New Tetrahedral Maps
I drew four variations of the Lee Conformal Tetrahedral projection in 2005, each with the polar vertex rotated 45º along a given meridian. When these maps are tessellated, the terminal singular points (seen at the North or South Pole in other versions of this projection) are apparent at the juncture of figure-eight loops formed by the duplicated 45º latitude lines. I did not construct these maps mathematically, instead interpolating Azimuthal Equidistant projection maps (plotted using GCM software, as used in The Atlas of Nowhere), which were centred on the polar vertex of the eventual tetrahedral design, and on the opposing centre of the triangular face on the other side. These working maps were plotted to the radial limit of the spherical triangle (~8040 km) or to the extent of the other three spherical triangles which complete the tetrahedral form (~14198 km). These circular maps, with the correctly generated graticule of latitudes and longitudes, were overlaid with concentric polar grids upon their centres, which were directly translated into the latitudes and longitudes of the Lee projection, in its normal aspect with the terminal vertices at 90º North or South. By setting the terminal vertices at a round-number (eg. 20, 40, 90, or 140) degree of longitude, the direction of the grid was easily established. The positions of the other three vertices along the edges of the completed triangular grid were reckoned from the limits of the Azimuthal Equidistant maps. In so many words, these maps were less calculated than hacked.
The projection-aspects of the four 45º offset maps I have made thus far are deliberately set to create singular points at inland locations, or near to coastlines, as much as possible (this limits the selection quite a bit, considering that the surface of the Earth is 75% water). In this way I generated what I felt were the most interestingly redundant geographic spaces.
When drawing these maps, I made a few departures from my earlier variants of the Lee Conformal Tetrahedral projection (XTPF, Multiplication/Projection): the graticules of Latitude and Longitude are drawn at larger intervals (less lines), and rivers are not detailed, while National boundaries are. The inclusion of colour-coded Nation-shapes as graphic objects helps considerably with recognition of even the most distended Continental shapes, for instance, in the case of an extreme shape- and scale-distortion of the Southern areas of Africa and South America, relative to the areas around the Sahara or the Amazon. In these instances the overall forms of the Continents, as depicted within the projection, differ considerably from their familiar shapes seen in conventional World maps. The more localized political boundaries exhibit this nonconformity to a lesser (or more gradual) degree, and their inclusion within the map helps with the recognition of the more extremely distorted Continental (or double-Continental) shapes, as the sum of their parts. Also, in terms of the graphic manipulation of these maps as vector-based objects, the more subdivided shapes of Nations were easier to align within different scaling, skewing and rotational transformations, required after copying them and pasting them among the different designs.
New Tetrahedral Map 1 The first of these maps was set with the main vertices at 45º N 60º W, fairly close to the Cabot Strait between Cape Breton Island and Newfoundland. As it turned out two of the other vertices along the edges of the triangle are also very close to coastal regions with a lot of colonial history, located near Taiwan, and South Africa (the other is out in the Pacific, which, given its size, is almost always the case with this projection).
New Tetrahedral Map 2 The second map was designed with the main vertices at 45º N 130º W, somewhere off the coast of Oregon. As the 130º W meridian continues along the edge of the map, it turns into the 50º E line of longitude as it runs over the poles, and the second singular point appears along its trajectory, in the Persian Gulf near Bahrain. A third singularity is found near Wagga Wagga, New South Wales, Australia, and the fourth, off the coast of Southern Brazil.
New Tetrahedral Map 3 The third variation was plotted with the main vertices at 45º N 80º W, in the middle of Georgian Bay in the Canadian Great Lakes. The second singularity occurs along the 100º E longitude, in Southern China, with the other two in the Southern Atlantic and South Pacific Oceans.
New Tetrahedral Map 4 The fourth map is the only example with a 45º S orientation, with the main vertices at 45º S 60º W, in the vicinity of the Falkland Islands / Malvinas. The second singularity is in Western Australia, with another in the Mediterranean Ocean, South of Crete, and the fourth point located in the Pacific.