Wednesday, July 14, 2010

Bi-dimensional Regression Revisited:
Studies in the geometry and form of the Medieval Portolan Chart

Introduction to my talk at the Library of Congress’ Conference
Re-Examining the Portolan Chart: History, Navigation and Science
May, 21st 2010

There are spaces in which the determination of position requires not a finite number, but either an endless series or a continuous manifold of determinations of quantity. Such manifolds are, for example, the possible determinations of a function for a given region, the possible shapes of a figure, and so on.
--Bernhard Riemann

What the historian of cartography should be concerned with is a systematic study of the factors effecting error, and seek to establish their cause and variability and the statistical parameters by which error is characterized...

--J.B. Harley

...secular magnetic variation is potentially as valuable in the history of cartography as the radiocarbon method in archaeology, though the calibrations have yet to be worked out.

--Tony Campbell, History of Cartography Volume 1

Click here to read the Washington Post Story on the LOC Portolan Conference

Those of you who know my more academic publications in the history of cartography realize that for the most part they tend to take an extremely phenomenological approach to cartographic objects. From my earliest publications on Fourier transforms and the Space Oblique Mercator projection through my current research on Topological existence theorems and mathematical constructivism in early computer cartography I have always been more interested in the conceptual and mathematical foundations of cartography than in any historical causalities or contingencies relating to maps themselves.

Calculated Distortion grid and vector displacements for the Library of Congress's 1320 Portolan Chart

Because of this phenomenological approach my paper this afternoon may be quite difficult for some of you (especially right after lunch) as it is extremely analytical and most of it is going to be concerned with very complex transformational geometry; discussions of things like Laplacian matrices and thin-plate splines. This being said, I promise you that if you keep your focus on the actual cartographic problems that I am trying to resolve much of the mathematics will dissolve into the background and in the end it is my hope that you will not only learn something about the geometric and mathematical structure of Portolan charts but also that this talk might serve as a methodological introduction to some of the computational techniques that I have helped develop and that I have been using in my cartometric research. These techniques have there beginnings with the work of Waldo Tobler whose paper and computer program called, Bi-dimensional Regression (see links section to read Tobler's paper) is where modern historical cartometry can be said to have started. Everytime I read this paper I am amazed at Tobler’s geometric insights and I find new inspiration in every re-reading. Using the integrated sums of the squares of the four partial derivatives was a real breakthrough and took incedible geometirc imagination.

Calculated rotation isolines for the Library of Congress' 1320 Portolan chart

My paper this afternoon will deal principally with three problems concerning the form of Portolan charts that have to date eluded solutions and whose logical structure goes directly to the heart of the geometric form that these early charts take. Borrowing a definition from the philosopher of science, Bas van Fraassen, “A representation [like a map] is made with a purpose or goal in mind, governed by criteria of adequacy pertaining to that goal, which guide its means, medium and selectivity”. In other words the form of a representation, in this case mathematical form, reflects the purpose for which the representation was created and hence my questions this afternoon are principally mathematical and not historical.

The first is the question of projection; Are Portolan charts purposely projected? Obviously, the fact they are the surface of a sphere drawn on the plane makes them projected, but the question is more specific; did their creators purposely project them in a consistent way and did these early mapmakers have any knowledge of the error they were introducing through this geometric transformation? This question is extremely difficult to answer because of the fact that any distortion that might be systematic from the projection is somewhat buried in the noise of the distortion caused by simple inaccuracy in the mapping of the coastlines. The most we can hope for is some statistical ruins that might be buried in the non-linear parts of the distortion...

The second has to do with the question of their apparent rotation; as has been pointed out this morning, the charts have various degrees of rotation; why do the parallels, at least in the area of Mediterranean Sea, appear to be rotated? Can we, by analyzing this rotation, gain some insight into the sources and measurement techniques used to construct the charts?

The third question concerns their evolution as geodetic maps; do they get more accurate with time? Do they maintian their accuracy even though many charts have obviously been copied multiple times. Are there any correlations that can be found in looking not only at modern comparisons but also in intrasample variations? (it is here that thin-plate splines are useful, see link to Booksteins paper in links section) Are there any structural changes that we can perceive through their history as a cartographic form?

Principal warps of the LOC's 1320 Portolan Chart

I am going to begin with a bit of a theoretical and historical primer into both the mathematical and philosophical justification for the computational techniques that I am using…many of them have a long history in cartographic analysis and I think it will help you to understand the motivations for some of my research here and on other maps…

For the complete slides of this talk click on the link on the right or on the small Portolan chart above it......