How long is a piece of string? We all know this proverbial and largely rhetorical question. We also probably assume that it has no right answer—indeed, that’s the point of this rhetorical question in the first place, namely to indicate that the issue under consideration does not have a meaningful answer.

Enter the venerable BBC. If you are keen on knowing how long the proverbial string is, here is a whole documentary on that question:

And also, enter cognitive science.

A recent article in the Psychonomic Society’s journal Attention, Perception, & Psychophysics provides a psychological answer to this question. In a nutshell, if you place the string on a surface and play with it until it defines an “open” object—that is, an object with missing boundaries—then that object will appear to be much bigger than it would appear if its boundaries were closed but its size unchanged.

So there. The string is as long as it makes the object appear big.

Let us consider the article by Tal Makovski in a bit more detail.

The researcher’s departure point was the fact that object recognition, and figure-ground segregation crucially rely on contour closure; that is, the fact that an object is most readily defined by boundaries all around it. To illustrate, consider the figure below:

Would you agree that the dots in the left hemi-ring appear slightly darker and more numerous than in the right hemi-ring? Except they are neither darker nor more numerous, as is readily revealed by enclosing the donut with boundaries:

Clearly, having a boundary matters.

But what are the effects of having—or not having—a boundary on size estimation? If a boundary is missing, does an object “lose” area because it is less well defined? Does perceived size remain unaffected? Or might it even increase?

Across four experiments, Makovski unambiguously showed that missing boundaries make objects appear bigger.

The figure below shows the stimuli and the procedure used in the studies. Stimuli A and B were closed and stimuli C and D were open. On each trial, a stimulus was shown diagonally offset from a standard stimulus (the box inside the blue screen area). The participants’ task was to adjust the size of the stimulus by changing its width or height (using arrow keys) until it was perceived to be equal in size to the standard.

The measure of interest was the ratio of the perceived size (width, length, and area) of the test stimulus relative to the standard. Any ratio below 1 would indicate that the stimulus was perceived to be smaller than the standard, any ratio above 1 reveals a larger perceived size than the standard.

In the first experiment, the ratios for area were .99 and .98 for stimulus A and B, respectively, and 1.19 and 1.14 for C and D. The strength of the effect is revealed by the next figure, which breaks the ratios down by the size of the standard stimulus—the standard came in three different sizes:

Clearly the open stimuli were perceived to be bigger than the standard or their closed counterparts irrespective of the precise shape of the stimulus (C and D are pretty different from each other, but those differences did not seem to matter). This pattern was also highly consistent across participants. Nearly everybody showed the effect, suggesting that it constitutes a new, quite fundamental visual illusion.

Accordingly, the result replicated in a second study that differed from the first one mainly by moving the test stimuli closer to the standard. Increased proximity might facilitate size comparison, thereby perhaps removing the illusionary size advantage of open objects. Except it didn’t.

The third experiment generalized the effect to other open stimuli, shown in the figure below.

You may have noticed one peculiar attribute of the stimuli used in the first three studies: They were all quite tall and narrow. The fourth and final experiment therefore examined what would happen if the stimuli were rotated so they were now wide and short.

The next figure shows these rotated stimuli from the final experiment.

Considering area judgments, the results were again the same as before: The ratio between perceived and actual area of the open objects was 1.08, whereas that ratio was 0.95 for the closed objects. Those numbers mirror the results of the first experiment.

However, there was an intriguing difference from the preceding studies when the perceived-area effect was broken down into its components: All experiments also measured the effect of height and width separately. Recall that participants adjusted both height and width independently, using the arrow keys, until they thought that the size of the stimulus matched that of the standard.

It turned out that in this final experiment, the distorted perception of area was driven mainly by the perceived length of the object, rather than its width. This represents a reversal from the first three studies, where the effect was largely due to an overestimation of width, whereas length estimates were fairly accurate.

Makovski concluded that the illusion is “dependent on the side of the missing boundaries, with missing vertical boundaries inflating width perception, and missing horizontal boundaries inflating length perception.” In other words, the object “oozes out” in whichever direction it does not encounter a boundary.

Given the acknowledged importance of boundaries in object identification, the results are quite surprising. Makovski’s results also have practical implications for graphic design, given that designers often combine open and closed objects.

Perhaps most interesting, though, are the remaining open questions: What causes the effect? Will it extend to 3D objects? What is the role of thickness of the boundary? Stay tuned for answers to those questions.

Psychonomics article focused on in this post:

Makovski, T. (2017). The open-object illusion: size perception is greatly influenced by object boundaries. Attention, Perception, & Psychophysics, 79, 1282-1289. DOI: 10.3758/s13414-017-1326-5.