Chesnut School Lane, Amersham.
A few days back when England was covered in a bit of snow, an image on the television screen caught my eye. An aerial camera was showing a highway from above–for a moment the snow covered trees and houses looked like they were engulfed in thick white clouds. Snow looks like clouds: this is so obvious that I thought nothing of it until later, when I started having a nagging feeling that it wasn’t so obvious.
Why should snow and clouds look the same to us? I mean, is there a way to find some way to measure the two on some aspect and say, right, this is why they look similar?
One candidate for measurement could be their fractal dimension, since the structure of clouds and snow is more amenable to fractal geometry than conventional geometrical measurements. Interestingly, the fractal dimension of snow and clouds are quite close as far as I could find out (close to 1.2).
Obviously, the above is a small subset of much broader class of illusions in nature that deceive us. What’s curious about this, to my mind, is the apparent fractal connection to our perception of illusions.
Note on fractals:
Fractals, first called by this name by Benoit Mandelbrot, are mathematical objects that have the curious property of fractional dimension, like 1.3 or 1.5. Dimension, as we normally know it, is the extension of an object in some measurable directions (includes imaginary ones). Dealing with objects this way–using conventional measurements–is effective. However, there are situations where other ways of representing objects is more intuitive and clear. For instance, consider how we could mathematically represent a leaf or a cloud. Instead of using the straightjackets of normal mathematics that uses Euclidean space (three spatial dimensions, height, width and depth), a more useful representation would be to represent it’s cloudness mathematically–the characteristics of clouds that make them all look alike. Some of these characteristics are surprisingly simple: the property called self-similarity where bits of cloud look the same as each other, another aspect is called scale-invariance where clouds looks the same from different distances. There are many objects in nature that are self-similar and scale-invariant. Coastlines, forests seen from above, leaves, veins and arteries in our bodies, broccoli and many more. These are fractals and the measure of their self-similarity and scale invariance is, in a way, their fractional dimension (this is only roughly true, but it is ok for a beginning, I think).