Conformal Cube: The Story of a Search
Daniel M. German I've able to construct the conformal projection of the cube. If your first reaction is to mutter: "The what of the cube?" or just "So?", and if you are interested, please follow these explanations...
You may have seen before panoramas projected in a cube: in flickr for example, there is a "foldables" group and I have some myself (including other polyhedra). These panoramas have all in common the use of the gnomonic or perspective projection: in this type of projection straight lines stay straight. This is great for architectural photos, but as you can see for the model on the right, you get broken lines in the sides of the cube (the columns of the cathedral seem to be broken).
It is possible in theory to devise a conformal projection that would show the sphere as a cube. However it is very difficult to find information on such a projection. The usual internet cartography sources, if they mention this projection, give no clue at all how it can be constructed. The only reference is in this PDF file where a planisphere projected conformally in the cube is given, only quoting John Parr Snyder's personal software. John Parr Snyder is the most precious reference for mathematical cartography available on the web, and though he died in 1997 his personal BASIC program used to generate maps is the only trace left of how he constructed the conformal cube... mostly undocumented, and of course not portable at all. Furthermore, the formulas used in that program are all forward transformations, whereas the projection of an image needs the inverse formulas.
But a reference was there! a 1976 paper in Cartographica titled "Conformal Projections based on Elliptic Functions" by L.P. Lee (another incredibly talented cartographer, "who retired in 1974 as Chief Computer in the New Zealand Department of Lands and Survey") seemed to be the source for these polyhedral globes. Daniel German was very very helpful in locating this rather obscure publication (and others), and I have to thank him for that!
The conformal projection uses an incredible 1872 theorem by Hermann Schwarz who showed that "the spherical triangles into which the surface of the sphere is divided by the planes of symmetry of the regular polyhedra can be conformally represented on the infinite half-plane by algebraic transformations of the stereographic projection on the central plane so that the vertices are represented by the points 0, 1, infinity on the real axis". Unfortunately the paper is in german and is probably even harder to get, so I will able to look how Schwarz showed this...
(the source image for this shot is from the Notre-Dame-de-Reims Cathedral in Reims, France)