Fractals, vision, and golf

I came across an article on vision at that is concerned chiefly with how digital imaging technology may and may not be able to provide a fix for damaged retinas.  While digital cameras can better mimic the human eye, the gaze of the camera, unlike the eye, is static and uniform.

As sensors in digital cameras fast approach the 127 megapixels of the human eye, clinical trials are under way to implant this technology directly into the retina. But Richard Taylor cautions that such devices must be adapted for humans, because of the special nature by which we see.


even the best optical theories suffered from a weakness: given that light rays bounce off a friend’s face, why can we not spot it immediately in a crowd – even though it is directly before our eyes? We are forced to conclude that the visual system is not passive but that it has to hunt for the information we need.

The eye does not employ the Euclidean design of cameras.  In other words, the eye’s photoreceptors are not arranged in a uniform, 2-dimensional array across the retina.  Instead the eye’s seven million cones are “piled into the central region of the retina.”  The eye has to move if the focus of our attention is to fall mainly on the fovea, which, while being only 1% of the retina, uses more than 50% of the visual cortex.  Richard Taylor and his colleagues at the University of Oregon investigated this movement or how we search for information in a complex scene.  By tracking the motion of the eye they found that

the eye searches one area with short steps before jumping a larger distance to another area, which it again searches with small steps, and so on, gradually covering a large area.

It turned out that the trajectory of the gaze is like a fractal, a 1-dimensional line that starts to occupy a 2-dimensional space because of its repeating structure. Weierstrass’ example of a function which is everywhere continuous but nowhere differentiable is actually a fractal.

One of the intriguing properties of a fractal pattern is that its repeating structure causes it to occupy more space than a smooth 1D line, but not to the extent of completely filling the 2D plane. As a consequence, a fractal’s dimension, D, has a value lying between 1 and 2. By increasing the amount of fine structure in the fractal, it fills more of a 2D plane and its D value moves closer towards 2.

The article explains how the D value is calculated and found that eye movements traced out fractal patterns with a D value of 1.5, which mimics the foraging patterns of an animal’s search for food.

Significantly, fractal motion (figure 1d, middle) has “enhanced diffusion” compared with Brownian motion (figure 1d, right), where the path mapped out is, instead, a series of short steps in random directions. This might explain why a fractal trajectory is adopted for both an animal’s searches for food and the eye’s search for visual information. The amount of space covered by fractal trajectories is larger than for random trajectories.

In this light, the author raises an interesting question:

what happens when the eye views a fractal pattern of  D = 1.5? Will this trigger a “resonance” when the eye sees a fractal pattern that matches its own inherent characteristics?

Experimental collaborations between psychologists and neuroscientists found that images matching the fractal dimension of the eye’s searching movement are ones that are most aesthetically pleasing.  And, even more provocative, that exposure to these images can reduce our physiological responses to stress by 60%.  Looking further,

preliminary functional-magnetic-resonance-imaging (fMRI) experiments indicate that mid-D fractals preferentially activate distinct regions of the brain. This includes the parahippocampal area, which is associated with the regulation of emotions such as happiness.

Examples of these 1.5-dimensional fractals are images of clouds, trees and river patterns (likely golf course landscaping). This contrasts with the perspective of some evolutionary psychologists that the surprising cross-cultural appreciation of the beauty of particular landscapes is grounded in an ancestral ideal of the Pleistocene savannas.

But, more importantly, it is an indication of the way our bodies and our thoughts are like two sided mirrors, or perhaps how the internal and the external inevitably match.  And, for me, it says again how mathematics can search out the forms they share.


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