I remember that we already had a similar type of research when dealing about the stariness of the hypercubes, i.e. that most of their body volume would be maintained close to their corners, because the ratio of hypervolume of hypercube and inscribed hyperball in its dimensional limit would diverge.

The main arguments in the FB discussion so stick to just one and the same dimension. In fact his question itself was concerned with 3D merely. Thus answers like ratio of polyhedral volume to volume of inball, or even to maximal inball (i.e. with radius upto the outermost faces) seem likely.

One clearly might want to dualize that very setup and consider the ratio of the volume of the circumball to the volume of the polyhedron. This ratio clearly generalizes to every dimension, and moreover would easily encompass all orbiform polytopes.

But then there is our interest in CRFs (in 4D and even beyond), which is the Johnson solids within 3D, not at all to mention the toroids, which all would fall short here.

So I brought in a further idea for a potential roundness measure, the ratio of volume to surface content. That one clearly maximises for the ball and similarily can be applied to 2D setups, then maximising for the disk. Quite generally we would consider here in a first run vol(P)/vol(∂P), where ∂P denotes the surface or boundary of P and vol simply is the according dimensional volume function. But this measure lacks the disadvantage that it obviously would be size depending. For the D-dimensional volume of the body content scales differently with the absolute size of P than the D-1-dimensional surface content.

This disadventage can be overcome quite simply. Thus consider vol(P)

^{dim(∂P)}/vol(∂P)

^{dim(P)}instead. This value now will be size Independent, and still is being maximized for the according dimensional hyperball. In fact this measure now serves very well for comparision of any polytopes within one and the same dimensionality. Sure, it is a global measure. Thus the seeming counter argument of an icosahedron with an attached needle, which still has nearly the same roundness amount as the icosahedron itself, does not count here. That very isolated excess is kind a Lesbeque null set, which thus does not change the global behaviours of both, the hypersurface and the hypervolume.

Still there is a further issue to be overcome. When we consider shapes of various dimensionalities, their values differ widely. In fact even the according values vol(O)

^{dim(∂O)}/vol(∂O)

^{dim(O)}for the hyperballs O themselves would decrease quite fast with the increase of the dimensions. This is why it would be much better to take refuge to a normalized setup instead. This is why I opted within that FB discussion for

where O again is the according dimensional hyperball, i.e. dim(O)=dim(P).roundness(P) = (vol(P)^{dim(∂P)}/vol(∂P)^{dim(P)}) / (vol(O)^{dim(∂O)}/vol(∂O)^{dim(O)})

Sure, the roundness values according to this definition are not too easy to calculate. You not only have to evaluate the body volume and the surface content of your own to be considered polytope only, moreover you will have to get the accoding values for the hyperball as well. And those body volume and the surface content values of the hyperball either are known recursively only, are provided explicitely in 2 separate formula for even and odd dimensions, or would relate to the Gamma function.

Nonetheless I managed since to evaluate the according values for all three, the dimensional series of regular simplices, of hypercubes, and of cross-polytopes (aka orthoplexes). Here they come:

roundness( 2D-simplex ) = (2D)! π^{D}/ (D! sqrt[(2D)^{2D}(2D+1)^{2D+1}])

roundness( 2D+1-simplex ) = D! (2π)^{D}/ ((D+1)^{D+1}sqrt[(2D+1)^{2D+1}])

roundness( line ) = 1 = 100 %

roundness( {3} ) = π sqrt(3) / 3^{2}= 60.459979 %

roundness( tet ) = π sqrt(3) / (2 3^{2}) = 30.229989 %

roundness( pen ) = 3 π^{2}sqrt(5) / (2^{2}5^{3}) = 13.241464 %

roundness( hix ) = 2^{3}π^{2}sqrt(5) / (3^{3}5^{3}) = 5.231196 %

roundness( hop ) = 5 π^{3}sqrt(7) / (3^{2}7^{4}) = 1.898165 %

roundness( oca ) = 3 π^{3}sqrt(7) / (2^{4}7^{4}) = 0.640631 %

roundness( ene ) = 5·7 π^{4}/ (2^{8}3^{8}) = 0.202982 %

...

roundness( 2D-hypercube ) = (π/4)^{D}/ D!

roundness( 2D+1-hypercube ) = π^{D}D! / (2D+1)!

roundness( line ) = 1 = 100 %

roundness( {4} ) = π / 2^{2}= 78.539816 %

roundness( cube ) = π / (2·3)^{}= 52.359878 %

roundness( tes ) = π^{2}/ 2^{5}= 30.842514 %

roundness( pent ) = π^{2}/ (2^{2}3·5) = 16.449341 %

roundness( ax ) = π^{3}/ (2^{7}3) = 8.074551 %

roundness( hept ) = π^{3}/ (2^{3}3·5·7) = 3.691223 %

roundness( octo ) = π^{4}/ (2^{11}3) = 1.585434 %

...

roundness( 2D-orthoplex ) = π^{D}(2D)! / (2^{3D}D^{D}D!)

roundness( 2D+1-orthoplex ) = π^{D}D! / sqrt[(2D+1)^{2D+1}]

roundness( line ) = 1 = 100 %

roundness( {4} ) = π / 2^{2}= 78.539816 %

roundness( oct ) = π sqrt(3) / 3^{2}= 60.459979 %

roundness( hex ) = 3 π^{2}/ 2^{6}= 46.263771 %

roundness( tac ) = 2 π^{2}sqrt(5) / 5^{3}= 35.310570 %

roundness( gee ) = 5 π^{3}/ (2^{6}3^{2}) = 26.915171 %

roundness( zee ) = 2·3 π^{3}sqrt(7) / 7^{4}= 20.500183 %

roundness( ek ) = 3·5·7 π^{4}/ 2^{16}= 15.606620 %

...

Now, what are your thoughts?

- - To Tadeusz' original question?

- and to all those reported initial answers?

- To my setup of that roundness measure definition?

- To the provided list of those calculated values?

- and esp. their dimensional behavior?

--- rk