> Surely then there's arbitrarily large numbers of polygons but not infinite, because if they're infinite then the spacing between points will be zero?
If we describe the set of these polgons as P, then |P| = |N|, for N referring to the Natural Numbers.
This is usually called "abzählbar unendlich" (countable infinite).
That's kinda my point, if there are aleph_0 polygons, then surely the length of a side has to tend to 0. As you increase the number of vertices then the difference between the points occupied by the vertices and those of a coincidental circle reduces, ultimately the vertices must describe a circle if they form a set of size aleph_0?
If you want the sides of the polygons to have a length then the polygons, to me, seem necessarily to be unable to have an infinite number of vertices, they could only have a very large number but one necessarily less than infinity.
Perhaps it's my misunderstanding how a curve works? If you form a second set from an infinite number of equally spaced points between two arbitrary points on a continuous curve (the first set) then it seems, to me [ie intuitively, always a risk in maths], that the second set must also be a continuous curve? The corollary of that would seem to be that if you have an infinite number of vertices for your polygon it also forms a circle, to not form a circle it simply has to have a number of sides less than infinite.
My contention would then be that either the set of polytopes in 2D is < aleph_0 or the set in 3D is 6 (ie applying the same rules makes circles and spheres both parts of the set of polytopes in the particular n-dimensional space).
If we describe the set of these polgons as P, then |P| = |N|, for N referring to the Natural Numbers.
This is usually called "abzählbar unendlich" (countable infinite).