I'm working with some icosahedron-based tessellations of triangles over the surface of a sphere.

Class I and Class II tessellations have a nice property where, cutting along the edges of the tessellated grid, I can split the surface into a series of parallelograms. This gives a neat way to associate points on the tessellated sphere with points in a simple rectangular array (which I hope to use for things like game logic and texturing)

Here's what it looks like for the Class I case: (starting from a corresponding Goldberg polyhedron)

Class I Geodesic split into parallelograms, warped back to rectangles

For Class II, we get one rhombus per edge of the icosahedron, three meeting in the middle of each icosahedral face - the same pattern as a rhombic triacontahedron

But many geodesics I might want to decompose are Class III:

Classes of tessellations, with possible decompositions sketched

So far I've only found similar decompositions for those of the form (n, n + 1) and (n, n + 2), sketched in blue on the right side above, but although these cover every vertex of the tessellation they do leave some faces uncovered (which is probably tolerable for my use case but not my first choice).

So the question I'm chipping away at now is:

Is there a method to decompose a Class III geodesic tessellation into a set of non-overlapping parallelograms, either in general or for special cases?

Or conversely, is there no such method and I'm on a wild goose chase?

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