How to make an octant display?
Have you heard of POV (persistance of vision) displays? This video shows such a display in action, it is a circle of LEDs that is spun around with the LEDs being controlled depending on rotational position. You could do the same, but switch off the display everywhere but that one eigth of the sphere you want to use.

Another option would be to project the image onto the inside (or outside) of a white, thin piece of spherical plastic or similar, using a simple projector. In this case, you would of course have to run a bit mathematics on the image before displaying it in order to not stretch it.