The Calculus BC class of 1995-96 mapped the earth onto the surfaces of polyhedra as an enrichment activity inspired by the polyhedral architecture of Buckminster Fuller.
There are several methods for mapping the earth's surface. The first is the method by which ordinary maps are made: projection. A Mercator projection (the most common type) is made as follows. Imagine you have a large sheet of paper and a transparent globe with a light in the center (sort of like an overhead projector; only it projects in all directions). Now wrap the paper around the equator of the globe, forming a cylinder with the globe in the center. The globe will project the map onto the surface of the cylinder. Using this method, the latitude and longitude lines would be at the proper angles, though distances would necessarily be distorted. As you can see on any map using a Mercator projection, Greenland appears the same size as South America, though it is actually seven times smaller.
Using the projection method with a polyhedron, you would place the globe inside the polyhedron instead of a cylinder. Of course, placing a transparent globe in the center of a paper model is not a very practical method.
On those polyhedra where you can draw an equator and prime meridian that symmetrically divide the solid (such as a cube with the poles at the center of a face), you can mark off longitude and latitude lines using equal distances. This is the second method. This way, every degree of longitude or latitude is the same length and can easily be marked on the surface of the polyhedron. On many other polyhedra, you can improvise (e.g. on a tetrahedron, though the equator may not divide the solid symmetrically, it is easy to find lines at which to mark off latitude and longitude lines).
Tetrahedrons, cubes, and octahedrons are all easy. What about the other polyhedra? We don't know of any satisfactory, systematic method to transfer a spherical map onto a polyhedra. But for example, with the rhombicosadodecahedron, there is only 1 mm of error if it is treated as a sphere during the mapping. This solid has 30 square faces, 20 triangular faces, and 12 pentagonal faces. In fact, it's the largest Archimedean solid (it has the most faces). The more faces a polyhedron has, the more closely it resembles a sphere, and the smaller the error.

For more on this topic, see the Polyhedra Page.


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