The Waterman Butterfly is one of the finest map representations ever produced, but it is uncommon. It is aesthetically beautiful and has minimal distortion in angles, distances and area across the entire globe. Mathematically it belongs to a general class of tetrahedral projections generated from an arrangement of close-packed spheres. The Waterman Butterfly is based on a truncated octahedral arrangement. A similar projection can be made using the same octahedral zones, but arranged to produce an M shape which is particularly useful for plotting multiple gridded datasets with minimal wasted space. Either way, the map can be cut up and folded to make polyhedra that is sufficiently similar to a globe.
You won't find the Waterman Butterfly in most mapping/GIS software so I've been looking for the mathematical description of the model or a code that I can easily translate. I desperately want to plot my own data on the maps myself. I finally found a code a few days ago on GitHub by Justin Kunimune. The original code is in Java and all my work is done in Matlab. I spent the last couple of weeks translating the code and am starting to produce viable maps (sample above). I'll be building a whole toolbox with a number of utilities to plot raster and vector data. I will probably post it back to GitHub myself.
Justin includes a number of map projections in his repository, many of which are fairly common. However, he also includes several versions of the Cahill projection which are quite similar to the Waterman projection. There are some minor advantages to the Cahill projection, namely less distortion (Cahill actually produced his projection decades before Waterman), but the differences are mostly cosmetic and the choice is personal preference. I may implement the Cahill projections as well so that I have options.
I'll update this post as the work progresses.