# How do we see area?

Yousif, S. R., Aslin, R. N., and Keil, F. C. (2020). Judgments of spatial extent are fundamentally illusory: ‘Additive-area’ provides the best explanation. Cognition. [Preprint available here]

Yousif, S. R., and Keil, F. C. (2020). Area, not number, dominates estimates of visual quantities. Scientific Reports, 10, 1-13. picture_as_pdf

Yousif, S. R., and Keil, F. C. (2019). The ‘Additive-Area Heuristic’: An efficient but illusory means of visual area approximation. Psychological Science, 30, 495–503. picture_as_pdf

Yousif, S. R., Alexandrov, E., Bennette, E., Aslin, R. N., and Keil, F. C. (Under revision). Children estimate area using an ‘Additive-Area Heuristic’.

Consider the image below. Which side of the image has more cumulative area? (In other words, if you were two print these two separate images on a sheet of paper, which would require more purple/blue ink: the right or the left?)

(Hover over the above image to see an example with rectangles)

If you are like most observers, you will perceive the left side as having more than the right side (and for the rectangles, you will perceive the right side as having more than the left). However, cumulative area is actually equal in both cases. So what's going on? Why do most observers see the left side as having more than the right side?

My work suggests that our impressions of area are best captured by true, mathematical area — but instead by 'additive-area'. 'Additive-area', in short, is equal to the sum of an object's dimensions rather than their product (think length + width instead of length * width). For example, below you can see the data from a task where we explicitly dissociate 'true' area from 'additive-area'.

As you can see, observers judgments of area are governed almost entirely by 'additive-area' and not true area. Thus, impressions of area are illusory — in a way that we can easily control and manipulate. In subsequent work, we explore how this may impact our understanding of the perception of other dimensions, namely number.

For one, we are able to show that these effects are not caused by number (see graph below). And in ongoing work, we are documenting how exactly accounting for perceived area (i.e., 'additive-area') can affect our understanding of the relationship between area and number. (For example, you may notice in the graph below that, when there is more number, observers are less likely to indicate that a stimulus has more area. This pattern demonstrates that number cannot explain our results, but it also contradicts many known findings regarding 'congruity effects' between area and number.)

By contrast, we are able to demonstrate that perceived area (i.e., 'additive-area') has a massive effect on number discriminations. For example, in the data below, you can see that 'additive-area' influences number discriminations to a large degree. In this case, observer's performance is better predicted by 'additive-area' than number itself!

These results are robust in several ways. Most importantly, we have shown that children as well as adults exhibit an 'Additive-Area Heuristic'. In the figure below, you can see data from 4-7 year-olds. While performance does increase with age, you will notice that even four-year-olds seem to discriminate area on the basis of 'additive-area' and not true area.

In ongoing work, we are trying to further understand how explaining area perception can influence our understanding of dimensions like number. We are also interested in how this may explain volume perception in the real world (and to summarize those results briefly: it seems that an 'Additive-Volume Heuristic' seems to best explain adults' impressions of volume). Stay tuned for more details!