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.

98.42° □ | △ 64.74° |

98.42° □ | □ 98.42° |

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° |

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.

93.86° ▯ | △ 81.93° |

90.36° □ | ▭ 93.86° |

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.

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.

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>.

Name | Faces | Diagonals | Sides | Angles | Maximum grid angle | Area | Diagram |
---|---|---|---|---|---|---|---|

Cube | 6 | 109.5° 109.5° | 70.5° 70.5° | 120° 120° | 109.47° | ||

Rhombic Dodecahedron | 12 | 90.0° 70.5° | 54.7° 54.7° | 90° 120° | 98.59° | ||

Deltoidal Icositetrahedron | 24 | 60.0° 54.7° | 45.0° 45.0° | 90° 120° 90° | 95.31° | ||

Rhombic Triacontahedron | 30 | 63.4° 41.8° | 37.4° 37.4° | 72° 120° | 93.18° | ||

Deltoidal Hexecontahedron | 60 | 37.4° 36.0° | 31.7° 20.9° | 72° 120° 90° | 92.23° | ||

Rhombic Enneacontahedron | 60 | 37.4° 27.0° | 23.4° 22.4° | 72.00° 110.70° 75.52° | 91.18° | ||

30 | 41.8° 16.6° | 22.4° 22.4° | 44.48° 138.59° | 90.78° |

- 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