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 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 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, 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 fractal 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

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. http://www.foldinglandscapes.com/ 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.

Fourier Finds Caesar:

A Study in the Physical Evidence of Roman Surveying and Land Usage Using Image Analysis and Periodic Functions

Landscapes are dynamic constructions, with each community and each generation imposing its own cognitive map on an anthropogenic world of interconnected morphology, arrangement, and coherent meaning.

--Kurt Anschuetz, An Archaeology of Landscapes

Finding the physical and epigraphical remains of Roman surveying and centuriation throughout the Roman world remains an area of research that currently engages only a few historians of cartography. In the past the practice of Roman surveying was studied by many important Roman historians like Theodore Mommsen and Max Weber[1]. There remain however, many difficult and unanswered questions about Roman cartography, and the lack of actual extant maps has made me begin to look elsewhere for information that might shed light on its origins and methods. My current research on this problem employs GIS and image analysis to historical aerial photography and remote sensing imagery. It is my hope that in the near future it will produce the first complete map of North African sites that shows both the extent and orientation of Roman mapping. Several authors, such as Rita Compatangelo [2], J.W.M. Peterson [3], D.J. Bescoby [4] have pioneered the use of various mathematical transforms in the analysis of remote sensing imagery for the purpose of finding new sites and orientations. I have started to apply these methods in combination with edge detection algorithms in order to calculate the extent of Roman surveyng and the various types of orientations associated with the physical remains of limites.

The physical remains of Roman centuriation take on a number of sizes and orientations, but are typically discovered through the outlines of the limites that seperated the various regions from one another. Limites or Limes (singular) can be defined as a man-made boundary or balk, that is uncultivated and wide enough to form a road or pathway, which divided centuriae or other land division units from one another. These can take many forms from simple paths all the way to larger structures like the main roads of the decumanus and kardo maximus that were centrally located in a surveyed region. The feature that makes these remains discoverable through the use of transformational techniques is the fact that they appear on the landscape as periodic phenomena. This simply means that the pattern of centuration repeats itself over areas of the landscape, showing up as linear freatures that appear in remote sensing imagery as periodic pattern of grids over fixed distances. One of the most useful ways to study periodic phenomenon, at least from a mathematical perspective, is through the use of Fourier Series and transforms [5]. These transforms model any periodic phenomenon that we might be ineterested in as a infintie series of cosine and sine functions of varying frequencies.

This sum can be expressed more conviently through the use of complex exponentials which are easier to work with algebraically. Using a discrete and algorithmic version of the Fourier transform, known as the fast Fourier transform (FFT), Peterson and Compatangelo, in truely ground breaking papers, showed that one could calculate the most common period found in a group of periodic linear features found on more modern maps. I say the most common period, because many of the linear features found on the landscape today are subdivisions of modern and medieval origin, and it is sometimes extremely difficult to determine the date of the features whose period the transforms are measuring. As an illustrative example, one can think of the linear features found in the landscape as a more complex superposition of the images in the figure shown below. In the figure we see that there are linear features that repeat themselves and that in each of the figures they have different periods of repetition. One of the figures also has a different orientation than the other two. What we see in the landscape is typically a combination of all of these in the same region and on the same remotely sensed image. Peterson used the FFT to generate periodograms that produced the most common harmonics in a series of regions dislaying linear features that he took from 19th Century Ordnance Survey Maps of Scole-Dickleburgh area in South Norfolk. What the periodogram does is allow one to pick out the frequency of the linear features and the larger harmonics. An simple example of this is shown in the figure below. The periodograms not only show the most common distances between the linear features found on the map or on the satellite image, but they also yield a series of harmonics that might have the physical meaning. Larger harmonics beyond the most common one may show subdivisions in the original survey or different grid patterns from the type one is looking for. Because we are not only interested in the distances between linear features in the landscape buy also in their orientation, we have employed a second technique known as a Radon transform. This technique has been used by Bescoby to detect Roman boundaries in aerial photographs in Albania. The strength of this method is that it allows for the calculation of the angle and hence the orientation of the series of linear features. When combined with the two-dimensional version of the Fourier transform, this allows a complete characterization of the grid formed by the limites of Roman surveying. The Radon transform can be expressed by the equation below and its operation can be seen in the graph shown in the figure. Finding the size and orientation of linear features in a landscape lets us compare what we have calculated with known patterns of centriation found in literary and epigraphical evidence, such as that found in the 5th or 6th century Corpus Argimensorum. According to Hyginus, one of the authors found in this compilation of Roman surveying texts, the typical layout for an area of surveyed land is shown below. The letters and numbers define the parcel of land and very often appear as eppigraphic inscriptions on surviving boundary stones. The main intersection shown in the figure is that of the kardo and decumanus maximus which are the beginning points of any Roman survey. A typical kardo or decumanus can be seen in the photograph below that I took in a heavily centuriated area around Carthage just north of Tunis. The distances that the Romans typically used and their various names are shown in the schematic, with a century measuring 2400 Roman feet or about 705 meters. Many of these grids would however have been further subdivided in a variety of schemes that are not easily dated using physical evidence. The research that I have been doing has concentrated its efforts on the non-coastal regions of North Africa, taking in parts of Tunisia, Algeria and Libya. Below are two satellite images of the areas around Dougga and El Jem in Tunisia both of which are the sites of important Roman towns and ruins.

In both of these photographs one can see a variety of linear features that may or may not be associated with Roman activity.

It would be interesting to know the extent to which the Romans actually produced maps of these areas considering their overall importance to the history of Roman colonization and occupation in the region. El Jem for example contains one of the best preserved Roman colesseums in all of Africa.

To apply these algorithms to remote sensgin imagery it was necessary to clean them up and enhance the linear features using edge detection algorithms. An example of this is shown in the figure below.

Once this is accomplished and the various transforms have been applied we can begin to compare the results with known grids based on our knowledge of Roman practice derived from the epigraphic and literary evidence.

Using ArcGIS I have generated maps with overlays showing the orientation and extent of the surveyed region under study. The map below shows a single division into centuriae around Enfida, Tunisia. The map below shows both a division into centuriae and into a second subdivison which probably dates from a later time. [1] There are many studies of Roman surveying. For a modern bibliography see Brian Campbell, The Writings of the Roman Land Surveyors, Society for the Promotion of Roman Studies, 2000.

[2] Rita Compatangelo, Un Cadastre De Peirre Le Salento Romain, Annales Litteraires de l'Universite de Besancon, 1989

[3] John Peterson, Fourier Analysis of Field Boundaries, in G. Lock and J. Moffet, CAA91: Computer Applications and Quantitative Methods in Archaeology 1991. BAR International Series s577. Oxford, 149-156.
[4] D.J. Bescoby, Detecting Roman land Boundaries in aerial photgraphs using Randon transforms, Journal of Archaeological Science (2006) 33, 735-743. See also, J. S. Bailly et'al "Agarian Landscapes linear features detection: application to artificial drainage networks" International Journal of Remote Sensing 29 (2008) 3489-3508 and E. Magli, et.al. Pattern Recognotion by means of the Radon transofrm and the continuous wavelet transform, Signal Processing 73 (9990 277-289.

[5] See any of the recommended books on Fourier Analysis on this blog or for a good introduction to the subject see L. Solymar, Lectures on Fourier Series, Oxford University Press, 1988.

His aqueducts and his cartography:
Frontinus, Roman law and the missing maps of the waters of Rome
The foundations of the science of land measurement lies in practical experience, since the truth about sites or area cannot be expressed without lines that can be geometrically measured.
--Frontinus, De arte mensoria
According to R.H.Rodgers, 'obscurity veils the early career of Julius Frontinus,' who in the year 97 was appointed curator aquarum of the city of Rome. Frontinus wrote two groups of texts that are important to us here in our study of Roman cartography; those being De Aquaeductu urbis Romae and a series of texts on Roman surveying. The work that remains extant on Roman surveying is found in the Corpus Agrimensorum and is very fragmentary. Karl Lachmann (shown below), who worked on the first edited edition of the text, believed that the full work comprised two books, the first consisting of De Agrorum Qualitate and De Controversiis, the second work containing De Limitibus, De arte Mensoria and some other more fragmentary texts from Urbicus, another writer on Roman surveying who may have copied his work from the now missing parts of Frontinus.


Besides his interest land surveying however, Frontinus is more well-known for his text on the aqueducts of Rome. (For more on this see the website www3.iath.virginia.edu/water/furst.html run by Katherine Rinne) In the text of De Aquaeductu urbis Romae he discusses the fact that he made maps used in the administration of the aqueducts. In the prologue to the book Frontinus refers to his work as a commentarius, and explains that it is a collection of data and other materials that he made primary for 'himself'. The contents of the book are quite technical and numerical, pertaining to sizes of individual aqueducts, the dates they were built, pipes and their sizes, the quantities of water delivered and legal matters relating to the right of private individuals to the use of public water. Although most of the material serves an adminstrative aim some of the text deals with methodological issues and it is these that are of interest for historians of cartography. In Chapter 17 of his book on aqueducts Frontinus writes:

Non alienum mihi visum est longitudines quoque rivorum cuiusque ductus etiam per species operum complecti. nam cum maxima huius officii pars in tutela eorum sit, scire praeposiutum oportet quae maiora impendia exigant. nostrae quidem sollicitudini non sufficit singula oculis subiecisse; formas quoque ductuum facere curavimus ex quibus adpareret ubi valles quantaeque, ubi flumina traicerentur, ubi montium lateribus specus adpliciti maiorem adsiduamque petendi ac muniendi vi exigerent curam[1].

Which translates as:

It has seemed to me not unfitting to include as well as description of the lengths and courses of each aqueduct, according to the classifications of construction. Because the greatest part of the duties of this position lies in the maintenance of the lines, the man in charge must know what thongs demand greater outlays. My sense of responsibility has not been satified with personal examination of particular items. I have also taken care to prepare maps of the lines, from which it is clear where there are valleys and how great they are, where rivers are crossed, and where channels attached to the sides of mountains demand greater and constant attention...for their repair.

Hence we learn here that Frontinus had detailed maps made of the aqueducts describing not only the lines themselves but also the topography of the surrounding countryside that they traversed. According to Harry B. Evans, in his Water Distributon in Ancient Rome, (Michigan, 1994), "Frontinus' mapmaking merits closer attention." Evans postulates that Frontinus' data, which he gives in the text on aqueducts, is in fact derived from the maps that he had made and that those sections of the text describing the actual lines are commentaries on the maps themselves. There are other indications in the text that Frontinus is using maps as he pinpoints some of the sources of the aqueducts by using exact spatial references to the existing road system outside of Rome.

Although none of Frontinus maps survive there are maps on inscriptions that show what aqueducts maps may have looked like. An example shown here (CIL 6.1261) displays in graphic form the lines of an aqueduct and the epigraphy gives indications of water flow and on what legal terms individuals may draw water from particular lines. The inscription contains the names of the people who shared the channel that came off the aqueduct, the volume of water that they where alloted, and the scheduled times that they could take that allotment.

The inscriptions translates as:

a. for Thyrsis, freedman of Augustus, two pipes from the second to
the...hour, on the fourth day before... b. for the freedman of C. Julius
Caeser, C. Bicoleus Rufus Squaterianus, one pipe... c. to the Aufidianum of Julius Hymetus, two pipes from the second
to the sixth hour... d. To Vibius...pipes, to C. Bicoleus, Freedman of C. Julius
Caesar,... pipes from the sixth hour until sunset...

There are several known examples of this type of inscription and another is shown below (CIL 14.3676). This inscription describes a shared channel related to the supply of water at Tibur, a rural area outside Rome. The stone containing the inscription is found built into the side of the Church of Saint Peter at Tivoli and it preserves a fragment of a map showing two channels. The inscription itself lists the people to whom the water is to go to, the amount of water they have been alloted, and the time of day when it may be taken. (for more on water rights see Cynthia Jordan Bannon, Gardens and Neighbors: Private Water Rights in Roman Italy, Michigan, 2009). As Evans says, all of this deserves further work....and can help us understand some of the lesser known aspects of Roman cartography and its application.
















[1] I have used the new edition of Frontinus by R.H. Rodgers, "De Aquaeductu urbis romae", Cambridge Classical Texts and Commentaries 42, 2004.