Friday, August 23, 2013

Through Pollock's Eyes:

Reflections on the Fractal Nature of Geographic Curves, Abstract Cartographic Spaces and the Last Pool of Darkness

The idea that one gets a better and better approximation of the length of a shoreline by measuring it in finer and finer detail is false; the series of approximations does not converge to an answer, it just gets bigger and bigger, to infinity.... ....................................................................---Tim Robinson, The Connemara Fractal

Many of my best mathematical reflections seem to come to me while wandering around in museums, especially the Museum of Modern Art in New York City. The abstraction of shapes and colors found in the paintings of many of the 20th century's greatest masters, all of which line the walls there, remind me of spatial and geographical forms that have been bled of reference, seemingly residual landscapes and spaces devoid of human action; the proverbial blank on the map or a vast bringing together of spatial silences. Recently, while standing in front of one of Jackson Pollock's great drip paintings, the connection that I have often felt between mathematical and geographic spaces and the planar surfaces created by the abstract expressionists suddenly crystallized in my mind as geometry. Fractal Geometry. I am of course not the first to think of this connection. J.R. Mureika for example, studied perceptual color space, and related it to the fractal forms of Pollack's paintings ("Fractal dimensions in perceptual color space: A comparison study using Jackson Pollack's art," Chaos 15 (2005))
Both geographic spaces and most of Pollack's drip paintings are fractal in nature. In contrast to lines on a map, which are one-dimensional, and are pure generalizations of reality, fractals consist of patterns that recur on finer and finer scales. Because of this 'scaling', fractals can build up natural shapes of immense complexity like coastlines, boundaries and even full landscapes. The Pollock drip paintings are similar and when looked at in finer and finer scales, such as those shown in the detail images below, are much more like geographical curves than cartographic maps actually are. His paintings, strangely enough, unlike maps, can be seen as scaled portions of the whole at larger and larger scales. Several scholars have looked at the fractal dimension of Pollock's paintings using one of the easiest, at least from a mathematical perspective, methods to determine dimension. A method known as box-counting. Mathematically, box-counting is relatively easy to calculate, and to program and its origins go back to at least the 1930s when it was known as Kolmogorov entropy after its Russian inventor. The dimensions calculated for Pollack's paintings show them to be somewhere in the range of 1.3 to 1.7, and unlike those of typical cartographic expressions, are actually closer to that of real world geographic curves.

Real geographical curves are so complex in detail that their lengths are often infinite, or to put it in a more accurate sense, indefinable. Many of these curves, such as those representing a coastline, are statistically 'self-similar', which means that each portion can be considered as a reduced-scale model of the whole, much like we see in a Pollock drip painting. [1] . This peculiar feature of self-similarity, which is an artifact of scaling (something Galileo would have loved), can be described mathematically as a type of dimension, which unlike normal curves, is fractional.
This fractional dimensional property of coastlines was first posited in a formal way by Benoit Mandelbrot in his classic article in Science from 1967, entitled, "How Long is the Coast of Britain ? Statistical Self-Similarity and Fractional Dimension."

Self Similarity of the von Koch Curve. It looks the same no matter what scale we look at it in.

In that article [2] Mandelbrot studied the form of geographic curves, and simpler examples like the von Koch curve shown above, for which the concept of 'length' has no apparent meaning. The von Koch curve is built up by an algorithmic procedure that at every stage in the operation the middle third of each interval or line segment is replaced by the other two sides of an equilateral triangle. These types of curves, to use Mandelbrot's language, can be considered as "superpositions of features of widely scattered characteristic sizes." As he puts it, "as even finer and finer features are taken into account, the total measured length increases, and there is usually no clear cut gap or crossover, between the realm of geography and details with which geography need not be concerned". [3]

Mandelbrot set of a quadratic function in the complex plane. Self-Similarity at every scale

One of the best reflections on this problem of crossover and scaling in cartographic thought comes from the writer and mapmaker Tim Robinson's essay, 'The Connemara Fractal' published in his Setting Foot on the Shores of Connemara and other essays. Robinson, who is one of the great writers on cartography and what for him is the solitary process of map making, writes in that he is "not very interested in maps from a technical point of he will move on to the more interesting questions of what it is like to make a map...insofar as I can untangle my memories of the process." The process of map making that Robinson is speaking is "a long walk," and "an intricate, knotted itinerary that visits every place within its territory." The idea of a long and very detailed walk that Robinson invokes was suggested to him by the extraordinary form of the southern coast of Connemara. Robinson tells us that, "it looks so complicated as to be unmappable; it is a challenge to be unraveled." ...a very Mandelbrotian series of images indeed.

Robinson crosses over to details that according to Mandelbrot "geography need not be concerned", when he starts looking at actual distances along the twisting and knotted coast of Connemara. What he finds are not mappable distances, but rather impossibly infinite curves. Robinson writes that the "distance from Ros a' Mhil to Roundstone is only about 20 miles, but the coastline in between is at least 250 miles long, even as estimated on a small-scale map." He continues, "when I wrote that I was ignorant of the work of Benoit Mandelbrot, who had proved that an outline as complex as a coastline does not have a definable length." Robinson first learned of Mandelbrot's work through a newspaper article sent to him by a reader of his essays on Connemara and he calls Mandelbrot's work "a disturbing doctrine---disturbing to one who fondly imagined he had walked a coastline with due attention to its quiddity....for [Mandelbrots paper] was an annihilating critique of my essay's imagery."

In most respects Mandelbrot's paper was not really a critique of Robinson's essay, but rather a profound theoretical statement that opens up a clearing for us to conceive of a very different, and inherently less positivist conceptual foundation for the cartographic spaces we create. For as Robinson says, the process of map making, like that of discourse must stop somewhere. To be true of the world of fractals is to be infinitely and indefinitely seeking a precise measurement of length or to possess the unexplainable talent of a Jackson Pollock. But to find our way in the real and empirical world it appears that you and I will have to be satisfied with our approximations and our finitude.

[1] For more on Pollack's paintings as fractals see, "The Visual Complexity of Pollack's Dripped Fractals by R.P. Taylor, at

[2] Benoit Mandelbrot, "How long is the coast of Britain? Science 156 (1967) 636-638.

[3] For more on the concept of fractal dimension and measurement see M.C. Shellberg and Harold Moellering's classic paper, "Measuring the Fractal Dimension of Empirical Cartographic Curves."

Tim Robinson is one of the most interesting cartographer's working today. His books on the Aran Islands and Connemara are must reading for anyone interested in the practice of cartography and its relationship to landscape. His short essay called, Interim Reports From Folding Landscapes, imagines the process of cartography in a way that few, if any, other cartographers have ever been able to describe in detail.

Robinson writes that, "A map is a sustained attempt upon an unattainable goal, the complete comprehension by an individual of a tract of space that will be individualized into a place by that attempt."

In Connemara, the last pool of darkness, Robinson reflects on the time Ludwig Wittgenstein spent in the remote parts of Ireland writing and thinking. Wittgenstein, as is well known loved these remote and barren places. These rocky shores seem to have enabled him, in a way not possible in places like Cambridge, to reflect on mathematics, logic and his sins. Landscapes like those of Iceland, western Ireland and the fjords of Norway were the places he worked best and so it seems with Robinson who like Wittgenstein (or not so like him) was a Cambridge trained mathematician.