INTRODUCTION II. REFERENCE FRAMES IN FLUID DYNAMICS III. REFERENCE FRAMES IN THE LAW IV. WHAT DO REFERENCE FRAMES MEAN FOR ENVIRONMENTAL LAW? I. INTRODUCTION
In 1569, the Flemish geographer, Geradus Mercator, published a new type of map. His innovation placed the Equator as its standard parallel, making lines of latitude and longitude intersect at right angles to one another. (1) Previous cartographers had realized that the Earth was round and placed continents as best they could but, as the Mappa Mundi of 1449 shown below makes clear, their maps were of no practical use with regard to the ocean. (2)
The Mercator Projection was particularly well-suited to its time because it allowed navigators to determine lines of constant true direction--the compass direction on the map connecting two points was the same compass direction that a ship would follow at sea. (3) At a time of maritime empires, far-flung voyages, and exploration, this map was just what captains needed to cross an ocean. Mercator's vision has endured and remains the standard map on classroom walls around the world, the conventional and accurate means to portray the surface of the Earth. (4)
At least that's the story, but it's not really true.
While maintaining accurate geographic direction between lines of latitude and longitude, the Mercator Projection map quickly starts to distort areas and shapes once one moves north or south from the Equator. (5) Consider, for example, the exchange on the popular TV show The West Wing White House staffers C.J. Cregg and Josh Lyman with Professor Sayles and his well-intentioned colleagues in the Organization of Cartographers for Social Equality, who want the White House to replace the Mercator Projection map in classrooms with the more accurate Peters Projection map:
SAYLES: [Showing the Mercator Projection map on the screen] Here we have Europe drawn considerably larger than South America when at 6.9 million square miles South America is almost double the size of Europe's 3.8 million.
HUKE: Alaska appears three times as large as Mexico, when Mexico is larger by .1 million square miles.
SAYLES: Germany appears in the middle of the map when it's in the northernmost quarter of the Earth.
JOSH: Wait, wait. Relative size is one thing, but you're telling me that Germany isn't where we think it is?
FALLOW: Nothing's where you think it is.
C.J.: Where is it?
FALLOW: When Third World countries are misrepresented they're likely to be valued less. When Mercator maps exaggerate the importance of Western civilization, when the top of the map is given to the northern hemisphere and the bottom is given to the southern ... then people will tend to adopt top and bottom attitudes.
C.J.: But ... wait. How ... Where else could you put the Northern Hemisphere but on the top?
SAYLES: On the bottom.
FALLOW: Like this.
[The map is flipped over.]
C.J.: Yeah, but you can't do that.
FALLOW: Why not?
C.J.: 'Cause it's freaking me out. (6)
C.J. is understandably upset when her world is turned upside down, but there is no obvious reason why the map should have north on top or, for that matter, be centered along the Equator. Indeed, the first question posed by professors in introductory geography courses often is the simple yet disarming, "Why is north up?" (8) The Mappa Mundi shown on the first page of this Article, for example, had south on top. Or imagine a map with the North Pole at the middle, projected outward from the Arctic. (9) This projection is disorienting. Finding Alaska takes some time.
In the past, this projection was largely irrelevant or a simple curiosity. The melting of the ice cap along with the continued discovery of natural resources at the North Pole, however, has made this projection increasingly relevant for understanding rapidly evolving geopolitics. More broadly, conceptualizing the world as centered on Western Europe and the Atlantic (as the typical projection implies) may be less relevant as the global economy pivots toward South Asia, (10) or, potentially, as there is greater global commerce across the North Pole than across the North Atlantic. (11)
Most often we use maps to represent the physical size of geographic space scaled to some projection, but a map need not just be a spatial projection of geographic relations. A map is a model--a representation of the spatial distribution of natural, political, economic, or any other type of information or data. The size of a projected area on a map may be scaled by income or by race rather than actual geographic area. (12) Within the United States, voting maps for political elections scaled by population indicate far different relations than the more typical projection of geographic scale. (13)
Maps powerfully provide a great deal of data in an accessible manner; but they cannot reflect reality. At its most obvious, geographic maps do not fully reflect the world because something is inevitably lost when projecting a three-dimensional object on a two-dimensional surface. More subtly, map projections inherently depend on a sequence of often unarticulated, or even unrecognized, assumptions. The geographic map not only loses some physical accuracy depending on the projection system, but also smuggles in a series of normative assumptions that are much less obvious--hence Professor Fallow's concerns about the primacy of Northern Hemisphere countries over the Global South. (14) Maps both clarify and distort.
Despite their limitations, maps also create opportunities. The way they frame the world allows questions to be asked that might otherwise be ignored or unseen. Looking at the Mercator map, for example, tells us virtually nothing about the North Pole. A polar projection, by contrast, immediately brings into stark relief the complicated jurisdictional conflicts among the northern countries. (15) These conflicts are invisible on a Mercator projection, where the North Pole is distorted out of all recognition.
These same challenges and opportunities are also present in the natural sciences. There, in addition to space, scientists must combine temporal and spatial information to develop an understanding of the biophysical dynamics that shape the distribution of natural resources, organisms, communities, and chemicals. In these cases, the projection or presentation of temporal and spatial information forms the basis for not just presenting information, but also how that information might be abstracted into relations that form the basis of analytical reductions, from simple correlations and analytical equations to complex computational models.
How objects move through space and time has been fundamental to the intellectual development and scientific theory of physical processes. Two eighteenth century mathematicians--Leonard Euler and Joseph Louis Lagrange--made seminal contributions to how movement is conceived in the natural sciences, specifically in the field of fluid mechanics. (16) They are not popular names today, but their different ways of framing the world still dominate how scientists conceptualize and measure physical phenomena.
Central to both of these mathematicians' early work was conceptualizing the seemingly simple problem of water flow and the velocity of water "parcels." Euler developed a theory of fluid mechanics that began with a clearly defined system of Cartesian coordinates--a mathematical description of the space of interest. (17) This theory allowed him to develop a powerful system of equations that could describe the distribution of force and velocity within a particular region of fluid.
In contrast, Lagrange's formulation of mechanics was not tied to any one coordinate system--rather, any convenient independent variable could be used to describe the system (e.g., distance along a river channel).18 Lagrange focused on specific actors within the larger system and analyzed how those specific actors (e.g., parcels of water) behaved wherever they might be. He then calculated the characteristics of the fluid that caused the particular movements of the object of interest. (19)
In simple terms, Euler's approach framed the system as a black box with inputs and outputs: a flux. The fluid moved in and out of that region and, by studying the characteristics of the collection of fluid parcels at the entry and exit, one could quantify the forces exerted within that region that might cause the changes observed at the boundaries. Lagrange's reference frame looked inside the box, tracing specific actors within the system: a flow. While the Eulerian and Lagrangian reference frames were derived from mathematics and physics for fluid mechanics, they have been adopted by natural scientists as the basic reference frames for the analysis of the movement of objects within the environment. (20)
A simple example of these different frames of reference can be seen in monitoring traffic. The standard approach is to use road-cams above bridges. This is an Eulerian approach, measuring flux from a fixed position. The new smart phone application, Waze, by contrast, allows each Waze user to transmit his or her location or speed on the road at that moment. (21) This is a Lagrangian framing, following individual actors rather than measuring a flux at a fixed point. Both frames provide an accurate view of traffic, but from very different perspectives.
The debates surrounding the Mercator Projection or the Eulerian and Lagrangian approaches take place in very different fields, but they are asking the same basic question: What is the most effective reference frame for conceptualizing, understanding, and analyzing space and movement in the natural world? While largely underappreciated, such basic conceptualizations are the first step in any analysis of a system, necessary priors to identifying the parts of the system, how those parts interact, what...
Turning the world upside down: how frames of reference shape environmental law.
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COPYRIGHT GALE, Cengage Learning. All rights reserved.
COPYRIGHT GALE, Cengage Learning. All rights reserved.