or horizontally oriented (factor H). Every "pure more-ness" of figures belongs to the next higher-order level (as in Case I), also subdivisions (for example, by proximity grouping) lead to a higher-order level (as in Case T). A cross-shaped (factor R) arrangement of figures E can become a cross-figure K. The cross in N can be formed by connecting four F-lines at right angles. Pattern L can be developed by combination of three Fs, or by forming a further (white) triangle within the infield of the (black) triangle E. Figure M is composed of F-figures, too, however, in a more "complex" manner. This complexity can be understood as being formed by joining seven figures F of unequal length (factor M), by actualization of certain angle relationships (tenth-level factor R) (or as two not fully closed triangles L that are connected by a nearly vertical line F; in the case of this interpretation, M would be a third-order figure.) The filled square G becomes an unfilled square O, by development of a second square within it. The same is true for the transition of the first-order figures A (via D) to the second-order figure P. Such a process can often be observed under actual-genetic conditions. In a fourth such case, A is transformed into Q2 via the intermediate stage of Q1. Q1 is to be thought of as a developmental stage in which a hole, in the center of the disk A, continuously enlarges itself. An actual genesis of a figure from a filled into a line figure was described for sensory perception in Wohlfahrt (1932) and Kleine-Horst (1989), and for ESP in Tischner (1920) and Ryzl (1982), in the last two cases by verbal reports, and in Kleine-Horst (1987) by graphic reports of the subjects.

Figure 6-1. Possible developments toward greater complexity

On the basis of a second-order figure, other forms can be developed only by actualizing other gestalt factors. The rectangle S, for instance, can  occur either by changing the square O's actualizing state of the factor M (measurement equality) or by differentiating the "elongated" contour with the help of the factors S, M, and R. Figures O and S can be transformed to R, by changing the actualizing state of the factors M and R.

 The third-order figure U can come from the second-order figure K by doubling the contours, or from L by quadrupling the triangle and arranging them in horizontal and vertical orientation. V is to be seen as a combination of four Rs, W by combining N and Q2, and the fourth-order X by combining U and V. If a number of figures, among which at least one is a second-order figure, as the figure of the highest order-level, form a superfigure, a third-order figure is established. Thus, X can be considered to be composed of the third-order figure U or V, and the second-order figure O.

The designation "order-level" does not say anything about the way in which a figure of a particular order-level has been formed. This is even true for simple groups of figures that the early gestaltists (Wertheimer 1923) assumed to be formed by (several) grouping laws. Thus figures, whose location differences are small, form a group better than figures whose location differences are large, as Fig. T shows. According to Wertheimer, also contemporaneity and similarity (whatever this means) are grouping factors. (See the gestalt laws "XFl" in Part 8.)

In all mentioned cases of an increase in factor-based complexity, a progression of the actualization of gestalt factors was established. In most cases, this progression consisted in the additional actualization of one, or more, higher-level factors. As an example: in Fig. 6-1A, the factor E was added, to form the "elongated figure" D, which is thus somewhat  more  complex  than  A,   in  the  sense  of  factor-based  complexity.  It is possible to receive a larger factor-based complexity also by only increasing the actualization of an already actualized factor.  Thus, H comes from A by an increase in the actualization of the quantity factor from value 1 to value 2. By actualizing the same factor yet more strongly at the value 6, for example, the figure I results, which is "more complex" than H. In the second-order figure I, also other factors can be in operation, depending on the kind of actual experience (This has already been shown in Part 5, in the Section "Experiencing the same stimulus at different levels"). With Q only, I perceive pattern I only as a "moreness" (at best as "sixness") of figures. With actualization of the line factor Ll, for example, a line of figures is seen, and this line is perceived as curved with the straightness factor S. Thus we can distinguish between the continuous increase of the factor-based complexity and its increase by steps.

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