More Generalizations (Page 3)
As it happens, this question is equivalent to asking
the size of the successive squares. The largest circle is tangent to the four
sides of the outermost square. Its diameter is therefore the same as any side of the square. This is easily seen if we look only
at the outer square and the first square inscribed within it:
Since the more complex design is produced by drawing the same
figure over and over at successively smaller scales, we could clearly take the
above figure, shrink it, rotate it, and place it within the inner square above.
This would introduce another circle, tangent to the sides of the inner square.
Using the Pythagorean Theorem, we can easily compute the length of the side of
the inner square and, thereby, the ratio of the outer and next inner circle. If
the length of a side of the outer square is 1, it is easy to see that the length
of a side of the inner square is the length of the hypotenuse of a right
triangle whose other two sides are 1/2. This value is
 —call
it r—or approximately 0.7071.
Because of the geometric relationships of the nested squares, we can easily see
that the ratio of successive diameters of the circles forms a geometric series:
s, sr, sr2, sr3, ... , where s
is the length of the side of the outer square.
Before going on, it is worth saying that the figure
above actually could be constructed with cardboard and thread. One can easily
imagine putting a photograph in the middle of it. |
