Most cartographic problems would disappear if the Earth were a polyhedron |

- 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

- 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.

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 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"*).

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

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.

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

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.

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

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 |

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.

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

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.

Gnomonic map on a regular dodecahedron; printable versions available |

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 |

Copyright © 1996, 1997 Carlos A. Furuti