Map Projections

Polyhedral Maps

Most cartographic problems would disappear on a polyhedral Earth

Introduction

• first mapping the sphere into an intermediate zero-Gaussian curvature surface like a cylinder or a cone, then converting the surface into a plane
• partially cutting the sphere and separately projecting each division in an interrupted map

1. inscribe the sphere in a polyhedron, then separately project regions of the sphere onto each polyhedral face
2. optionally, cut and disassemble the polyhedron into a flat map, called a "net" or fold-out

Intuitively, distortion in polyhedral maps is greater near vertices and edges, where the polyhedron is farther from the inscribed sphere; also, increasing the number of faces is likely to reduce distortion (after all, a sphere is equivalent to a polyhedron with infinitely many faces).  However, too many faces create additional gaps and direction changes in the unfolded map, greatly reducing its usefulness.

Polyhedral maps are completely unrelated to "polyhedric" projections, used in several variants circa 1900 for large-scale mapping; essentially, they mapped the spheroid in small separate trapezoidal regions which, if joined, would comprise part of a polyhedron.

Common Polyhedra

If the polyhedral faces cover (i.e. tile or tesselate) the plane when juxtaposed, the map can be useful even in its unfolded form.  Any triangle or quadrilateral tiles the plane, like a regular hexagon does, but the regular pentagon does not.

The five regular or Platonic polyhedra (whose faces are identical regular polygons, and with identical angles at each corner) are natural candidates for polyhedral maps, although distortion is usually unacceptable in the tetrahedron. 

Solid	Common names	Faces (always regular)	Edges per vertex
	Regular tetrahedron, regular triangular pyramid	4 triangles	(3 x) 3
	Regular hexahedron, cube	6 squares	(3 x) 4
	Regular octahedron	8 triangles	(4 x) 3
	Regular dodecahedron	12 pentagons	(3 x) 5
	Regular icosahedron	20 triangles	(5 x) 3
	Truncated octahedron	8 hexagons, 6 squares	4, 6, 6
	Cuboctahedron	6 squares, 8 triangles	3, 4, 3, 4
	(Small) rhombicuboctahedron	18 squares, 8 triangles	3, 4, 4, 4
	Truncated icosahedron	12 pentagons, 20 hexagons	5, 6, 6
Basic features of a few regular and semiregular polyhedra; the octahedron, icosahedron and cuboctahedron have been applied to commercial maps, like a different form of truncated octahedron.

Some semiregular and uniform (whose faces are regular polygons and vertices are congruent) polyhedra have also been considered for projection.

The idea of using solids as maps goes back at least as far as A. Dürer, even though he did not actually design more than fold-out drafts as part of a general treatise on perspective (1525, revised in 1538). 

The most frequently used design for polyhedral faces is the gnomonic projection, followed by conformal approaches. In gnomonic polyhedral maps, like in all gnomonic designs, great circles (like the Equator and all meridians) are transformed to straight lines, unless where broken at face edges.

Tetrahedral Globes

The tetrahedron is generally regarded as ill-suited for mapping, due to exaggerated distortion near the vertices. It was used by Botley with a gnomonic projection (1949). A former design by Woolgar (1833), based on a stereographic projection, was not exactly polyhedral, since face edges overlapped.

North polar (star-like) aspect of Lee's conformal projection in a regular tetrahedron. Lee's original map is centered on the South Pole.

In response to what he perceived as a critique by Fisher of inordinate distortion in (probably gnomonic) tetrahedral maps, L.P. Lee created a conformal design (1965); compared to other unfolded polyhedra, he pointed as advantages the small number of gaps, the reduced number of cuts in continents given a proper arrangement, and the possibility of tesselating the plane. The projection is conformal everywhere but in the tetrahedral vertices (corresponding to the corners and middle of edges in the flattened form), which also display considerable area exaggeration. Lee arranged his map in a south polar aspect, leaving all four vertices in oceanic areas.

The 3-point variant of Berghaus's star map is incidentally foldable as a tetrahedron, although its development is unrelated to any method aforementioned.

Despite the common name, Bartholomew's tetrahedral projection is actually a star-like composite, unrelated to polyhedra.

The concept of truly tetrahedral pseudoworlds was used by the Dutch artist M.C.Escher in his fanciful engravings Double planetoid (1949) and Tetrahedral planetoid (1954).  Tetrahedral "globes" suggest a new meaning for Isaiah 11:12 ("He will assemble the scattered people of Judah from the four quarters of the earth").

	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.
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.
The Quadrilaterized Spherical Cube was similarly designed for celestial data and is presented here for comparison only.
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.

Cubic Globes

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).

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 rearranged as so: 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

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

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

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

A Modified Collignon Map

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.

Mapping to Truncated Octahedra

Using the Semiregular Truncated Octahedron

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

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

Graticule for Waterman's projection shows the large hexagonal faces and triangles comprising the smaller square faces. Antarctic inset uses the same projection. Original image copyright © 1996, Steve Waterman. Full-color, complete wall maps of the world are commercially available.

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, is the foundation of this projection. All square faces are split in four right triangles, except the one with the South Pole, which also borrows narrow slices of adjacent hexagonal faces, thus keeping Antarctica whole. The opposite face is split, leaving interruptions in Arctic islands.

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.

In its basic form, this projection is neither conformal nor equal-area. The butterfly lay-out combines legibility and low distortion.

Icosahedral Maps

Gnomonic map projected onto an icosahedron; "central" meridian 30°W printable versions available

With the highest face count among regular polyhedra, the icosahedron was long a favorite for world maps. Icosahedral maps were mentioned in patents awarded to J.N. Adorno (1851) and J.M. Boorman (1877).

R. Buckminster Fuller (made famous by geodesic domes and other innovative engineering ideas) designed several polyhedral (like other of his creations, formally named Dymaxion™) maps, at first on a cuboctahedron (1943), later adopting the icosahedron. All were patented and heavily promoted; some icosahedral versions further subdivide a few triangular faces, thus almost completely avoiding split land masses. Most Fuller maps employed arbitrary projections, usually with constant scale along face edges.

Another icosahedral map was briefly made popular by Irving Fisher and Osborn Miller (1944); it used the gnomonic projection, with poles usually located at opposite vertices.

Maps on a dodecahedron

Gnomonic map on a regular dodecahedron; printable versions available

Rhombicuboctahedral Maps

Two assembled rhombicuboctahedral pseudoglobes, with poles centered on opposite square or triangular faces

Rhombicuboctahedral map fold-out, "central" meridian 0°; more faces mean lesser distortion, but also less continuity. Printable version available

www.progonos.com/furuti    March  3, 2008