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Cubic Globes

Cubic gnomonic map
Cubic gnomonic map
Cubic gnomonic map Assorted unfolded gnomonic cubic maps, graticule spaced 10°. From top to bottom: face-centered poles and face lay-out 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.
COBE Quadrilaterized Spherical Cube map Faces of the COBE Quadrilaterized Spherical Cube are by convention arranged in a T. Although not intended for world maps, it is interesting comparing its distortion pattern with that of the gnomonic map in Reichard's lay-out.
Quadrilaterized Spherical Cube map The Quadrilaterized Spherical Cube was similarly designed for celestial data and is presented here for comparison only.
Cubic HEALPix map HEALPix map for H = 4 with the triangular facets rearranged into a cube; scale is 20% greater than for gnomonic maps.
This map is exactly equal-area, but unlike the previous projections, its distortion pattern is completely different in the equatorial and polar faces.

Although mapping into the regular hexahedron (an ordinary cube) is prone to pronounced distortion, the nonsensical notion of “Earth-in-a-box” has long fascinated me. Once I plotted and folded such a map by hand alone, using an arbitrary compound of cylindrical and Collignon-like 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 equal-area 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 equal-area 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 equal-area.

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.

Octahedral Maps

Cahill's Butterfly Maps
Butterfly gnomonic map

Gnomonic map in Cahill's “butterfly” lay-out, central meridian 20°W; printable versions available

Butterfly Collignon map

A map using Collignon's projection is easily modified to fit the faces of a regular octahedron; central meridian 20°W

Starting in 1909, Bernard Cahill patented several maps based on the octahedron, using gnomonic, conformal or arbitrary projections. All were based on eight equilateral triangles which could be arranged in several ways, the commonest called a “butterfly map”. Here it is presented in the gnomonic form with poles in opposite vertices, cutting meridians every 90°. Other variants are conformal or equal-area, but include additional interruptions or slightly curved edges.
Apparently no Cahill map was ever much popular, even after thirty years of promotion by the author. More recently, Gene Keyes has revived interest on this projection with a revised version and different land arrangements.

The butterfly lay-out, superficially resembling the conoalactic projection, benefits from the continental distribution much like star projections.

A Modified Collignon Map
Collignon's curious projection can be modified to an octahedral “butterfly” variant in three straightforward steps:
  1. interrupting the diamond-shaped version along three meridians, creating eight triangular lobes
  2. changing both horizontal and vertical scales in order to make lobes equilateral while keeping area constant
  3. rearranging the lobes around the North pole.
The second step can be omitted yielding a slightly different map which folds into an elongated irregular octahedron. Either map is still equal-area but, of course, pseudocylindrical only at each lobe.
Using the Semiregular Truncated Octahedron
Butterfly gnomonic map

Gnomonic map on a semiregular truncated octahedron, central meridian 20°W, with each square face split in four pieces (each octant is a regular hexagon surrounded by three right-angled isosceles triangles); printable versions available

Of all regular polyhedra, projecting the world gnomonically on an octahedron is the simplest task since all meridians and the Equator are mapped into straight lines; therefore octant boundaries can be easily deduced from map coordinates.

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.

Waterman's Projection System
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. Full-color wall maps of the world are commercially available.
Bottom: gnomonic projection adapted to Waterman's polyhedron.
S.Waterman's grid
Gnomonic projection on Waterman's truncated octahedron
Gnomonic projection on Waterman's truncated octahedron

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 sphere-packing — 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 51/2) 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 lay-out combines legibility and low distortion.

As adopted in commercial maps, Waterman's projection is neither conformal nor equal-area. 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.

HomeSite MapPolyhedral Projections - part 1Polyhedral Projections - part 1Polyhedral Projections - part 3 — June 16, 2018
Copyright © 1996, 1997 Carlos A. Furuti