Quote:
Originally Posted by Anthony
Well, an apeirohedron isn't a closed surface, so it can't be similar to a sphere. I'm not actually sure it's possible to define a fair (all sides equal) polyhedron with countably infinite sides (you can define an unfair one by selecting an infinite series that adds up to a finite value, such as (r^n) where r is a number between 0 and 1).
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My geometric intuition is that as the number of sides of the polyhedron goes to aleph sub null, the area of each side goes to 0, and so does the length of the boundary of each side. So it seems as if all the sides would be geometrically identical. In effect, each would be a geometric point. That seems to be a sort of regular polyhedron; you would lose the irregularities in taking the sidedness to aleph sub null.