## Saturday, March 31, 2018

### [xjommsdj] Polyhedral map projections

Project the globe onto a cube then print the 6 faces onto 6 pages.  People understand square-shaped maps.  There are several ways to project 1/6 of a sphere onto each square face.

Of course, the four squares around the equator can be joined and printed onto a (long) flat page.

Each square could cover slightly more than 1/6 of a sphere, allowing overlap with neighboring squares for convenience.

Projection onto a cube is equivalent to views from the 6 vertices of the regular octahedron.  We could view from more than 6 points: what are some nice symmetric collections of points on a sphere?  Tammes problem.  We prefer symmetry so readers can understand how different pages relate to each other (especially in distance).  12 pages seems doable; maybe 20 corresponding to the vertices of the regular dodecahedron? The region covered will no longer be naturally square; use circles or the shape of the polyhedral face.  Azimuthal equidistant projection is nice for circles.

12 corresponding to the faces of the rhombic dodecahedron, equivalently edges of a cube, seems nice.  The cube edges correspond to the short diagonals of each rhombic face.  Maybe do two-point equidistant map projection based on the long diagonal.