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Computing Space II: Taking Waldo Tobler’s Geography 482

Today’s post is the second in a continuing series called,”Computing Space,” which will highlight the lives and work of many of the mostly unknown cartographers, geographers, mathematicians, computer scientists, designers and architects who had a hand in the birth of today’s computer cartography.

When working with the archives and personal papers of the pioneers of early computer cartography one has to be open to surprises. In the 1960s and 70s the development of computer cartography was not only the realm of geographers and mapmakers, but was also inhabited by computer scientists, mathematicians, architects, and designers, along with a whole host of spatially oriented physical and social scientists hoping to apply the new technology to particular problems. For this reason, the archives that make up the history of this period are filled with everything from papers and notes relating to differential and algebraic geometry, to color theory, algorithms, topology, the theory of computation, novel data structures and, even in one case, reflections on the maddeningly complex incompleteness theorems of Kurt Godel, (one of my favorites) and the logical symbolism of the Austrian philosopher, Ludwig Wittgenstein.

So it was really no surprise when I was going through the archives of Roger Tomlinson, who many have called the father of Geographic Information Systems, and which have recently come to the Library of Congress, that I came across a reading list and problem set from Waldo Tobler’s (1930-) innovative and groundbreaking University of Michigan class, Geography 482, popularly known, as Analytical Cartography.

First page of the reading list for Waldo Tobler's Geography 482. Roger Tomlinson Papers, Geography and Map Division, Library of Congress.

First page of the reading list for Waldo Tobler’s Geography 482. Roger Tomlinson Papers, Geography and Map Division, Library of Congress.

Tobler himself described the class in a short paper  that details its philosophy and why he called it “Analytical.”

A popular title would have been Computer Cartography. This did not appeal to me because it is not particularly critical which production technology is used. […] The substance is the theory that is more or less independent of the particular devices; equipment becomes obsolete rather quickly anyway. Mathematical Cartography could have been used for the title, but this already has a definite meaning and I had in mind more than is usually covered under this heading. Cartometry, the study of the accuracy of graphical methods, is another available term, but has a rather narrow meaning. One could also speak of Theoretical Cartography. This did not appeal to me on two grounds. It would frighten students, who are always concerned that they learn something practical. Secondly, the precedent is not very attractive. Max Eckert, for example, wrote a great deal about theoretical cartography but did not solve many problems. I wished to emphasize that mathematical methods are involved, but also that an objective is the solution of concrete problems.

One of the most amazing things about his assigned reading list is the variety of material from across the spectrum of cartography and computer science of the time. While one would of course expect to see Robinson’s Elements of Cartography used in the class, the inclusion of Minsky’s Perceptrons, Goldman’s Information Theory and Cornsweet’s Visual Perception are a surprise.

Waldo Tobler at University of Michigan. Image Courtesy of the University of Michigan Faculty History Project

Waldo Tobler at University of Michigan. Image Courtesy of the University of Michigan Faculty History Project

Perceptrons, for example, written by Marvin Minsky, who was the co-founder of MIT’s Artificial Intelligence Laboratory, was first published in 1969, and is an introduction to computational theory; in particular, to a type of machine learning using neural networks. The book contains a number of detailed mathematical proofs showing, at least mathematically, that simple computers can be made based on neural models; hardly the subject one would find on a reading list in a geography course today. When I recently asked Tobler about the reading list, he explained that he wanted his students to get an extremely broad idea of what was happening in the computational side of the subject and to get a feel for the deep mathematical results coming out of computer science. Something missing from today’s curriculum.

Mathematics and computational theory are at the heart of Tobler’s class for a reason. He explains,

As regards the substance, rather than the course title, the major difference is perhaps only that a somewhat more general view is taken of the subject [of cartography]. It is appropriate to introduce students to what is already in the literature, to introduce similar concepts that occur in other branches of knowledge, and to suggest new directions. Embarrassingly frequently cartographers claim as unique problems those which also occur in other fields and which may even have been solved there. […] But clearly the application of mathematical methods to cartography is growing rapidly.

Tobler’s class was rigorous both in its treatment of topics and in the intensity of the mathematics presented. The exercises and problem sets were even more technical and treated subjects like profile and contour generalization (what might go by the the title Morse theory today) and curvature and projection calculations that involved fairly advanced differential geometry.

For Tobler, teaching what was happening in the present was only part of the story….it was the future he wanted to capture and to prepare his students for. He thought (and continues to think) deeply, conceptually, and mathematically about cartography and the technology that would revolutionize it. He sums up his philosophy writing,

What is easy, convenient, or difficult depends on the technology, circumstances, and problem. The teaching of cartography must reflect this dynamism, and the student can only remain flexible if he/she has command of a theoretical structure as well as specific implementations. The spirit of Analytical Cartography is to try to capture this theory, in anticipation of the many technological innovations which can be expected in the future; wrist watch latitude / longitude indicators, for example, and pocket calculators with maps displayed by colored light emitting diodes, do not seem impossible. In a university environment one should not spend too much time in describing how things are being done today. The course outline presented here tries to avoid this bias.

Below, I leave readers with an exercise from Tobler’s class involving the calculation of Gaussian curvature. One should note the annotation in upper right hand corner of the page, “difficult, but only because not all of the details are specified–no particular difficulties, but acquaintance with differential geometry would be helpful.”




Task #3 Curvature Computation from Tobler’s Geography 482. Roger Tomlinson Papers, Geography and Map Division, Library of Congress.

I await the submission of all your solutions….


  1. aldo celso smaniotto
    December 28, 2015 at 12:48 pm

    I think this is very interesting
    We hope this is very interesting for always

  2. Bruce Pigozzi
    January 14, 2016 at 8:24 am

    I met Waldo in his UoM office back around 1971. The first thing that struck me was the dozens of empty LEGGS pantyhose containers scattered around the room. Responding to my very puzzled look Waldo said “Great for contemplating spherical forms!”

  3. Eric
    April 14, 2016 at 5:03 pm

    That’s my dad!

    He made computer programs to remap a map onto odd objects like ties and apples. You could navigate from them. What was really impressive was all his old books of maps. He could not pass by a used book store without going inside. I remember the musty smell of used book stores he’d drag me into as a kid.

    He still loves his maps.


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