Nope. Just multiply Volume and area with the same value, so you get the same coefficient. 10qmm / 10mm is the same as 20qmm / 20mm or 1qmm / 1mm

No, it is not true that two 3D bodies of different shapes cannot have the same volume-to-area ratio unless both have exactly the same volume and area.

The statement you provided is incorrect. There are cases where two different 3D shapes can have the same volume-to-area ratio without having the same volume and area. This is known as the isoperimetric inequality. The isoperimetric inequality states that among all closed curves or surfaces with a fixed perimeter or area, the circle or sphere has the maximum volume or area, respectively.

For example, consider a cylinder and a sphere. Although they have different shapes, it is possible to find specific dimensions for the cylinder such that its volume-to-area ratio is equal to that of a sphere. The same principle applies to other shapes as well. It's important to note that this is only possible for certain combinations of shapes and specific dimensions.

So, the statement that the volume-to-area ratio rule applies to all 3D geometric shapes, except solid spheres, is incorrect. The isoperimetric inequality allows for exceptions to this rule.

Consider two rectangular parallelopipeds of sides (6,3,1) and (10,2.5,1) respectively. Their respective surface areas are 54 and 75. Again their volumes are 18 and 25. See, both the cases the ratios are identical.

We assume the answer is yes, it is.

Two 3D bodies of different shapes cannot have the same volume to area ratio unless both have exactly the same volume and area.

This rule applies to all 3D geometric shapes, cubes, cones, pyramids, half-spheres, ...etc.

However, it does not apply to the case of solid spheres.

The answer to this question exists, in a way, in the universal laws of thermodynamics.

The rarely mentioned Zero Law of Thermodynamics states that:

Two contacting bodies of any shape and material kept inside an isothermal insulated container will be in a state of dynamic thermal equilibrium.

In other words, the heat flow from one to the other is the same as vice versa.

Keeping in mind that the heat flow from/to a body is proportional to its surface area and the heat capacity of the body is proportional to its volume, then the rule:

Two 3D bodies of different shapes cannot have the same volume to area ratio unless both have exactly the same volume and area,

is validated.

In my previous answer I have given examples of two cuboids of sides (6,3,1) and (10,2.5,1). Both the cases area divided by volume is 3. Now consider a cube of sides 2. Here also the ratio is 3. Furthermore, consider a cylinder of radius and height 4/3. Then area and volume are 64π/9 and 64π/27 respectively. So, again the ratio is 3.

** Answer II-Continued

This simple geometric rule has at least three rigorous i-iii proofs, none of them mathematical.

A. Einstein once said that solid geometry is nothing but physics and went through three decades of fierce debates with the old iron guards.

The story mentioned above is a problem that we do not recommend.

Moreover, we hope to receive decisive mathematical proofs from superpowered mathematicians who have not yet participated in the answer to the current question.

Brief:

i- A thermodynamic proof provided by the zero law of thermodynamics which was explained in the previous answer.

ii- Experimental evidence from plotting experimental results of cooling curves for different metals in different forms [1].

iii- A physical statistical vision which results from the theoretical analysis of the results of the cooling curves via the statistical chains of the chains of the B-transition matrix [1].

In ref 1 it was stated in full letters:

[i-Perform the experimental results of the temperature cooling curve at the center of mass CM of the tested object and thus find the half-time decay value in real time, i.e. T1/2 , T1/4, T1/8. . etc [1,2].

ii- Calculate the statistical characteristic length of the tested object Lc via the semi-imperial formula (2), [1,2]Lc = {6*Volume of the object V / Area of ​​the object A}. . . .(2)The statistical factor f derives from another semi-imperial formula, f = Pie /2. = 1.571 In fact, the characteristic length in itself is of great importance since the experimental temperature of the real-time cooling curve at the center of mass CM is described by,

T(t)=T(0).Exp (-D.t/Lc^2) . . . . . (3)

Equation 3 is simply a consequence of defining the exponent of the cooling curve as the heat left per second dU/dt divided by the heat stored U.

Equations 2 and 3 suggest an important geometric physical rule,

[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]}.

Ref 1:

I. Abbas, Researchgate / IJISRT review .A Rigorous Experimental Technique to Measure the Thermal Diffusivity of Metals in Different 3D Forms, August 2022

*** Answer III-Continued

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