Elsewhere in this thread, 'proee' links to what might be an even more interesting technique, which restrains the object in a dodecahedral "cage" (to allow for precise angular positions) and then measures the amount of liquid necessary to create a set predetermined rise in liquid level. http://www.romansystemsengineering.com/hypothesis.html.
Combining some aspects of the two, it might make sense to start with the object at the bottom of an empty container (in a cage or otherwise restrained) and add liquid at a known constant rate (as for a titration). Then generate a 2D graph of time against liquid height for a number of known angles, and solve in the same manner as this paper describes.
I believe this is isomorphic to the draining mechanism described in their paper.
I also wonder if instead of using discrete angles and multiple fills (or drainings) one could just tilt the container, possibly even slowly rotating it continuously. Add a squirt, measure the liquid level for a 360 rotation, then add another.
Edit: just saw your other comment suggesting similar things!
Just a quick solution of a million-dollar math problem (https://en.wikipedia.org/wiki/Millennium_Prize_Problems#Navi...) and you're on your way!
Yes, to be sure you are right. It was just a silly (and, as you point out, mathematically inaccurate) joke.
How about shining a normal light though and just inverting the shadow calculations.. hmm.. has that been done?
Not sure how practical it is right now, but I wonder if you could do this with air volume at a high enough delta measurement resolution you might get some amazing results.