Math Regular polygon

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Now move to a regular polygon. We lose just a little bit of
generality, because we see that only with an even number of sides can
we call one SIDE opposite another; but otherwise all the definitions I
can see yield the same result. We haven't clarified the definition at
all.

Now make the polygon slightly irregular, and we're forced to make some
decisions. There probably won't be any parallel side, so that's out.
Halfway around the perimeter, or cutting the polygon into equal
halves, may give you an intuitively "opposite" side, but may also give
a vertex, help in math ; leaving the choice uncertain - and both ways would be very
hard to calculate. Actually, once the sides have different lengths,
such a definition applies only to points not sides; and that's the key
to our choice. Since we're talking about sides, our definition ought
to relate to sides.

So we go back to the most basic possible
definition, one that relies only on counting sides - count half the
sides, and you're at the opposite side. This is the "topological,"
rather than "metric" definition - one that doesn't depend on
measuring any distances, but only on how the sides are connected. For
some special purposes a different definition (especially for 'opposite
point') might be useful, but since we're accustomed to thinking of
polygons topologically, this feels so natural to most mathematicians
that we don't bother mentioning it. more on math forum.