Assorted unfolded gnomonic cubic maps,
graticule spaced 10°. From top to bottom:
facecentered poles and face layout used by Reichard (the choice
of central meridians is conjectural); poles on corners
(printable
versions available); poles slightly displaced from
face centers in order to reduce land cutting; again, cuts reduced
with one face split into four triangles (after a suggestion by
Chris Maynard).
Area exaggeration is strong near face edges and especially corners; the Equator and all meridians are straight except at edges. 






Although mapping into the regular hexahedron (an ordinary cube) is prone to pronounced distortion, the nonsensical notion of “Earthinabox” has long fascinated me. Once I plotted and folded such a map by hand alone, using an arbitrary compound of cylindrical and Collignonlike projections. Fortunately now I own a computer... Modern advertisers have also often employed the concept.
Different arrangements of six gnomonic square faces (usually four in equatorial and two in polar aspects) were used by Christian G. Reichard (1803) and other cartographers, in world and celestial atlases. An oblique version was used by C.A. Schott (1882). None of those are equalarea or conformal.
More recently, cubic globes were revived for astronomical and cosmological research. COBE (Cosmic Background Explorer), a satellite probe launched in 1989, measured in high detail incoming microwave radiation from the entire sky — the conceptual celestial sphere. The COBE mission sparked considerable scientific interest, as its results promised to reveal structures dating from the origin of the universe. Of significance here are the schema selected for efficiently storing the substantial amount of data returned to Earth (ever improving instruments onboard more recent missions provided even more detailed and voluminous information). Since radiation density is an important factor when analysing raster data, the chosen projection should emphasize areal preservation; on the other hand, shapes should not be too distorted — otherwise different regions could not be easily compared.
The chosen compromise design, the COBE Quadrilaterized Spherical Cube (Chan and O'Neill, 1975) or “sky cube”, is based on the gnomonic projection but the center of every face is expanded in order to make an approximately equalarea map. The distortion pattern is identical in all six faces, which by convention are arranged in a “T” shape with four equatorial faces in a row and two polar faces above and below on the far right.
A related design, the Quadrilaterized Spherical Cube (O'Neill and Laubscher, 1976), involves more complex computation, but is exactly equalarea.
Also associated with analysis of cosmological raster data, the HEALPix projection was not intended as polyhedral, but some of its forms can trivially be so rearranged: the H = 4 case yields a cube; the rescaled H = 6 a hexagonal prism, and the rescaled H = 3 a triangular prism capped by two regular tetrahedra.
The butterfly layout, superficially resembling the conoalactic projection, benefits from the continental distribution much like star projections.
The same projection applied to a truncated octahedron reduces area distortion in part of the map since the six original vertices are clipped, or more properly “flattened” into square faces closer to the inscribed spherical surface. Splitting each new face into four right triangles introduces very few additional interruptions in land masses compared to the original butterfly map, except in North America and Northern Asia. Several of Cahill's butterfly maps were based on this solid, with the square faces further subdivided into triangles which were spread along the map for a more symmetrical look and lesser splitting of land masses.
Right: a 5° graticule for
Waterman's projection with an inset for Antarctica;
image courtesy Steve Waterman. Below: Waterman's projection, central meridian 20°W. Fullcolor wall maps of the world are commercially available. Bottom: gnomonic projection adapted to Waterman's polyhedron. 

More recently, Steve Waterman devised a polyhedral projection addressing both distortion and partitioning of land masses, and also based on a truncated octahedron.
Waterman studied spherepacking — the old mathematical problem of juxtaposing identical spheres in the smallest possible volume — and compiled a list of polyhedra whose vertices are defined by the centers of a set of packed spheres. One of those, called W5 (loosely defined as the convex hull of a cluster of unit spheres in cubic close packing whose centers are not farther away from the origin than 5^{1/2}) is the foundation of this projection. It resembles the Archimedean truncated octahedron, but the edges of its square faces are only half as long as the hexagons' longer edges: therefore, in a continental arrangement similar to Cahill's butterfly map, cuts in northern shorelines are not so deep. All square faces are conventionally split in four right triangles, except the one with the South Pole, which also borrows narrow slices of adjacent hexagonal faces, thus keeping Antarctica nearly whole.
Like in other polyhedral maps, the lobes can be rearranged depending on the map's purpose. For instance, the left and right map halves can be joined at the South Atlantic edges, or they can be swapped for a conterminous North or South Pacific Ocean. Similarly as in Cahill's maps, the butterfly layout combines legibility and low distortion.
As adopted in commercial maps, Waterman's projection is neither conformal nor equalarea. Equatorial scale is constant; all meridians are broken straight lines and scale along each one is linear, but taking the Equator as reference, only the four boundary meridians at 45° East and West of each lobe's central meridian are standard lines.