What projection would I use to map the northern night sky onto a geodesic dome?
The mathematically ideal scheme would be a separate gnomonic projection
(see Wikipedia or
IMO)
for each tile of the dome.
Many planetarium programs implement this projection as an option.
Fortunately, each tile boundary is a straight line on both adjoining maps.
Unfortunately, continuity across tile boundaries requires some care in choosing each tile's scale and center of projection.