Xref: utzoo sci.astro:5794 comp.graphics:8700 Path: utzoo!utgpu!watmath!watserv1!watcgl!awpaeth From: awpaeth@watcgl.waterloo.edu (Alan Wm Paeth) Newsgroups: sci.astro,comp.graphics Subject: Dymaxion World Globe (Re: formula for star projection) Message-ID: <12446@watcgl.waterloo.edu> Date: 27 Nov 89 22:14:47 GMT References: <6143@shlump.nac.dec.com> <7521@ulysses.UUCP> Reply-To: awpaeth@watcgl.waterloo.edu (Alan Wm Paeth) Organization: U. of Waterloo, Ontario Lines: 44 Fuller's Dymaxion projection is Gnomonic. The geometry of the projection is easy to describe (this is seldom true -- for instance, the Mercator projection is cylindrical but is derived analytically, not geometrically). In short: place a globe inside an Icosahedron and project features (from a point of projection at the center of the globe) out to each face. The Gnomonic (tangent plane) projection is nice in that *all* great circles are represented as straight lines on the plane, but it suffers severe distortions away from the single point of tangency as points halfway around the globe project onto the point at infinity -- a full hemisphere cannot be represented with finite paper. This can be overcome by developing the surface of the sphere onto a circumscribing, encasing object thus limiting the map extents. Fuller chose an icosahedron -- that polyhedron with the largest number of regular, congruent faces (20 equalateral triangles). If laid flat and used as a world-map there are discontinuities -- the map is "interrupted" to use cartographer's jargon. If a traversal along any face crosses an edge you must then find the next connected sheet. If the net is folded up into its 3D icosahedron you have a good approximation to a sphere, with the Gnomonic projection "making up the difference" in a nice way: the geodesics are correct. By this I mean that a taut string between two points on either a sphere or the Dymaxion icosahedron represent the shortest path and on both models the ground track of the string is exactly the great circle route. /Alan Paeth PS - if you have access to map projection software you will need to locate the map centers (points of tangency) at the twenty face centers. These points are in fact the location of the *vertices* of the dodecahedron (and vice verse -- these are "Dual Platonic Solids"). These twenty vertices are at the locations: (0, +-i, +-p), (+-p, 0, +-i), (+-i, +-p, 0), (+-1, +-1, +-1) where p (phi) is the golden section = 1/2 * (sqrt(5) + 1) and where i (phi inv) is 1/phi = phi - 1= 1/2 * (sqrt(5) - 1) [Interesting note: the last eight coordinates define the vertices of one cube "hidden" within the dodecahedron, there are four others) The dodecahedron is resting along an edge, as viewed along the Z axis (see also HSM Coexeter's _Regular Polytopes_, p 53). Of course, the vertices may be rotated about freely -- Fuller chose an orientation which minimized the edge cuts across land masses by placing the triangle corners in oceans. As I recall, in Fuller's globe the North Pole lies on neither a vertex nor a face center. Brought to you by Super Global Mega Corp .com