SOLSTICE: An Electronic Journal of Geography and Mathematics. (Major articles are refereed; full electronic archives available). Persistent URL: http://deepblue.lib.umich.edu/handle/2027.42/58219 
Download
the linked kmz file and load it into
Google Earth to drive around the model as you read this article. Mathematical Geography and Global Art: The Mathematics of David Barr's 'Four Corners Project. 
Easter Island (Pacific Ocean):  27.10556º latitude, 109.425º longitude 
Kalahari Desert (Africa):  27.51º
latitude, 24.1º
longitude 
Greenland
ice cap: 
72.64º
latitude, 41.92º
longitude 
Irian
Jaya (New Guinea): 
2.11º
latitude, 137.39º
longitude 
Figure 1a. Map of antipodal lands from Monograph #1, 1986. 

Figure 2a. A view of four corners for the location of a tetrahedron in the Earthsphere. Easter Island. 
Figure 2b. A view of four corners for the location of a tetrahedron in the Earthsphere. Easter Island to the Kalahari Desert. At most two corners can be viewed simultaneously in Google Earth when the skin of the Earth is visible. 
Figure 2c. A view of four corners for the location of a tetrahedron in the Earthsphere. Easter Island to the Kalahari Desert. At most two corners can be viewed simultaneously in Google Earth when the skin of the Earth is visible; here, the skin from Figure 2b has been removed with all else the same. Notice that now all four corners are visible. 
Figure 2d. A view of four corners for the location of a tetrahedron in the Earthsphere. Rotate the globe to better view the corners simultaneously. 
Figure 2e. A view of four corners for the location of a tetrahedron in the Earthsphere. Add edges to get a better view of the embedding of the sculpture in the globe. John Nystuen asked also to have some sort of continental context, for orientation, added to the image. Figure 2f illustrates one approach to that. 
Figure 2f. A view of four corners for the location of a tetrahedron in the Earthsphere. Add edges to get a better view of the embedding of the sculpture in the globe and reference them to continental outlines, imported as a shapefile, so that they, too, show through the Earthsphere. Easter Island is the bright red vertex closest to the center; the vertex to the left and on the other side of the globe is on Irian Jaya. The vertex in the Kalahari Desert is on the back side of the globe (east/west pattern in Africa is thus inverted). 
Figure 2.g. An
equilateral triangle cut from a piece of paper was used as an aid in
locating the collection of points which could serve as vertices of the
proposed sculpture. By placing its corners on the circle drawn on
the azimuthal map, one can rotate the triangle and discover the
collection of points which satisfy the condition that they all fall on
dry land. This procedure confirmed that Barr's own discovery of
the location set {Easter Island, Kalahari Desert, Greenland Ice Cap,
and New Guinea} was feasible. Again, only half the sphere is
visible so that one must imagine the Easter Island location in this map
centered on its antipodal point in the Thar Desert.

Figure 3a. Screen capture from Google EarthGrid. See link in Reference section. 
Figure 3b. Cube. 
Figure 3c. Octahedron. 
Figure 3d. Dodecahedron. 
Figure 3e. Icosahedron. 


Figure 5a. Easter Island, on the outline map, marks its antipodal point in the Thar Desert in India. These antipodal points are antipodal vertices of the octahedron. The remaining four vertices all lie on a great circle in equatorial position between these two antipodal poles. 
Figure 5b. Easter Island, on the antipodal skin map, marks its antipodal point on the outline map in the Thar Desert in India. These antipodal points are antipodal vertices of the octahedron. The remaining four vertices all lie on a great circle in equatorial position between these two antipodal poles. 
Figure 6a. The interval along the equatorial great circle (bisecting the antipodal poles of Easter Island and the Thar Desert. The yellow outline represents the Earth's continents. The colored skin is the antipodal lands map. The only candidate locations for vertex locations for the remaining four vertices are locations along the great circle where landmasses from both maps intersect. In this interval, it appears that there are no such candidates. A closer look might reveal otherwise; however, this scale is the scale in use herethere are some near misses in Australia. Thus, because there is no candidate in this interval, the octahedron cannot fulfull both of Barr's initial conditions. 
Figure 6.b. Again, another interval with no candidate points. 
Figure 6c. Yet another interval along the great circle with no candidate points. 
Figure 6d. The final interval also offers no candidate points. Thus, there is also no problem at the boundary points. 

Figure 7a. The dodecahedron is embedded in the sphere, with one vertex at Easter Island. The yellow semitransparent triangle represents three vertices adjacent to the vertex at Easter Island. All must be simultaneously placed on land. As the animation suggests, even if Antarctica is admitted as "land," such simultaneous placement is not possible. Thus, a dodecahedron could not have been selected. One might, however, wish to take a closer look lest some small islands serve as possible locations. The Google Earth .kmz file permits such a closer look. Depending on the scale chosen, simply chop the motion into suitable intervals from island to island to show that simultaneous placement of these three vertices on land is, or is not, possible. 
Figure 7b. In
the original text rough drawings and cardboard cutouts were used for
analysis. The Google Earth capability expands not only the
visualization level of the analysis but also its precision.

Figure 8a. Both Easter Island and its antipodal point in the Thar Desert serve as poles for the embedding of the icosahedron in the sphere. Each is surrounded by a set of five adjacent vertices, all of which must be land based. While one might imagine such a possibilty surrounding the Indian location, it is clearly not possible for the Easter Island set. The magnitude of the Pacific Ocean makes it impossible, at least at this scale and perhaps at all others! Take a further look in the .kmz file. 
Figure 8b.
In
the
original text rough drawings and cardboard cutouts were used for
analysis. The Google Earth capability expands not only the
visualization level of the analysis but also its precision.

.
Solstice:
An Electronic Journal of Geography and Mathematics,

Congratulations to all Solstice contributors. 