Geodesic Icosahedra |

For these pages, a "geodesic icosahedron" is a polyhedron derived from an Icosahedron by subdividing each face into smaller faces using a triangular grid, and then applying a canonicalization algorithm [1] to make the result more spherical. The triangular grid is applied to a face by identifying three vertices in the grid that form an equilateral triangle, and mapping them to the face's three vertices. If the grid's edges do not line up with the face's edges, then any edges and triangles 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 triangles 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 Icosahedron using the smallest non-unit equilateral triangle in the grid, which has a side length of √ 3 grid units. This mapping adds one new vertex at the center of each original face, and after canonicalization, the result is a Pentakis Dodecahedron. The next smallest non-unit equilateral triangle in the grid has a side length of 2 grid units, and the associated mapping adds one new vertex at the midpoint of each original edge. After canonicalization, the result is a Pentakis Icosidodecahedron.

All geodesic icosahedra formed this way can be enumerated according to
the relative position in the grid between two vertices of the equilateral
triangle, 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 a
different direction after a 60° turn. Using this notation, the simplest
non-trivial case (using an equilateral triangle with side length equal to
√ 3
grid units) is [1,1], and the next simplest is [2,0].

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