Is it related to triple integrals and the fact that three is the smallest dimension in which Rn arises as a special case of the Barnett construction.
The underlying metric signature on the generating Barnett space of R3 varies non-smoothly between permutations of bases.
If you take the permutation group of the 3 bases, there is a homomorphism between that group and the cyclic group C3, meaning that integrals over spaces with 3 basis vectors are not symmetric.
If you take a cochain complex generated by infinitesimal volume elements, and you try to access homology groups, they are all trivial except for the first group in the sequence, which is not Z like you would expect, but is actually SO(3).
Furthermore, you can form three categories, each containing modules generated by infinitesimal elements of a space, and you can prove that if there is a functor between two of the categories, that there is no functor connecting the third one. this proves that many triple integrals are literally impossible to solve.
The underlying metric signature on the generating Barnett space of R3 varies non-smoothly between permutations of bases.
If you take the permutation group of the 3 bases, there is a homomorphism between that group and the cyclic group C3, meaning that integrals over spaces with 3 basis vectors are not symmetric.
If you take a cochain complex generated by infinitesimal volume elements, and you try to access homology groups, they are all trivial except for the first group in the sequence, which is not Z like you would expect, but is actually SO(3).
Furthermore, you can form three categories, each containing modules generated by infinitesimal elements of a space, and you can prove that if there is a functor between two of the categories, that there is no functor connecting the third one. this proves that many triple integrals are literally impossible to solve.
