Fascinating that higher dimension manifolds are more restrictive!
Though in a _very_ handwavy way it seems intuitive given properties like that in TFA where 4-d is the only dimension where the edges of the bounding cube and inner spheres match. Especially given that that property seems related to the possible neighborhoods of points in d-4 manifolds. Though I quickly get lost in the specifics of the maths on manifolds. :)
> However in four dimensions something very interesting happens. The radius of the inner sphere is exactly 1/2, which is just large enough for the inner sphere to touch the sides of the cube!
Though in a _very_ handwavy way it seems intuitive given properties like that in TFA where 4-d is the only dimension where the edges of the bounding cube and inner spheres match. Especially given that that property seems related to the possible neighborhoods of points in d-4 manifolds. Though I quickly get lost in the specifics of the maths on manifolds. :)
> However in four dimensions something very interesting happens. The radius of the inner sphere is exactly 1/2, which is just large enough for the inner sphere to touch the sides of the cube!