I came across a paragraph that intrigued me. It's not formulated in the most mathematical way, but I'll do my best to write it more rigorously. The statement is:
Consider a cube prescribed with a family of real analytic functions on each face s.t. the shadow of the object inside the cube is cast precisely onto each face (light is shown from opposite faces) and maps directly onto the analytic functions. In some sense the functions encode the shape of the object (3-manifold) inside the cube. Is the 3-manifold necessarily unique, assuming you explicitly make a choice for the functions on the boundary?
So let's make this more precise. Here's my argument:
Assume that there is a 3-manifold that can be constructed via a family of real analytic functions $f_n(x)=x^n$ s.t. $n \in \Bbb R^+$ on each face of the cube.
If one revolves the family $f_n(x)$ for each $n$ you'd get a series of surfaces that look something like this:
Now, we need to orient the series of surfaces inside the cube in the proper way. The solution is to place the cusps (endpoints) of the surfaces at two corners of the curve furthest apart.
And now, the projection onto any face of the cube yields our $f_n(x)$ graphically depicted below:
Then the 3-manifold can be defined as the disjoint union of the infinite series of surfaces.
I don't think I've shown uniqueness though.