Draw a circle.
Draw a square around it. Perimeter = 4
Remove corners. Perimeter is still 4 !
Remove more corners. Perimeter is still 4 !
Repeat to infinity.
Pi = 4 !
Problem Archimedes ?
If you approximate a curve
by a series of straight lines,
how do you know
when you use smaller and smaller straight lines,
in the limit,
the sum of lenghs of these small straight lines
will be the length of the curve ?
I suspect it has something to do with
differentiability.
If you approximate a curve
by smaller and smaller corners,
in the limit,
the 1st order differential of the resulting curve
is no where continuous.
Whereas if you aproximate the curve
by the hypotenuse of the small corners,
the resulting curve's 1st differential
will be continuous when the hypotenuses
become smaller and smaller.
Thus the lenghts of the hypotenuses
will be able to approximate the length of the curve.
I will leave you to fill in the missing details
on why a continuous 1st order defferential
will enable the lengths of a series of
small straight lines to be able to approximate
the length of the curve.
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