On Perceptual Grouping

 Adapted from Max Wertheimer, Untersuchen zur Lehre von der Gestalt, II. Psychologishe Forshung, 1923, 4,301-305.

      Now we turn to Wertheimer's consideration of perceptual grouping. What is most important about this work in the present context is that Wertheimer attempted to see what our mind does against a consideration of what our mind might have done. In the work we review here, Wertheimer explores the way we typically perceptually group elements. But in order to determine whether the mind is exerting a "bias" he considers some of the ways these same elements could have been grouped but weren't. The behaviorists typically weren't this analytic. Consequently, it was the gestaltist that kept trying to point out these pesky problems that can arise when you are a bit more analytic about what you are up to.

     The examples that Wertheimer constructed were very simple; most of them consisted of a set of dots. The purpose of these examples was to aid in the understanding of the different factors that influenced the grouping or composition of elements into wholes. The gestaltist had suggested what they called unit forming factors that influence how elements are grouped or organized in wholes. These unit forming factors were:

  • similarity;
  • proximity;
  • common fate;
  • good continuation;
  • set; and
  • past experience.

     The examples that follow are mainly concerned with the first two factors. The first example shown below consists of a row of 10 dots. Think of labeling these dots from left to right as a, b,...,j. Below this row of dots is shown two ways in which the dots might be organized. The one on the left is in terms of groups of two as indicated by ab/cd/.... where the elements that are close to each other are grouped into a unit. An alternative basis for grouping, depicted on the right by a/bc/de/... involves grouping the leftmost and rightmost elements of adjacent pairs together; e.g. bc. This is a logical possibility; and, if you work at it you may be able to "see" that organization. But, it is not the organization that we would typically see; nonetheless, we can imagine a mind that would be biased to see this latter organization of the dots.


     We have shown only two ways in which the dots might be organized. In fact, there are many possible ways. There are 10 dots in this example. If we think of these 10 dots as being members of a set, then what we have called a grouping is simply a partition of this set. The number of possible partitions of a set of n items is given by something called a Stirling number, and these numbers can get large very rapidly. To get some idea of why this is the case, consider something called the power set. The power set is simply the set of all possible subsets of a set. For example, the power set of the the set {x,y,z} is the set { {x,y,z}, {x,y}, {x,z}, {y,z}, {x}, {y}, {z}, {} }. Note that a set is a subset of itself and every set has the empty set, {}, as a subset. Now, a partition of a set is simply a selection of a set of subsets such that no element occurs in more than one of the subsets selected and each element occurs in one of the selected subsets. For example [ {x,z} ,{y} ] is a partition. Note that there are 8 sets in the power set of this set of three elements. In general, a set of size n has 2 to the n subsets. The set of 10 dots below has 2 to the 10 subsets, and then there are all the ways to choose from these 1024 subsets to form partitions! In case your interested in seeing a list of the power set of the 10 element set {a,b,c,d,e,f,g,h,i,j}, click here.

     The next example below is similar except that now the dots are arranged in a diagonal. Again, two of the many possible groupings are shown with the one that we are biased to use appearing on the lower left.


     The figure below has increased the number of dots to 15, but, despite the enormous increase in the number of possible groupings, we don't see this as very different from the 10 dot figure above


     In order to help you visualize these groupings, the two groups above have been animated as well as a random grouping. To see each of these animations, click on the selections below.


     In the next example below, it is clear that the distance between the dots influences whether we organize it as rows or columns.


     In the top matrix, all of the dots are equidistant and you can probably organize them in a number of ways. But the matrices below show that the similarity in "color" is a another factor that influences the way we group these elements.


     In the examples below, we tend to group the dots as forming two lines; e.g. in the left and middle figures we tend to see the A and C segments grouped as a single "line" and the B segment grouped as a line; rather than the A, B, and C segments each as a line.

     The next two figures, one composed of dots and the other of continuous lines further demonstrate our bias in grouping such figures.


     The next example illustrates the factor that the gestaltists referred to as common fate. Here the arrows pointing in a common direction tend to be grouped.

     Finally, the examples below provide evidence for the effect of past experience and context on the grouping of elements. Notice that in the top figure the middle lines are grouped into single whole, but in the next figure they are grouped into two elements.
And, finally the image below shows only the center element which is, of course, the same in both examples.

Productive Thinking...the Gestalt Emphasis

© Charles F. Schmidt