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Polyhedral Maps

A cubic globe
Most cartographic problems would disappear on a polyhedral Earth

Introduction

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

Solid Common names Faces (always regular) Edges per 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 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 a few regular and semiregular polyhedra; the octahedron, icosahedron and cuboctahedron have been applied to commercial maps, like a different form of truncated octahedron.

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.

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.

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.

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.

Gnomonic projection on a tetrahedron
Both area and shape distortion become extreme when the gnomonic projection is applied to a tetrahedron, except at the center of each face.
Lee's conformal projection on a tetrahedron
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.
Snyder's equal-area projection on a tetrahedron
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.
Gnomonic projection on a tetrahedron
Single polar face of tetrahedron at identical scales: gnomonic (left), Fisher/Snyder equal-area (center), Lee's conformal projection (right).

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 (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").



HomeSite MapInterruption DevicesMap Projections - ContentsPolyhedral Projections - part 2  www.progonos.com/furuti    May 30, 2011
Copyright © 1996, 1997 Carlos A. Furuti