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If I had a water bottle filled with water and I squeezed it the shape will not change because I am constraining both the volume and surface area. If I allow air into the bottle meaning volume can change, the bottle can now be deformed.

This leads me to believe that it is impossible for two polyhedrons with the same volume and surface area to not be congruent.

But if this was the true, then supposedly every single polyhedron of any complexity could be described with some parametrization, which requires only volume and surface area to describe every point. This notion seems absurd.

Can someone help, here?

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Take any two tetris blocks (put into 3D if you wish) other than the square and you get a counterexample.

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  • $\begingroup$ Nice, that works well $\endgroup$ – Joe Clinton Aug 27 '19 at 14:30

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