awpaeth@watcgl.waterloo.edu (Alan Wm Paeth) (11/28/89)
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.