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I'm going to take a step away from the entire question of mapping entirely and ask a simpler question. Consider a circle drawn on your stock variety euclidean plane. we choose some arbitrary line in the plane and let L be the set of all lines in the plane paralell to our choosen line. for any line in L that intersects the circle we put an element s in a set S such that s is a scalar that represents the length of the line segment inside the circle if the line intersects the circle at 2 points and is 0 if the line intersects it at one. If we now throw away the circle and have only the information in the unordered set S then we won't have enough information to reconstruct the circle even if we know how the information was collected. But if we dictate some rules concerning what shapes are possible (analogous to physical principles) then, we can make the rules so that the circle is the only possible shape. (at least I hope we can make the rules like that but I wouldn't bet too much on it considering weirdnesses like banach-tarsky)

The same sort of question is analogous to taking a set of crossections of the universe ala frames and then leaving them in a jumbled heap. Assuming that the universe does in fact operate on some sort of physical principles there is a limit to the ways we can reconnect these frames into a world that still makes physical sense. But what physical principles are necessary to leave only one possible world or does such a set of principles non existent? Are the physical principles of say relativity enough?

Getting a square would actually be an improvement, since squares and circles are topologically equivalent. With the condition that topology is preserved, you'd at least get some sort of connected shape. Without it, all bets are off, since topology is the most fundamental of notions.

Two words: Zeno's Paradoxes.

http://en.wikipedia.org/wiki/Zeno's_Paradox

"Finally, in the arrow paradox, we imagine an arrow in flight. At every moment in time, the arrow is located at a specific position. If the moment is just a single instant, then the arrow does not have time to move and is at rest at that instant. Now, in following instants, it then must also be at rest for the same reason. The arrow is always at rest and cannot move: motion is impossible."

and

"Aristotle denies that time is composed of "nows", as implied by Zeno's argument. If there is just a collection of "nows" then there is no such thing as temporal magnitude. Therefore, if Aristotle is correct in denying that time is composed of indivisible nows, then Zeno is wrong in saying that the arrow was stationary throughout its flight despite saying that in each now the moving arrow is at rest."

Done.

Not really... neutral