Optimal orientations of discrete global grids and the Poles of Inaccessibility

Spatial analyses involving binning often require that every bin have the same area, but this is impossible using a rectangular grid laid over the Earth or over any projection of the Earth. Discrete global grids use hexagons, triangles, and diamonds to overcome this issue, overlaying the Earth with equally-sized bins. Such discrete global grids are formed by tiling the faces of a polyhedron. Previously, the orientations of these polyhedra have been chosen to satisfy only simple criteria such as equatorial symmetry or minimizing the number of vertices intersecting landmasses. However, projection distortion and singularities in discrete global grids mean that such simple orientations may not be sufficient for all use cases. Here, I present an algorithm for finding suitable orientations; this involves solving a nonconvex optimization problem. As a side-effect of this study I show that Fuller's Dymaxion map corresponds closely to one of the optimal orientations I find. I also give new high-accuracy calculations of the Poles of Inaccessibility, which show that Point Nemo, the Oceanic Pole of Inaccessibility, is 15 km farther from land than previously recognized.