Geodesic Cubes |

For these pages, a "geodesic cube" is a polyhedron derived from a Cube by subdividing each face into smaller faces using a square grid, and then applying a canonicalization algorithm [1] to make the result more spherical. The square grid is applied to a face by identifying four vertices in the grid that form a square, and mapping them to the face's four vertices. If the grid's edges do not line up with the face's edges, then any edges and squares from the grid that lie partially outside of the face are merged with their counterparts on adjacent faces. The counterparts are easily identified because the grid is mapped the same way on each face. Edges and squares merged this way are not coplanar with any face of the original polyhedron, but this is of no concern because the shape of the final polyhedron is dictated by the canonicalization algorithm.

In the simplest non-trivial case, the grid is mapped to each face of the Cube using the smallest non-unit square in the grid, which has a side length of √ 2 grid units. This mapping adds one new vertex at the center of each original face, and after canonicalization, the result is a Rhombic Dodecahedron. The next smallest non-unit square in the grid has a side length of 2 grid units, and the associated mapping adds one new vertex at the center of each original face and one new vertex at the midpoint of each original edge. After canonicalization, the result is a Deltoidal Icositetrahedron.

All geodesic cubes formed this way can be enumerated according to the
relative position in the grid between two adjacent vertices of the
square, which is expressible using coordinates [*h*,*k*],
where *h*≥*k*, and traveling from one vertex to the other
involves walking *h* units in one direction and *k* units in an
orthogonal direction. Using this notation, the simplest non-trivial case
(using a square with side length equal to
√ 2
grid units) is [1,1], and the next simplest is [2,0].

References: | [1] | Canonical Polyhedra (George Hart) |