Fractals, Calculus and Jackson Pollock

“You’ve got to deny, ignore, and destroy a hell of a lot to get at truth.” –  Jackson Pollock

I enjoy viewing fractals and especially like the fact that they are more than a line, they show a surface too. (I do take pleasure in a creature showing itself.) Also, and perhaps most importantly, they do not lend themselves to being differentiated, as in calculus.

Wikipedia says: “…fractals are usually nowhere differentiable.[2][6][8] An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.”

Listen to this: “In calculus, a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain. As a result, the graph of a differentiable function must be relatively smooth, and cannot contain any breaks, bends, cusps, or any points with a vertical tangent.”

You will recall that it is maintained that Jackson Pollock uses these creatures, fractals, in some of his paintings, so that the image to some is more prestigious since our painter, artist is now a scientist and enlists the difficult art of design versus just flinging paint. Seems to me that flinging paint has its methods and just as well stands on its own merits. There is of course many possibilities of influence as in chaos or the poets hand, that is, the artist as instrument. I think the end point is in our imaginative judgment, so leave it to us.

For those interested here is a good article on Pollock’s painting and his use of fractals.

http://www.nytimes.com/2006/12/02/books/02frac.html?_r=0

http://usatoday30.usatoday.com/tech/science/2006-12-24-pollock-authentication_x.htm

Fractal Brushup from Discovery Magazine article:

In Jackson Pollock’s drip paintings, as in nature, certain patterns are repeated again and again at various levels of magnification. Such fractals have varying degrees of complexity (or fractal dimension, called D), ranked by mathematicians on a series of scales of 0 to 3. A straight line (fig. D=1) or a flat horizon, rank at the bottom of a scale, whereas densely interwoven drips (fig. D=1.8) or tree branches rank higher up. Fractal patterns may account for some of the lasting appeal of Pollock’s work. They also enable physicist Richard Taylor to separate true Pollocks from the drip paintings created by imitators and forgers. Early last year, for instance, an art collector in Texas asked Taylor to look at an unsigned, undated canvas suspected to be by Pollock. When Taylor analyzed the painting, he found that it had no fractal dimension and thus must have been by another artist. Photographs courtesy of Richard Taylor

Lead image:ackson Pollock’s “Going West” (ca. 1934-1935). From the Smithsonian American Art Museum