hmm its not direct sum of of 2 groups there. the dimension concept, its infinite length of direct sum there. i mean to depict sphere definition. if one direct sum component is dihedral group alike thing, then here is second direct sum component which is similar to it e.g. rotated 10 degrees in y-z axis alike. 

so such notion has already has inherently all possible directions inside. 

and plus physics of particle physics have that 90 degree sep. have such newtonian mechanics like pulling in y direction does not effect x-z axis like result, do just mean that those particles in particle physics their mathmatical structure somehow have such vanishing effect but this does not mean our universe is constrained to 3d. 

of course those particle systems could work other like to not vanish in 90degree in newtonian side. surely that would be another version of universe regimen. but that universe wouldnt be less or more dimensional then what our universe is imho.

i mean there is extraneous coupling between dimension concept and force system in these theories. I find it incorrect/incomplete.

Ok one time I would study the string theory that is capable of showing such regiments of universe could exist with different particle physics behaviours. thats logical. 

but I only as only read introduction of, only reject the definition set of 1d strings there. there is nothing like 1d in universe imho. 

but eventually their math structure might be correct, I mean such formulization might actually work out to cover the universe mechanistics. I mean someform of isomorphism between or homomorphism between how universe actualy is and how string theory defines it might be existing. I mean string theory might be a correct isomorphism of how universe actually is. even imho being wrong at their conceptual definition sets like 1d strings. I think I believe I mean there is nothing like 1d at all. but of course such definition set and its mathematical structure might have eithe rhave isomorphism nor homomorphism of how universe actually is. so string theory might be correct isomorphically (might be isomorphic or homomorphic to how universe works )  but I do reject even basic definition set there that they place something like 1d strings in introductionary knowledge as read of. I think there is nothing like 1d in real universe.  but those formulations might be a correct isomorphism to how universe actually is. evenmost possibly being conceptually wrong.  but its maths structure might be  isomorphic and correctly defining universe in that.

these be my ideas of dimensions concept.