Fractal geometry and the mind

At some point, you’ve probably looked at a Rorschach blot with your friends or in class, searching for patterns in patches of ink. Developed by Hermann Rorschach in the late nineteenth century, these inkblots were used for decades in psychological tests to examine a patient’s mental state, with psychologists extrapolating information about their subject’s emotional stability, personality, and thoughts based on what they reported having seen. Now mathematicians have connected the versatility of Rorschach’s blots with their inherent mathematical properties, suggesting that they exploit the propensity of the mind to see patterns in objects that exhibit low fractal complexity.

Fractals are infinitely complex patterns that look the same at every point of magnification. Nature is full of them — from lightning, in which each fork branches into smaller, identical forks, to snowflakes, where the arrangement of ice crystals along the perimeter of the snowflake can be modeled almost exactly by fractal patterns.

Like these natural phenomena, Rorschach blots can also be modeled by these mathematical objects. Researchers at the University of Oregon traced the edges of Rorschach’s original ten blots and assigned each a D value, a measure of their fractal complexity. By using historical data about how many patterns patients typically saw in each blot, as well as collecting new data from test subjects, they were able to correlate a blot’s D value with the number of images people perceived in it, or ‘percepts.’

All of the blots had D values between 1.1 and 1.3, which is fairly low — a value of 1.0 represents a straight line, while 2.0 represents a design so complex that it appears to be completely filled in. Surprisingly, both original patients and modern test subjects saw more percepts in the blots with the lowest D values, suggesting that shapes with a low fractal complexity are easier for us to imagine images in.

The researchers also found a correspondence between the D value and percepts in the art of Jackson Pollock, a twentieth century painter famous for his abstract drip paintings. Pollack, who said he constantly tried to “stay away from any recognizable image” in his art, made paintings with fractals of subsequently higher D values throughout his career, starting at values of 1.1 and ending at values of 1.7. It seems that he, like Rorschach, had an intuitive understanding of how our brains interpret images.

(featured image courtesy of Hermann Rorschach, Public Domain)