In fact the surprising geometric rule:
two 3D bodies of different shapes cannot have the same volume to area ratio unless both have exactly the same volume and same area,
is a physical rule produced by the laws of nature and at the same time have exceptions but only when applied outside its scope of validity.
However, these exceptions confirm the rule rather than deny it.
Hai, If the region is in spherical spaces then it is true.Example,hemispheres
Experimental measurements and theoretical statistical results of the B-matrix strings proved the rather challenging rule[1]:
[two 3D bodies of different shapes cannot have the same volume-to-area (V/A) ratio unless both have exactly the same volume and area].
On the other hand The same rule: 3D bodies of different shapes cannot have the same volume/area ratio unless they have exactly the same volume and the same area is only conditional?
It is a physical rule produced by the laws of nature and at the same time you can find many exceptions, but only when applied outside its scope.
The domain of validity of this rule is the same domain of validity for the heat diffusion equation as a function of time with Dirichlet boundary conditions.
It is also the same domain of validity of the imperial formula of Sabines for the reverberation time in audio rooms (TR= constant *V/6A).
In conclusion, L, W and H (length, width and height) of the object under consideration should be in proportion.
These exceptions therefore confirm the rule rather than contradict it.
Ref 1:
I. Abbas, Researchgate / IJISRT review .A Rigorous Experimental Technique to Measure the Thermal Diffusivity of Metals in Different 3D Forms, August 2022
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