What projection would I use to map the northern night sky onto a geodesic dome? I'm trying to create a geodesic dome with the stars of the northern night sky inscribed on the outside. The highest point of the dome would be the celestial north pole. How do I create a map that projects onto the "net" of the geodesic dome, such that when I assemble the dome, the stars are in the correct positions? I have an equirectangular map that I have rotated 90 degrees north so it is centered on the pole.

My existing map is a high-resolution jpeg file in an equirectangular projection. I have some experience creating and manipulating panoramic images, but much less experience converting between map projections! I would like to use a computer program to convert this projection into a printable template if possible.
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.