Map Projections: Polyhedral Maps

Map Projections

Polyhedral Maps

Most cartographic problems would disappear if the Earth were a polyhedron

Introduction

Several approaches were presented for reducing distortion when transforming a spherical surface into a flat map, including:

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

Both techniques are combined in polyhedral maps:

inscribe the sphere in a polyhedron, then separately project regions of the sphere onto each polyhedral face
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 polyedron 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.

Common Polyhedra

If the polyhedral faces cover (i.e. tile or tessellate) 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 (all regular) Face edges/vertex

Regular tetrahedron Regular tetrahedron, regular triangular pyramid 4 triangles (3 x) 3

Cube Regular hexahedron, cube 6 squares (3 x) 4

Regular octahedron Regular octahedron 8 triangles (4 x) 3

Regular dodecahedron Regular dodecahedron 12 pentagons (3 x) 5

Regular icosahedron Regular icosahedron 20 triangles (5 x) 3

Truncated octahedron Truncated octahedron 8 hexagons, 6 squares 4, 6, 6

Cuboctahedron 6 squares, 8 triangles 3, 4, 3, 4

Rhombicuboctahedron (Small) rhombicuboctahedron 18 squares, 8 triangles 3, 4, 4, 4

Truncated icosahedron Truncated icosahedron 12 pentagons, 20 hexagons 5, 6, 6

Basic features of common polyhedra; the octahedron, icosahedron and cuboctahedron have been applied to commercial maps.

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

Solid	Common names	Faces (all regular)	Face edges/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 common polyhedra; the octahedron, icosahedron and cuboctahedron have been applied to commercial maps.

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 method for projecting faces uses the gnomonic projection for each section, followed by conformal approaches.

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 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").

Cubic Globes

Gnomonic cubic map, graticule spacing 10°, poles in opposite corners; printable versions available

Although mapping into the regular hexahedron (an ordinary cube) is prone to strong distortion, the nonsensical notion of "Earth-in-a-box" has always attracted me. Once I plotted and folded such a map by hand alone. Fortunately now I own a computer...

Different arrangements of six gnomonic square faces were used by Reichard (1803) and other cartographers, mainly for celestial atlases.
Like in all gnomonic maps, great circles (including the Equator and meridians) are transformed to straight lines, except where broken at face edges.

Cahill's Butterfly Map

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

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.

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

Modified Collignon Map

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

Collignon's curious projection can be modified to a "butterfly" variant in three straightforward steps: interrupting the diamond-shaped version along three meridians, creating eight triangular lobes; changing both horizontal and vertical scales in order to make lobes equilateral while keeping area constant; and 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.

Mapping to a Truncated Octahedron

Gnomonic map on a 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

Projecting the world gnomonically on an octahedron is a fairly simple task since all meridians and the Equator are mapped into straight lines, therefore octant boundaries are easily deduced from map coordinates.

The same projection applied to a semi-regular 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.

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 maps.

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 Fisher and Miller, ca. 1944.

Maps on a dodecahedron

Gnomonic map on a regular dodecahedron; printable versions available

Perhaps the most globe-like of all five regular solids is the dodecahedron (its volume differs the least from that of a inscribed sphere; on the other hand, the icosahedron has the bigger volume/surface ratio, and its volume best approximates that of a circumscribed sphere); unfortunately its faces don't tile a plane so most faces in a fold-out would be connected by only one or two edges, causing too many gaps.

Rhombicuboctahedral Maps

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

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

In comparison with the previous solids, the rhombicuboctahedron looks pleasantly roundish due to a larger face count. However, its unfolded form makes evident the problem of finding a suitable distribution of features in multiple faces without too many cuts.