Just wanted to get some more work-in-progress images out. I'm calling the Immortal indeed done now (took out the socket). Turns out, he can in fact stand on his feet without breaking the legs off, but for longer term survival he will be mounted on a nice basalt socket via a metal support rod held by brass shells.
This is what my extra desk currently looks like (and will, for a while longer):
In the foreground you see the primary thing being worked on, the polyhedral shield segment (as outlined here).
Basically I drilled three big parallel holes into the chunk of blue alabaster and split it - which was surprisingly straightforward, and also worked flawlessly as intended. Then, I shaped the still fairly irregular piece into about a third of a sphere, flattening the underside. When the thing was roughly spherical, I started contructing hexes with a compass on its surface, ending up pretty close to a soccer ball pattern. The spherical parts above the hexes were then removed, ending up with a polyhedral shape.
After having it lie around in the basement for a couple of weeks, I hollowed the thing out with an angle grinder and dremel for the finer parts over the last two weeks, cutting away all partial hexes. The technique is to basically make parallel incisions and break away all material in between. A bit tedious, but fairly precise, reasonably quick and (much to my surprise) totally safe. The stone is only a few millimeters thick in the center of some hexes, intended for maximum translucency. We'll see how that pans out with polishing.
My original plan was to have the shield mounted on the socket behind the Immortal, but now decided it looks better with the shield in front, with the Immortal aiming over it. The shield will be fastened to the socket by smaller versions of the same brass mounts as the immortal itself (can be seen lying at the Immo's feet in the wide shot). I cut a hole in the shield, where the forward leg will point through (or towards, at least).
I'm concious of the fact that the shots of the shield are not spectacular at the moment, but you gotta leave room for improvement, right?