## Spherical Grids

A civilisation on an oceanless planet which was habitable at all latitudes might wish to live in a single global megacity with a gridiron street plan. But a global latitude/longitude grid converges rapidly at high latitudes. One alternative is to project a cube on to the sphere and make a nominally square grid in each face. This becomes distorted near the vertices of the cube: the quadrilaterals have 120° angles where they touch the vertex. A better alternative is to project a rhombicuboctahedron on to the sphere. In a grid on each of the 18 square faces, the biggest angles are then smaller. One approach to the 8 triangular faces is to extend a rectangular grid from each of the 3 sides and truncate it along lines joining each vertex to the centroid of the triangle. This creates a series of irregular polygons along each line, but the total area of these polygons decreases rapidly as the subdivision of the grid increases. Here, where the largest grid squares have sides of 1 degree of latitude and longitude at the equator, it is only 1.84 percent of the sphere. On earth, these grid squares are about 111 kilometers wide. The red lines indicate a major grid discontinuity, the blue lines indicate a minor grid discontinuity. If the blue lines surrounding the triangles are ignored, the remaining lines give the outline of a chamfered cube projected on to the sphere. A chamfered cube consists of 12 hexagons and 6 squares. Our rhombicuboctahedron above is shown in an orthographic projection. A stereographic projection is conformal and preserves angles, so it shows the amount of angular distortion of the grid near the vertices (fewer subdivisions show this more clearly):- A Lambert Azimuthal Equal Area projection preserves areas, so it shows the amount of size distortion of the grid near the vertices:- The angles around each vertex are:-
 98.42° □ △ 64.74° 98.42° □ □ 98.42°
So the grid squares around each vertex are actually rhombi with internal angles of 98.42° and 81.58°. This shape distortion is probably more than is desirable.

The square faces can be thought of as three overlapping bands each circling the sphere. If these bands are made narrower, the squares will get smaller and the triangles will get larger. Now 12 of the square faces have become rectangles and the angles around each vertex are:

 97.44° ▯ △ 72.77° 92.36° □ ▭ 97.44°
Here is the stereographic projection:- The grid in the rectangles has not yet been updated to a square grid but the vertex angles are correct. There are no grid squares with bigger vertex angles in the triangles.

As all of the square or rectangle grid angles have reduced, there seems to be no reason not to reduce the width of the bands to zero, so that the polyhedron becomes a rhombic dodecahedron, which has 12 faces.  Unfortunately, however, there is now a grid rectangle with an angle of 98.23°, almost as big as in the original rhombicuboctahedron, shown at the ends of the magenta strip. The reduction of band width can be stopped at the point where the maximum size of any of the grid angles is at a minimum. The angles around each vertex are shown below (there is a maximum angle of 93.87° in the triangles).
 93.86° ▯ △ 81.93° 90.36° □ ▭ 93.86°
Here is the stereographic projection:- The closeness of the largest grid angle in the rhombic dodecahedron projection to that in the original rhombicuboctahedron projection suggests trying the other rhombic polyhedron which has 30 faces, the rhombic triacontahedron. The grid lines are drawn parallel to the diagonals of each rhombic face.  The largest grid angle is now only 93.18°, close to the midpoint of the side of the rhombus. Another advantage is that all the faces are now of identical shape and size. Here is the stereographic projection:- Each gridded face could be regarded as a separate "country", with the irregular polygons on the borders being the "no-man's land" between them.

## General Scheme

We now have a general scheme for drawing grids on spherical quadrilaterals which have perpendicular diagonals. Some candidate spherical tilings are tabulated below. Dimensions are measured in degrees along great circles of the sphere. Areas are fractions of the area of the sphere. The table names each polyhedron projected on to the sphere. Each name links to the page for the corresponding polyhedron at Visual Polyhedra, David McCooey's wonderful site. Each small diagram links to a detailed zoomable PDF of the gridded sphere produced by Generic Mapping Tools>.

Cube 6109.5°
109.5°

70.5°
70.5°
120°
120°
109.47°16 Rhombic Dodecahedron 1290.0°
70.5°

54.7°
54.7°
90°
120°
98.59°112 Deltoidal Icositetrahedron 2460.0°
54.7°
45.0°
45.0°
90°
120°
90°
95.31°124 Rhombic Triacontahedron 3063.4°
41.8°

37.4°
37.4°
72°
120°
93.18°130 Deltoidal Hexecontahedron 6037.4°
36.0°
31.7°
20.9°
72°
120°
90°
92.23°160 Rhombic Enneacontahedron 6037.4°
27.0°
23.4°
22.4°
72.00°
110.70°
75.52°
91.18°0.74460 3041.8°
16.6°

22.4°
22.4°
44.48°
138.59°
90.78°0.25630
The projected Rhombic Enneacontahedron has the smallest maximum grid angle but, since it is composed of two differently shaped spherical quadrilaterals, the grids on adjacent faces do not align on their edges. This would make it difficult to connect any road networks on each grid. The best solution to the problem then seems to be the projection on the sphere of the Deltoidal Hexecontahedron.

## Credits

• Polyhedron images from Wikipedia are unchanged and covered by the GNU Free Documentation License
• The spherical maps were produced by the professional-grade Generic Mapping Tools via the PyGMT interface.
• The Python scripts using PyGMT were written by me. The transformations mapping the faces of the rhombic triacontahedron were obtained from Table E.2 of the compendious Divided Spheres by Edward S. Popko.
• The data used to create the table entries was obtained from the coordinate text files from Visual Polyhedra.

Christopher B. Jones 2023-05-10