This is the most exciting and beautiful new map we’ve seen in quite a while. It is a 14-sided polyhedral projection addressing both distortion and partitioning of land masses. The butterfly layout combines legibility and low distortion (comment on ODT maps website).
Here is a more technical explanation from Wikipedia.
The Waterman “Butterfly” World Map Projection was created by Steve Waterman and published in 1996. It is an octahedral transformation of a globe, reviving the Butterfly Map principle first developed by Bernard J.S. Cahill (1866-1944) in 1909. Whereas Cahill’s approach was that of an architect, Waterman derived his design from his work on close-packing of spheres.
This involved the interpretation of a spherical extraction from cubic closest packed spheres, into a corresponding convex hull. Then for its projection; straight lines were used to define each 5 x 5 section onto this convex hull.
Projection employed an equal length delineation approach for the equator. Latitudes were drawn in three straight line sections ( in each octant ) – from pole to fold-line, fold-line to largest line parallel to equator, and then from there to the equator. The largest line parallel to the equator also has equal length delinations. One particular Waterman Polyhedron best served to minimize land sinues ( breaking up of land masses ) and was therefore chosen.
You may find HERE Waterman’s website.
Also, more HERE on map projections.