At 05:47 PM 9/4/2006, Schroeppel, Richard wrote:

On other subjects, specifically land-based tetrahedra: Have we looked for land-based octahedra or cubes?

In 1996, when I found the land-based tetrahedron that Jim Propp mentioned, I also looked for an octahedron. There seemed to be no hope for an island-free answer, and the only suitable island vertex I came up with was Palmyra Atoll. At the time I thought there might not be much land there at high tide, but apparently there is about a square mile of it. The location has since been acquired by the Nature Conservancy, which has information and pictures on their website: http://www.nature.org/wherewework/asiapacific/palmyra/ . (The antipodal point of Palmyra has also undergone some changes in the intervening decade; the country was then called Zaire, now it's the Democratic Republic of the Congo.) Here are the vertices of the octahedron: 5.87 N, 162.10 W Palmyra Atoll 5.87 S, 17.90 E D.R. Congo, near Kikwit 75.00 N, 49.55 W north central Greenland 75.00 S, 130.45 E Antarctica, Wilkes Land 13.76 S, 73.54 W Peru, west of Cuzco 13.76 N, 106.46 E northeastern Cambodia This assumes a spherical earth; unlike the tetrahedron, finding a geometrically precise octahedron on the real, non-spherical Earth will require points not quite on the surface. In particular, the points in Greenland and Antarctica are closer to each other than the other antipodal pairs. I found these polyhedra by examining an old-fashioned paper atlas, using a computer only to do the trigonometry. Finding a land cube this way seems unlikely. It's tricky enough to find any land tetrahedron; finding a cube amounts to finding a tetrahedron for which each the antipodal point of each vertex is also on land. It would be interesting see whether more and better results could be achieved using a contemporary GIS. -- Fred W. Helenius fredh@ix.netcom.com