Map Projections
 Most cartographic problems would disappear on a polyhedral Earth

## Polyhedral Maps

#### Introduction

Several approaches have been 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:
1. inscribe the sphere in a polyhedron, then separately project regions of the sphere onto each polyhedral face
2. optionally, cut and unfold the polyhedron into a flat map, usually called a polygonal "net"

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.

Solid Common names Faces
Regular tetrahedron, regular triangular pyramid 4 triangles
Regular hexahedron, cube 6 squares
Regular octahedron 8 triangles
Regular dodecahedron 12 pentagons
Regular icosahedron 20 triangles
Truncated octahedron 8 hexagons, 6 squares
Truncated icosahedron 12 pentagons, 20 hexagons
Cuboctahedron 6 squares, 8 triangles
Icosidodecahedron 12 pentagons, 20 triangles
(Small) rhombicuboctahedron 18 squares, 8 triangles
Rhombic dodecahedron 12 diamonds
Basic features of selected polyhedra; all faces are regular polygons except in the rhombic dodecahedron. The octahedron, icosahedron and cuboctahedron have been applied to patented or commercial maps, like a different form of truncated octahedron. Intuitively, increasing the face count makes a polyhedron closer to a sphere while reducing the simplicity and usefulness of the unfolded map.

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 and Polyhedral Projections

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. Some semiregular and uniform (whose faces are regular polygons and vertices are congruent) polyhedra have also been considered for projection, most notably truncated octahedra, the cuboctahedron and the truncated icosahedron.

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

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

As usual with interrupted designs, an important issue with polyhedral maps is choosing the projection aspect and arranging the faces to avoid cutting important features on the map. More recently, as part of his framework of algorithms for myriahedral projections (2008), van Wijk obtained optimal face lay-outs for all Platonic solids minimizing continental cuts.

 Both area and shape distortion become extreme when the gnomonic projection is applied to a tetrahedron, except at the center of each face. North polar (star-like) aspect of Lee's conformal projection in a regular tetrahedron, scaled to match the gnomonic map's size. Lee's original map is centered on the South Pole. Area distortion is great in the six nonconformal points. When applied to a tetrahedron (again, scaled to match the gnomonic map's size), the cusps in Fisher/Snyder's equal-area projection are very evident along three radials of each face. Single polar face of tetrahedron at identical scaling factors: gnomonic (left), Fisher/Snyder equal-area (center), Lee's conformal projection (right).
##### Equal-area Designs

A paper by A.D. Bradley (1946) described an equal-area projection on an approximate icosahedron (the map edges did not exactly matched the polyhedron faces's); it also mentioned an equal-area design by the economist Irving Fisher which exactly covered the polyhedron. Fisher had previously designed and applied for the patents of a gnomonic map on an icosahedron.

Fisher's method, based on Lambert's azimuthal equal-area projection, was generalized by John P. Snyder (1992) to all Platonic solids, plus, near exactly, the truncated icosahedron. It is more easily explained, with no loss of generality, with a polar aspect centered on a regular polygonal face, with radius extending to each vertex. The "natural" boundary of Lambert's map is curved, either missing or exceeding the polygon. Therefore, the meridians are adjusted until all end at the edges; this breaks areal equivalence, so azimuths, i.e., meridian spacing, are modified to compensate. Finally, distances are proportionally calculated along each meridian to keep areas constant.

A shortcoming of this method is that cusps — changes of direction, visible as graticule breaks — are introduced along the lines connecting vertices to each face center; they are more conspicuous the larger relatively the faces are.

Other equal-area solutions have been proposed for specific solids like the Quadrilaterized Spherical Cube and HEALPix projection in the cubic form.

#### 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.

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 (those six singularities correspond 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, placing all vertices in oceanic areas.

 A tiling of Lee's projection on a tetrahedron. Centering on the south pole moves all singularities to the oceans.

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). Polyhedra, tilings, cycles, the infinite and paradoxical figures are recurring themes in works of the prolific Dutch artist. 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").

 www.progonos.com/furuti    March 30, 2015