So we know that \(\pi=3.14159...\), but here is an interesting example of
how playing with infitite sums can give weird results. If we have a circle
of diameter \(1\), its circumference is \(\pi\times 1=\pi\).
Take a square of width \(1\) and repeatedly fold in the corners like in the picture.
By changing the shape, you don't change the perimeter, as you always still use horizontal
or vertical lines at right angles so save or gain no distance. Adding up all the little
straight lengths, it looks like we get the same shape as the circle of the same size... right?
What is going on?
We see that by repeatedly folding the corners in of the square, it converges
to the shape of the circle. So the perimeter of the original square (\(4\)),
which remains the same through the 'folding', must be equal to the circumference
of the circle...
Now we know that's not true, in fact the two shapes never exactly match, the
deformed square is always going to be crinkly, not smooth like the circle, and
that is where the extra length hides!
You can show lots of seemingly impossible equalities with this method though,
by using a triangle instead of a circle, we can show that