For a new puzzle suppose that a circle has area 1. Within
it is inscribed a square with all four corners
on the
circumference of the circle. Within this square is inscribed
another circle with its circumference
touching the midpoints of
the sides of the square. Within this circle is another square.
Within this square
is another circle and so on. This process of
squares within circles within squares is continued until 8
circles
(including the first) have been drawn. What is the area
of this 8th circle? This problem can be worked with
ordinary
geometry. If you have had experience with summing infinite
series you may wish to work on the
following variation. Suppose
that the process of squares within circles within squares is
continued forever.
What is the sum of the areas of all of the
circles and squares created?