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Given a $3d$ object in $\mathbb{R}^3$, can we fully determine its shape with a finite number of projections (or with $2D$ slices)? It seems clear to me that there are objects that would require an infinite number of projections to determine their shape (and similarly for slices). Say, for example, a fractal $3D$ star.

Now if we restrict ourselves to solid objects constructed from unit cubes, can we then find a finite number of projections that fully describe the object? How about a finite number of $2D$ slices?

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    Do you limit yourself to convex shapes ? This does not look the case if you consider solids made of unit cubes ? In the case the object is not convex, you can easily imagine objects with very hidden parts... Even with 2D convex shapes, there is a classical counterexample [Reuleaux triangle(https://en.wikipedia.org/wiki/Reuleaux_triangle) which has alway the same apparent diameter, i.e., has apparently the same aspect as a disk. – 2017-02-01

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