Let's map on an icosahedron.
Each polyhedral face uses the gnomonic projection for ease of implementation.
The maps above were rendered using a custom version of D3, modified to support clipping against arbitrary spherical polygons.
This was necessary because unlike the octahedron, the icosahedron cannot be unfolded exactly along a whole meridian.
The polyhedral plugin automatically generates an outline in geographic coordinates, which can be passed directly to D3 as a clip polygon.