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 coverge 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 crystalized 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 in 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 when looked at through finer and finer scales, such as those shown in the details 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 perspectuve, method to determine dimension which is called box-counting. The method is relatively easy to calculate and program and goes 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 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, undefinable. 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 a Pollock drip painting. Because of this one can immediately sense how geographical curves (boundaries for example and coastlines) are similar to the drip painting fragments shown below [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 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 its effect of 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 mapmaking, writes in the essay that he is "not very interested in maps from a technical point of view...so 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 mapmaking for Robinson is that of a long walk, "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 unmapable; 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, what he finds to be possibly infinite distances. 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 Mandelbrots work "a disturbing doctrine---disturbing to one who foundly 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 many respects Mandelbrot's paper was of course not really a critique of Robinson's essay, but rather a profound theoretical statement that opens up a clearing for us to concieve of a very different, and inherently less postivistic conceptual foundation for the cartographic spaces we create. For as Robinson says, the process of mapmaking, like that of discourse must stop somewhere. To be true of the world of fractals is to be infinitely and indefinately 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, et.al. at http://pages.uoregon.edu/msiuo/taylor/art/fractalexpr.html
[2] Benoit Mandelbrot, "How long is the coast of Britain? Science 156 (1967) 636-638. http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf
[3] For more on the concept of fractl dimension and measurement see M.C. Shellberg and Harold Moellering's classic paper, "Measuring the Fractal Dimension of Empirical Cartographic Curves." http://mapcontext.com/autocarto/proceedings/auto-carto-5/pdf/measuring-the-fractal-dimensions-of-empirical-cartographic-curves.pdf
