If I remember correctly we had this discussion before concerning Pi.Math is logic, but it must start with some assume or defined objects. A "point" is one of these that by definition (an axiom) has no size. Thus if you have a line segment between point A and B which are only 1 micron apart, there are an infinite number of point, all different form each other on that 1 micron line segment. Even more startling is fact that the number of points between A' & B' is an infinity just as large as if A' & B' were a trillion light years apart. You "common sense" is wrong as this can be proven. Basically this proof goes like this. given any point on AB, P, there is one and only one point. P',on A'B' that correspond to P (and conversely). For a specific example, consider point x which is closer to A than to B by the fraction 41/ 333 or 0.123123123123123123.... there is the point x' (the only one) that is also 0.123123123123123123.... closer to A' than to B'.

As far as to which of the 1/3 slices of pie the center point "belongs to" the choice is arbitrary. If initially it belongs to piece #2 and someone say: No, it should belong to pieces #1, then I say OK. move it. Do you think after then move that the area of piece #2 has decreased and that of piece #1 has increased?

If yes, it just shows you are illogical, of have forgotten the axiom defining what points are.

the result of this is to refer to boundary (topology)

http://en.wikipedia.org/wiki/Boundary_(topology)

which resolves the issue apparently.

And it is similar to the issue of reducing a spheres diameter by infinity - what do you actually end up with? [sort of thingo]

zero or non-existence