Disc with internal structure such that to an observer who is part of it the disc would appear to have an infinite area.

Question

I apologize if this has been answered in a way that is just beyond me. I relatively ignorant. I seem to remember a discussion in a book of a disc that had the following property: if there were a 2 dimensional being walking from the center toward the circumference the distance would appear infinite if as it walked it and the space kept shrinking. Is it possible to define such a structure mathematically?