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Dymaxion Map (2)
Another of Fuller's inventions, in response to a very different problem than architectural design, is based on similar geometric principies. The problem is to draw a flat map of the world without the gross distortions inherent in the Mercator projection. In the early 1940s, dismayed by this widely accepted map's inaccurate depiction of our worlda visual lie presenting Greenland as three times the size of Australia, when exactly the reverse is trueFuller was determined to discover a better solution.
Let's consider the design problem. Visual data must be reliably translated from the surface of a spherical "whole system" onto a flat display with only one side. To understand how the Mercator projection attempts to accomplish this task, imagine wrapping a large rectangular piece of paper around a transparent globe, forming a cylinder that touches only the equator. The geographical outlines are then projected directly outward to the cylindrical paper, as if by a light source inside the globe casting omnidirectional shadows. As a result, the visual information is accurately translated to the paper only at the equator; some distortion exists slightly above and below the equator, and it increases radically as one goes farther north and south on the map. In many versions Antartica is left out altogether, even South America is smaller than the gigantically distorted Greenland, and certain land areasdepending on which country has produced the mapmust be split in half to turn the cylinder into a flat poster.
Unwilling to accept such distortion as necessary, Fuller started from scratch. If a spherical system is to be translated onto a flat surface, what is the most efficient and direct solution? It's a geometry problem; the relevant "generalized principle" involves the polyhedral system which best approximates a sphere with only one type of face. (The latter consideration insures an evenly distributed projection.) That system, which is of course an icosahedron, is the basis of a reliable and simple solution.
Imagine a globe with the edges of a spherical icosahedron superimposed on its surface by thin steel straps. Chapter 14 described planar polyhedra expanding into spherical polyhedra, as if drawn on balloons; we now visualize the reverse process. The steel arcs slowly unbend into straight-edge chords, while the curved triangle faces flatten out into planar equilateral triangles. The overall shape change is relatively slight (consider for comparison a spherical tetrahedron undergoing the same operation, or a sphere turning into a cylinder as in the Mercator projection), and the global "whole system" is preserved. As the globe transforms into an icosahedron, twenty spherical triangles with 72-degree corners become planar triangles with 60-degree corners; 12 degrees are squeezed out of each angle. (With three angles per triangle, 12 degrees times 60 angles equals none other than our old friend the "720-degree takeout.") Because each triangle of the spherical icosahedron covers a relatively small portion of the sphere and is thus fairly flat, the distortion during this transformation is minimaland in fact invisible to the untrained eye. Moreover, the polyhedral projection automatically distributes the distortion symmetrically around the globe's surface and thereby insures that the relative sizes of land masses are accurate. (This is why a regular polyhedron is a preferred vehicle; different types of faces would distort slightly different amounts during the transition from spherical to planar.) Finally, all geographical data are contained within the triangular boundaries; there is no "spilling" of information or need to fill in gaps with "extra" land as in the Mercator.
The next step is straightforward. Unfold the icosahedron to display its twenty triangles on a flat surface. The result is a map of the entire world with little distortion of the relative shape and size of land masses, and no breaks in the continental contours (Fig. 16-1). No nation is split and shown on two opposite sides of the map as if separated by 20,000 miles.
That last step required more work than is immediately apparent, however. It took Fuller two years of experimenting to find an orientation in which all twelve icosahedral vertices land in the oceanan essential requirement if land masses are not to be ripped apart. Observe in Figure 16-1 that many of the vertices are extremely close to shore. One can imaglne the frustrating task of searching for twelve water locations; moving a vertex away from land on one side of the globe would instantly result in a number of vertices bumping into land somewhere else. Contemplating the five or six angular gaps which are precariously close to land masses, one suspects that Fuller's final arrangement may be a unique solution to the problem.
The Dymaxion Map, awarded U.S. Patent 2,393,676 in 1946, is an unprecedented cartographic accomplishment, which was made possible by a straightforward application of geometry. This map is therefore another superb example of the design-science approach. Fuller considered the problem outside the context of traditional map-making; rather than attempting to work with and refine history's previous best solution, he started over. He sets the example of considering a design problem as a whole system.
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