Waterman butterfly projection
The Waterman "Butterfly" World Map is a map arrangement created by Steve Waterman. Waterman first published a map in this arrangement in 1996. The arrangement is an unfolding of a globe treated as an octahedron, evoking the butterfly map principle first developed by Bernard J.S. Cahill (1866–1944) in 1909. Cahill and Waterman maps can each be shown in various profiles, typically linked at the north Pacific or north Atlantic oceans.
As Cahill was an architect, his approach tended toward forms that could be demonstrated physically (i.e. his flattenable rubber-ball map). Waterman, on the other hand, derived his design from his work on close-packing of spheres. This involves the interpretation of a spherical extraction from cubic closest-packed spheres into a corresponding convex hull, as demonstrated in the following pictures. (Illustrated is the W24 hull, but the projection is based on the W5.)
Then, for its projection, straight lines are used to define each 5 × 5 section onto this convex hull. According to annotations on modern versions of the map, the projection divides the equator equally amongst the meridians, though Popko states the projection is the gnomonic, an incompatible description. Latitudes are drawn in three straight-line sections in each octant: from pole to fold-line; from 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 delineations. Waterman chose a specific Waterman polyhedron and central meridian to avoid interrupting land masses.
Like Buckminster Fuller's 1943 Dymaxion Projection, an octahedral butterfly map can show all the continents uninterrupted if its octants are divided at a suitable meridian (in this case 20° W) and are joined, for example, at the North Atlantic, as in the 1996 version.
See also 
- B.J.S. Cahill Introduces Butterfly Map of the World (1909)
- Edward S., Popko (2012). Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere. Taylor & Francis. pp. 20–21. ISBN 9781466504295.
- Carlos A. Furuti, Map Projections: Polyhedral Maps
- Rotating Waterman as globe Izidor Hafner
- Flat Worlds, Earth Literacy, and Map Alphabetism (Write-up comparing other world maps.)
- Positive News Youth Views site (Waterman selected as their signet.)