I am trying to find an efficient way to determine if two polyhedral solids are similar or not. So I am wondering if dis-similar solids that have the same number of faces can have the same surface area to volume ratio? If that is impossible then I have my litmus test. :)
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$\begingroup$ Since the ratio depends on scaling, you need to at least normalize the situation somehow (otherwise you can take any two polyhedra and scale them up or down until ratios match). For example: Can dis-similar polyhedral solids have the same surface area as well as the same volume? But I do suspect the answer is still yes, i.e. that your test won't work. $\endgroup$– MiltenJul 5, 2021 at 17:04
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It won't work, sorry :-(
Take for example a big cube with a smaller cube glued onto one of the faces. You can slide the small cube around on the face to get dissimlar polyhedra without changing surface area or volume. This idea can also be done with many other shapes.
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$\begingroup$ Thank you. True. :( still trying to find a straightforward method to determine if two objects are similar. $\endgroup$– W.J.Jul 6, 2021 at 13:28
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$\begingroup$ Been thinking about your answer that it won’t work. What if the first filter is that all surfaces need to have a similar surface 1:1 correspondence. In your example of the small cube moving around, the surface it sits on has a hole and that hole will move so the two surfaces (with holes considered) would not be similar so it would fail the first test. $\endgroup$– W.J.Jul 10, 2021 at 9:33
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$\begingroup$ Do you mean that the polygonal faces should be pairwise similar? Or that the polyhedra should be foldable from the same net? The first case is easy to disprove: take a big cube with a small cube on two faces; the small cube can be on neighbouring or opposite faces, giving different polyhedra. I’m pretty sure the other case also doesn’t work, but I’ll have to find a counterexample. $\endgroup$– MiltenJul 10, 2021 at 11:23
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$\begingroup$ The bellows conjecture (which is proven) gives counterexamples with equal nets but only non convex polyhedra. $\endgroup$– MiltenJul 10, 2021 at 11:50
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