The cuboctahedron is a quasiregular solid, comprising six square faces and eight equilateral triangular faces — a mix of features of the cube and regular octahedron. It is the base of the first polyhedral map introduced by Richard Buckminster ("Bucky") Fuller, well-known advocate of geodesic domes.
|Pure gnomonic projection on a cuboctahedron in two symmetric hemispheres.|
|Reconstruction of Fuller's projection on the unfolded cuboctahedron, in the conventional form as it appears (minus graticule) in Fuller's patent of 1944/1946.|
|Approximate reconstruction (minus graticule) of Fuller's map on the cuboctahedron with identical projection but reduced continental cutting as also included in Fuller's patent of 1944/1946.|
In Fuller's maps, a cuboctahedron is first inscribed into a sphere. Its edges only are gnomonically projected outward, creating spherical squares and triangles whose edges share the same length since every polyhedral vertex is tangent to the sphere. Due to the gnomonic property, every spherical edge is a geodesic arc.
Each spherical triangle is then transformed to a flat triangle on the map with identical edge lengths by Fuller's projection, a relatively simple procedure. Given a point on the spherical face, for every edge another great circle is found passing on the point and at identical distances from both ends of the edge. Inside the flat triangle, a smaller triangle is defined by the intersection of three lines, each parallel to an edge at a distance proportional to the geodesic distance between the corresponding great circle and spherical edge; the desired point on the map is the center of the smaller triangle. Square faces are mapped similarly, but the procedure is simplified because only two great circles are enough for defining any point.
Consequently, for every face the scale is identical in both spherical and mapped edges, and distortion grows towards the center, but is limited by the interrupted nature of the map. The projection is neither equal-area nor conformal.
Fuller published several variants of his Dymaxion™ cuboctahedral map (it is difficult to see any cartographic meaning in the name, which stands for Dynamic Maximum Tension; in fact, it may be more of a promotional device, since it was given to several of Fuller's concepts, including prototypes of a car, a house and a bathroom) with different arrangements of the continents. In 1943, a cuboctahedral map with poles centered on the square faces which could be cut, folded and assembled as a solid was drafted by Richard Edes Harrison and published by Life magazine. In 1944 Fuller applied for a U.S. patent, awarded in 1946, featuring two flat maps: one, that more conventional arrangement with the Equator along the diagonals of four square faces and poles on the two other ones, and an oblique version, also previously published, which shifted continents in order to minimize cuts, except in Antarctica. This was to be the definitive cuboctahedral Fuller lay-out, until superseded by his icosahedral version a few years later (see below).
With the highest face count among regular polyhedra, the icosahedron has long been a favorite for world maps. Icosahedral maps were mentioned in patents awarded to J.N. Adorno (1851) in the U.K. and J.M. Boorman (1877) in the U.S., though giving very little detail (or importance) to the actual projection process.
|Gnomonic projection on an unfolded icosahedron. Central meridian as it appears in Irving Fisher's patent of 1945/1948.|
|Reconstruction of the Dymaxion map using Fuller's projection on an icosahedron, scaled up in order to match the gnomonic version's size. Two of the triangular faces are subdivided and rearranged in order to avoid cutting Australia/Antarctica, Korea and Japan.|
|Icosahedral map using the gnomonic projection but with Fuller's continental lay-out.|
|Icosahedral map using the gnomonic projection and an approximate reconstruction of J.J.van Wijk's lay-out which minimizes continental interruptions (like Fuller's) while avoiding face subdivisions.|
|Fisher/Snyder's equal-area polyhedral projection on an icosahedron, likewise scaled up to match the gnomonic version's dimensions. Central meridian as in Fisher's patent.|
|Icosahedral map combining Fisher/Snyder's equal-area polyhedral projection with Fuller's continental lay-out. Use the graticule, especially the shape of polar parallels, as a reference aid for comparing distortion on the three projections.|
|A single 5° graticule face, centered on a pole, of icosahedral maps in Fuller's Dymaxion (left), gnomonic (center) and Fisher/Snyder's (right) projections, at identical scaling factor.|
Two other U.S. patents, more cartographically-oriented, were applied by Joel E. Crouch (filed in 1944, granted in 1947) and by the economist Irving Fisher (1945/1948; his map was previously published in 1943), featuring icosahedral pseudoglobes with poles at opposite vertices. That both patents were granted to apparently similar subjects in such a short period of time is maybe explained by both being concerned at least as much with mechanical tabs, links and clips for easily attaching and detaching the map faces as with the projection method itself. In fact, while Fisher dedicated some paragraphs to the defense of his choice of the gnomonic projection, Crouch just suggested it as the most straightforward to apply, but mentioned others would be just as acceptable. Fisher's design, of the two without a doubt the best-known today, was marketed as the Likaglobe.
Another icosahedral map proposed by Fisher is equal-area; the method was generalized by Snyder for other polyhedra. In the icosahedron's relatively small faces, the projection's cusps are not especially visible.
After initially criticizing Fisher's choice of polyhedron, ca. 1954 Buckminster Fuller himself abandoned the cuboctahedron in favor of the icosahedron for his Dymaxion map, retaining essentially his same projection neither conformal nor equal-area but without the complication of two different face shapes. Fuller also continued the emphasis on continental areas, with an arrangement which, after two faces are further subdivided, avoided cutting shorelines. Renamed the Dymaxion Air-Ocean World Map, the design is currently promoted by the Buckminster Fuller Institute, which holds trademarks related to the map and its lay-out. In contrast with the original design, it was never patented.
Unfortunately, popular descriptions of Fuller's design tend to exaggerate its qualities: it is frequently claimed to be the most accurate flat world map available, and free of any visible distortion. Aside the "most accurate" qualifier, which of course depends on the map's purpose, his projection incurs in noticeable shape and areal distortion everywhere (even on the face edges, where only distance is preserved) although, due to its interrupted nature, limited by the face extents. While Fuller's projection has a much lesser range of area distortion than the icosahedral gnomonic's, its maximum angular distortion is larger.
Indeed, it can be argued that most benefits of the icosahedral Dymaxion map result from its minimal-cut continental distribution, rather than from the projection method. That very same arrangement is, nonetheless, criticized as being unfamiliar and a potential source of confusion. To counter this, proponents argue that a radically new lay-out was deliberate and essential to Fuller's concept of a "Spaceship Earth™" with a single, unified, uncut inhabited land. Moreover, when presented as detachable faces, the map may be taken apart and reassembled in several "correct" ways (a claim common to most proponents of polyhedral maps) — another face of Fuller's humanist vision.
|A gnomonic projection on a regular dodecahedron.|
Perhaps the most globe-like of all five Platonic solids is the regular 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 are connected by only one or two edges, causing too many gaps.
A patent, applied for in the U.S. in 1937, was granted in 1939 to James A.Smith regarding a dodecahedral pseudoglobe, preferably using the gnomonic projection. The patent application emphasizes how the faces could be either assembled and secured by screws attached to a hub with twenty radial spikes, or, for educational purposes, laid flat in several different arrangements. It also mentioned how each face could be printed on both sides with different features, even a celestial map, and how much more portable and inexpensive the device would be, compared with an ordinary globe.
|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|