As soon as the hotel has taken in even a small infinity of patrons, this layout is not possible - a circle must have a finite circumference (and if we did try t

As soon as the hotel has taken in even a small infinity of patrons, this layout is not possible - a circle must have a finite circumference (and if we did try to generalize our definition of circle it would result also in an infinite radius). Maybe a hypersphere with finite bounds but infinite dimensions? My geometry is fairly poor but I think this is what is called, for reasons not directly related to the hotel paradox, a Hilbert sphere.