Excerpt from
L.Kleine-Horst: Empiristic theory of visual gestalt perception. Hierarchy and interactions of visual functions (ETVG), Part 5, III

Forming the geometric coordinate system

1.The gestalt factor "elongatedness/ extendedness" (E)

In gestalt factor T, as derived from Case Q2, new direction relationships were established, when compared with Case Q1. The eyes rotate not only in a certain direction, but also to a certain extent, according to the retinal interval between the peripherally projected stimulus and the fovea, measured by the Zl-system, in order to govern the eye movement. Our question is now: which in Case Q1 non-existent spatial interval relationships between the eye muscle innervations result in Case Q2, and which memory contents are formed hereby?

What is true for the direction relationships between two figures (Q2), is true also for their spatial interval relationships. First, contrary to Case Q1, the interval in Case Q2 is no longer "egocentric", i.e. measured from the fovea to the one perceived figure, but is "geometric", i.e. measured between the two perceived figures. Here, too, a "counter course" is given, a counter course of the spatial interval: B is as far away from A as A from B.  Second, the directions V and H are involved not only in the direction relationships T, but also (in connection with the relationships T) in the spatial interval relationships, because the second-order figure "twoness of figures" (Fig. 5-3.3) is mostly somewhat "elongated". A "twoness of figures" possesses two extensions, whose orientations are "absolutely different" from each other, i.e. perpendicular to each other. This "difference between the extendedness in two different orientations" is what the baby memorizes. The actualization of this new memory content leads to the gestalt quality "elongatedness". This term may be the designation also for both the gestalt factor producing the gestalt quality and the gestalt stimulus actualizing the gestalt factor. "E" is the appropriate symbolic expression of "elongatedness". Factor E is thought to be located at the eighth level, beside factor T, because both factors have the value Q2 of the quantity factor as prerequisite. One has, however, to take into consideration that, in a certain sense, the factor E has also the factor T as prerequisite, since it refers to the two "absolutely different" orientations. (This might be a reason to differentiate Level 8 into two sublevels, with E above T. When one, however, divides T into V and H, then V and H can be accepted as being immediate conditions of E.)

The qualitative-informative phenomenal effect of the factor E is the experience of "elongatedness". Its quantitative-informative effect is the detection of the "veridical" magnitude of the figure's elongated- ness. A figure appears more elongated the larger the ratio of the larger extension to the smaller extension in two perpendicular orientations. The formative effect of factor E consists in an "illusion": the enhancement of the perceived elongation. The normative E-effect occurs with an extreme ratio of extensions which is, for example, given in the case of a "line".

Here we must again be aware that one and the same stimulus can lead to different experiences, according to the levels being actualized. An observer (a reader of these lines, for example) who regularly views the world around him with a fully actualized factor hierarchy, has some problems to imagine a perception that occurs in the cases when his factor hierarchy is not fully actualized. The following maybe a small help to understand those lower-level perceptions.

If a figure is objectively not at all elongated, but extends equally into all orientations - like a disk, for example - the factor E can detect this relationship by being actualized at value zero; the gestalt quality it produces is "non-elongatedness" or "compactness" . Now every figure possesses a certain extension, a certain location difference of its contours perceived across its infield. This sort of "extendedness", at the fifth level, corresponds to the actualization of the factor Dl, at the second level, and the actualization of the higher-level functions Gml-, Ll-, and Fl-, but has nothing to do with the experience of oriented double extendedness  at the eighth level. A disk appearing at the fifth level as a "figure of a certain extension (or size)", appears with actualization of the factor E at the eighth level as "non-elongated". An ellipse, too, can be seen at the fifth level, but only as a "figure of a certain extension" (Dl), and is thus phenomenally not distinguishable from a disk. Only at the eighth level, with the aid of the actualized E-factor, can the ellipse be experienced as an "elongated figure". (Its bilateral symmetry, however, is still unperceived, because symmetry is an experience that requires the actualization of the higher-level form factors S,M, and R).

Another example for the different experience of the same stimulus with the actualization of factors at different levels is the "line", which is a "borderline" when perceived with the factor Ll, and a "line figure", when perceived with the factors up to and including E. An outline square, for example, consists of four such line-figures, that in their entirety, of course, at the same time form the Fl-contour (the "outline") of the square figure. The square arrangement of the Fl, E-lines makes the outline square into a second-order figure, but the E-lines themselves are (elongated) first-order figures (see Part 6).

All factors mentioned here operate not only between two figures but also within one figure. As an example, when the small sides of a rectangle decrease in length, as shown in Fig. 5-5, the rectangle appears "thinner". The thinner a stimulus is, the stronger it actualizes the factor E. It is not the absolute extension of a figure that is detected by E, but the relative extension, i.e. the ratios of "length" to "width". With the actualization of the factor E, the "width" of a non- elongated figure, too, is perceivable.  With "length" and "width" we have two extensions: a major and a minor extension (if they are indeed different). To designate the first term "length", and the second "width", is irrelevant: both are "extensions", detected at the eighth level. In the detection of the degree of elongation (i.e. the quantitative-informative phenomenal effect of the gestalt factor E), the quantity factor seems to play a certain role: only relatively lesser degrees of elongation, i.e.extension differences in two perpendicular orientations, can be detected rather well and veridically, approximately up to a ratio of extensions of 1 to 5, 6, or 7.  We seem to once again be confronted with the limitations of the detection of numbers, as number detection begins to become unclear  at approximately 5. It seems that the same ratio of approx. 1 to 6 establishes a limit also for experiencing rectangles as rectangles: if this limit is exceeded, i.e. if the ratio of the rectangles' width to their length is smaller than approx.1:6, one tends to perceive "bars" or even "lines", albeit "thick" ones, rather than rectangles, with which the observer of Fig. 5-5 can sympathize. The numbers below the rectangles indicate the reverse ratio: length to width. (I think that I have even read an experimental study that reported this ratio, but I do not know where.)


Figure 5-5. Influence of the ratio "length to width" on the experience of "rectangle" or "(thick) line"


2. The gestalt factor "straightness" (S)

When deriving both the V/H-factors and the T-factor, we started from the relationships between the eye muscle innervations, that are necessary to successively focus on a number of perceived figures.  Until now, we  have dealt with the cases of Q1 and Q2. Now we shall look at the relationships in Case Q3. At Level 8, the detected orientations between each two out of three figures are given. In Fig. 5-6.1, the orientations AB and BC are depicted. If one rotates the orientation AB in B "by 63° counterclockwise", one receives the orientation BC. Reversely, when one starts at orientation BC, and rotates the orientation in B "by 63° clockwise" we get orientation AB. In the two other cases, where the orientations are rotated in A and in C, exactly the same relationships are in control, each, however, with different angles of rotation. As the horizontal left/right or the vertical up/down reversals are in effect behind the reversal of the direction of rotation, and as the rotations cancel each other out, due to the reversal, when calculating the average, we get a rotation average of "zero".

Figure 5-6.1. Eye movements between three simultaneously perceived figures (Case Q3)

This means, the baby implicitly memorizes "equal orientations between each three figures", although, in most concrete situations, this equality of orientations is objectively not at all given. When the baby then "implicitly remembers" this memorized relationship, it perceives three  figures in the arrangement as shown in Fig. 5-6.2. Not only is the orientation AB equal to BC, but also is the orientation AC equal to AB and BC, i.e. all three orientations coincide. We designate such an arrangement "straight". Thus we designate the new memory content and gestalt factor, formatively producing straightness, the "straightness factor" and assign it the symbol "S".

If we see a "straight line", we believe it to possess two "points" standing out: its "ends". From a phenomenological point of view, this is true. But from the functionological point of view, the line possesses three "points" standing out: the two phenomenologically perceivable "end-points", and a point that is not phenomenologically but only functionologically existent. "Straightness" can only be established if one moves the gaze, really or virtually, from the one end of a figure (e.g. a line figure) to the other, pausing somewhere in between for a while. Only in this case can one establish that the orientation between one end and the intermediate stop point, and the orientation between the stop point and the other end, are the same.

Note: the orientation and form factors allow objects to be perceived all the more in respect to the memorized relationships, i.e. here "straightness", the less they are actualized by the gestalt stimuli, and the less attention is directed toward them. The stronger factor S is actualized the more veridically an objective orientation relationship is perceived, for example: the bend in point B of Fig. 5-6.1. One should, however, not be perplexed by the factor designation "straightness", into believing that one would experience more "straightness" the more the "straightness" factor is actualized. It is also not true that "non-straightness" is mostly perceived under these conditions. It is much more likely that, the stronger the actualization of S, the more the objective difference between two consecutive orientations is experienced, i.e. veridically perceived. This would also mean the experience of a "prägnant" straightness, if the stimulus is objectively straight, i.e. if the respective orientation difference is zero.

Thus, the qualitative-informative effect of the factor S is the experience of "straightness" and "bentness" (and possibly also curvedness). The quantitative-informative phenomenal effect of the factor S consists in the perception of the degree of non-straightness with the zero-case of "straightness". Straightness (non-straightness) can be defined as equality (inequality) of the orientations of two extensions in the case that both extensions possess a common end; this statement is true for all extensions between any three figures. The formative effect of the factor S is given when a line, or at least two consecutive extensions, appear "more straight", in the sense of less different in orientation, than corresponds to the objective orientation difference. When "perfect" straightness is objectively established, the impression of "Prägnanz" occurs, which is the normative effect of the factor "straightness".

3. The gestalt factor "measurement equality" (M)

The gestalt factor S was developed by memorizing the orientation relationships in Case Q3. Now we have  to reflect on whether a new gestalt factor must develop when the relationships between extensions are memorized. If we look at Fig. 5-6.1 and consider the consecutive extensions A-B and B-C, then we are able to claim: the second extension (B-C) is larger by X than the first extension (A-B). One can start also from C-B and state: the second extension (B-A) is smaller by X than the first extension (C-B). The one case occurs as often as the other; it is always the mean of relations, in this case: the mean of the relations between extensions, or spatial intervals, that is memorized. The mean is: equality of the two intervals, i.e. the first interval is equal to the second interval, regardless of which interval we take to be "first" and "second".  This relationship is valid for two extensions within a threeness of figures. Let us call this memory content, which is effective as a gestalt factor, "measurement equality", and assign it the symbol "M". The qualitative-informative effect of the factor M consists in the experience of difference or sameness of extensions, lengths, or spatial intervals; its quantitative- informative effect consists in the perception of the degree of length (interval) difference, with "measurement equality" at zero. The formative effect of M leads to the "misperception" of a certain length (interval) difference as "less different", i.e. "more equal". If two (or more) extensions are objectively equal, then the factor M will be "normatively" actualized, i.e. the experience of "measurement equality" will be particularly strong, "prägnant", as in the case of an equilateral triangle (Fig. 5-6.3). A simultaneous normative effect of both S and M is not possible: in Fig.5-6.4, measurement equality is only partial, as AB equals BC, but neither equals AC.  As the factor M refers to two extensions, which require three figures to exist, it is thought to be located at the same (ninth) level as the factor S.  What is true for S, is also true for M: the stronger the sensory stimulus, and the stronger the attention, the less factor M can formatively be actualized, i.e. the more the veridical measurement relationships are perceived.

4. The gestalt factor "rectangularity/parallelism" (R)

Fig. 5-7 shows the oriented extensions between four (Q4) figures. In addition to the relationships in Cases Q1, Q2, and Q3, two pairs of new relationships between the eye muscle innervations are established in Case Q4. Common to both is that the two ends of the oriented extensions, depicted by lines, do not connect to each other, as the extensions do in the cases of Q2 and Q3. One line of the pair ends at A and D, the other line of the same pair ends at B and C. There is no connection between these lines at all. In the other pair of oriented extensions, one extension ends at A and C, the other at B and D. Both lines - in both pairs - do not connect at their ends, but the lines of the last mentioned pair cross each other. Such "crossing" and "non-crossing" relationships between eye muscle innervations are established only with the simultaneous appearance of (at least) four figures. Now we have to examine whether new memory contents are to be expected to come into existence, and if so, which?

In the crossing case, it is easy to recognize that the mean of the established angles is 90°, since the sum of all four angles is 360°.  Even the mean of the angles with which the extensions connect at a figure, is 90°. An "angle" is nothing other than the orientation difference of two connecting extensions, and a "right angle" is characterized by the "average orientation difference of all possible crossing or joining extensions". This relationship is "implicitly memorized" by the baby. It is evident as well that the average orientation deviation of two non-crossing extensions is zero, i.e. "on average", they are "parallel" to each other. Both relationships, "rectangularity" and "parallelism" are thought to be antagonistic functions of the same gestalt factor. This factor shall be designated "rectangularity/parallelism" with the symbol "R"; the function "rectangularity" is assigned the symbol R+, the function "parallelism" the symbol R-.

Figure 5-7. Eye movements between four simultaneously perceived figures (Case Q4)

The factor R was derived from both the orientation and extension relationships between eye muscle innervations. The qualitative- informative effect of the factor R consists in the experience of "angle" with the zero case of "non-angle". The veridical experience of the angle's size corresponds to the quantitative-informative R-effect. The normative effect consists in the experience of "Prägnanz", when oriented extensions connect or cross at right angles (R+), or are parallel (R-) to each other. The formative phenomenal effect of the R-factor causes oriented extensions to appear more perpendicular, or more parallel, than they objectively are. The gestalt stimulus for both R functions consists of two non-consecutive oriented extensions, i.e. such that are located beside each other. However, the more the lines tend to, or actually do, cross each other or join, the more they form a gestalt stimulus for the function "rectangularity" (R+). The farther away the lines are from crossing or joining each other, the more they form the "parallelism" (R-) gestalt stimulus.

"Some formative effects of the coordinate factors"

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