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.

Epigraphic Evidence for Large-Scale Roman Mapping

What survives of their treatises [of the Roman surveyors] can appeal to few readers now, but so diverse are the manuscripts that preserve it, so many the names associated with its preservation, that no text opens the window wider on the transmission of Latin literature from Antiquity to print…
--L.D. Reynolds
Texts and Transmission

Besides the epigraphic cadastres from the colony of Orange in the South of France, a small fragment of which is shown in the figure below, there is other epigraphic evidence that the Romans actually made detailed maps of their territories. Although extremely rare, there are several examples of epigraphic inscriptions where explicit mention is made of the word "map'.

















In the Corpus Agrimensorum, a compilation of Roman Surveying manuals from the 6th century, there are several words used for map. Writing in the text the surveyor Siculus Flaccus says,

“The maps are given various names: some are set up on wooden tablets, others on bronze, still others on skins, although ‘map’ is their generic term, they are sometimes called ‘territory’, ‘centuriation,’ ‘demarcation,’ ‘limitation,’ ‘grid-pattern,’ figures…”
Hence Latin words such as Forma, tabula, pertica, typon, and metatio all appear to mean map.

Epigraphic evidence from Tunisia shows other examples of the word Forma being used in this fashion. In the Corpus Inscriptorum Latinarum (CIL) we find two examples in which maps are mentioned as having been made or that are being referred to. The figure below shows CIL 22788, an inscription from Henchir Chenah, that is carved on four sides of stone.

CIL 22788

The part of the inscription that we are interested in reads:

sec]undu(m) [f]orma(m) missa(m) sibi ab posu[it]

and records a boundary settlement made "in accordance with the map".

A second inscription from Henchir ez Zoubia, CIL 23910, shown below records a longer inscription referring to a boundary stone set up between the land of two communities.

The inscription reads:

positum sic [secum] dum forman [um mar]

This refers to the fact that the settlement was set up once again "according to the map". From the remainder of the inscription we might imply that this was done by a soldier (perhaps a surveyor) attached to the XIIi Urban cohort based in Carthage.

CIL 23910

Modeling Roman Land Use and Environment:
Epigraphy, Servitudes, and Game Theory

...we have all too often lacked, or failed to consider, conceptual frameworks of theory in which to examine Man's relationship to his environment, the manner in which he weighs the alternatives presented, and the rationality of his choices once they have been made....
---Peter Gould

In the study of Roman agricultural patterns it is important to have a conceptual framework in which to place the fragmentary information and evidence that is available from epigraphy, Roman law, and landscape archaeology. For the past few months I have been experimenting with Game Theoretical Models and the concept of Nash Equilibrium trying to see what type of land use models would arise.

The basis of game theory was first laid down in the late 1940's by the mathematician John von Neumann and the economist Oskar Mogenstern in their now classic book

the Theory of Games and Economic Behavior. In the book von Neumann gives the proof of the Minimax Theorem, which is central to game theoretic reasoning and that he first approached in 1928. In the 1944 book, von Neumann placed the theorem within the context of linear inequalities and the theory of convexity, which was later updated with more formal proofs of equilibrium states by John Forbes Nash.

My current work on modeling land use and some of the environmental decisions made by Roman farmers takes its real start however, from a conversation that I had with Waldo Tobler, Emeritus Professor of Geography at California, Santa Barbara, about 8 years ago. I had just read Peter Gould’s paper on African farmers in General Systems Theory, a paper that would later lead me to his seminal work, Man against the Environment. I knew that Tobler was close to Gould and that he was also playing around with some game theory during these years, and so I asked Tobler about the paper. What was most impressive to me in all this was not really Gould’s mathematics, but rather his vision of what game theory might be able to do in geographical sciences, that even simple matrix games had a spatial component that few geographers had thought to utilize.

One of the things that Gould wrote and that struck me as profound was that , “we have all too often lacked, or failed to consider, conceptual frameworks of theory in which to examine Man's relationship to his environment, the manner in which he weighs the alternatives presented, and the rationality of his choices once they have been made.” The rationality part instantly jumped out at me. As you may or may not know, the idea of rationality is an area of hot debate when it comes to questions of the Roman economy. There are many scholars, especially after Finley’s seminal book called The Ancient Economy, who believe that to consider Roman farmers and landowners as ‘rational’, in the sense of their maximizing the yield from their farms and thinking about market forces, is to project too much of a modern conception of a market economy onto the past. More recently however, some scholars like Dennis Kehoe, Cynthia Jordan Bannon and D. W. Rathbone, using legal inscriptions and the everyday account books of farms that survive as papyrus fragments, have started to use economic models and things like the theory of the commons to talk about Roman markets and agricultural estate management. Each of them in their own way incorporates many of the terms and categories of game and decision theory in their analysis. Perhaps the best book that accepts and summarizes the presence of ‘rational’ actors in the Roman economy is a book by Paul Erdkamp, entitled, The Grain Market in the Roman Empire: a social, political and economic study. Erdkamp puts forward many different models in the book, and summarizes the economic theory in his historical examinations and reconstructions. His is the sort of book that makes you anxious when you read it, as it gives you a good idea of how much you do not know and how long it takes to make any real progress in this area.

My own models are simply extensions of this kind of work. One group of Gould’s papers, from which my research certainly takes its inspiration, was written by him in the 1960's. His papers, "Wheat on Kilimanjaro: the perception of choice in game and learning model frameworks," and "Man against His Environment: a game theoretic framework", were among the first attempts to use the concepts of game theory and Nash equilibrium to look into agricultural land use. These papers, and a few others, were also discussed in an early review article on these methods written by David Harvey, "Theoretical Concepts and the Analysis of Agricultural Land-Use Patterns in Geography." It is in fact from Harvey’s book, Models in Geography that my concept of geographic model derives.

Harvey asserts, in his review article on agricultural land use, that at the time he was writing, many geographers tended to ignore theoretical breakthroughs from other disciplines, mainly on the "grounds that they proved too abstract to help in the search for unique causes of specific events." To counter this he quotes from William Bunge, whose book Theoretical Geography transformed geography and opened up an analytical window for the field, suggesting a more theoretical and inherently mathematical approach to the study of geographical and spatial distributions. To me Bunge’s book is the most important work of geography in the 20th century and I still mine it for inspiration.

Most of Harvey's paper is dedicated to outlining the requirements for a set of theoretical and conceptual elements to constitute a model in geography. A model, according to Harvey, requires a set of relationships to be established that somehow link the input, status and output variables in a specific way. This linkage must quantify the model mathematically in order for it to be tested. For Harvey, the relationships of the variables in the model can be of three distinctive types:

1. Deterministic relationships which specify cause and effect sequences.
2. Probabilistic relationships which specify the likelihood of a particular cause leading to a particular effect.
3. Functional relationships which specify how two variables are related or correlated without necessarily having any causal connection at all.

For agricultural models Harvey makes a distinction between two types of frameworks, one in which the underlying structure is normative and therefore, describes what ought to be under certain assumptions. The second, is descriptive, and describes what it is that exists. These distinctions are extremely important when we try to interpret game theoretical models, especially in something as difficult to conceptualize as the Roman economy.

In his early research Gould, using a normative game theoretic model, studied a group of African farmers around Kilimanjaro and analyzed how they decided what to plant in varying environmental conditions. Gould understood the patterns of land-use and the choices made by farmers are the result of decisions made either individually or collectively and that it might be useful to try to model those decisions in a game theoretical framework. In Gould's models the environment is one player and the farmer is another. Each of the players is faced with a number of different strategies the solution of which is the game's equilibrium. Using simple matrix games he was able to construct cartographic representations of various equilibrium alternatives that could be compared to what was in the fields.In his early research, Gould studied a group of African farmers around Kilimanjaro using decision theory to analyze how they decided what to plant in varying environmental conditions. Gould understood the patterns of land-use and the choices made by farmers are the result of decisions made either individually or collectively and that it might be useful to try to model those decisions in a game theoretical framework. In Gould's models the environment is one player and the farmer is another. Each of the players is faced with a number of different strategies the solution of which is the game's equilibrium.

Importantly, Gould realized that the game theory of the time was still algorithmically primitive and that his results determined neither how the farmers actually behaved nor how they should have behaved in an absolute sense, but rather how they should behave if they want to achieve particular results. In strategic games, such as the one Gould proposed in his papers, Nash equilibria are a set of actions amongst the payers that lead to a steady state. It is a position in the game in which each player holds the correct expectation about the other player and behaves and acts rationally according to his choices. Gould uses the simple graphical solutions to the matrix games he creates which I found so attractive early on in Harold Kuhn’s lectures. For more on Kuhn and John Nash watch the video of a recent seminar they gave together at Princeton, here.

The concept of equilibrium is not so straightforward here as one might think, and it can be interpreted in several ways. For example, when we say that a physical system is in equilibrium we might mean that it is in a stable state, one in which all the causal forces internal to the system are in balance. This is the traditional economic meaning of equilibria. The variables are dynamic however, and the balance between them that makes up the equilibrium can be thought of as networks of mutually constraining relations. Equilibria can then be considered as endogenously stable states of the model. Some scholars however, interpret game theoretic equilibria as being explanatory of the process of strategic reasoning alone. For them a solution must be an outcome that a rational agent would predict using the mechanisms of rational computation alone. The meaning of equilibrium states is still a matter of discussion in the literature of game theory and has interesting philosophical implications to how we view and interpret what the models tell us outside of their mathematical formalism. (For more on the interpretation of game theoretical results see Ariel Rubinstein's seminal paper Comments on the Interpretation of Game Theory or the Philosophy of Game Theory by Grune-Yanoff.

The current models I am working with are of course much more complex than anything Gould could have considered, as he lacked both the mathematics and the computing power. New techniques like quantal response functions, which allow us to look at probable actions, are much more powerful and yield much more interesting results. They were first introduced by McKelvey and Palfrey in the late 1990s and considered mathematically for the possibility that the players will make mistakes and therefore they give more realistic results than anything Gould imagined, at least we hope they do.

The power of models in historical geography is that you can look at many different scenarios and compare them with the little actual historical data you have. I would never assert that what I am doing actually gives me any definitive answers on what decisions Roman farmers made or how they planted, rather they show me what possibilities there were and how to rank them. Most importantly however, they greatly inform my thinking about the Roman economy in its most empirical form, and since I do not have the disciplinary constraints on my ideas that an economic historian might, I can push the limits of the models for purely theoretical and curiosity reasons.
My hope is that these methods will yield an 'experimental' historical geography, an acceptance of simulation as a method in historical studies. These simulations have the potential to shed light on the decision alternatives that face farmers and estate owners acting within primitive or developing economies. They give us a glimpse into how historically farmers interacted with their environment on a mainly cognitive level, allowing us to consider the choices they made and how their decisions affected the landscape around them. This to me, and to other geographers before me, like Gould and Harvey, is certainly a central geographical question.

For those interested I am using a software package that can calculate the Nash equilibria for games with large numbers of players, or in this case environmental variables called GAMBIT.
http://www.gambit-project.org/doc/index.html
It is an open source program and you can have a great deal of fun experimenting with variables and how they change the equilibrium outcome.