Curve-stitch Designs

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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:

Same picture as previous one with only two nested squares

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 Square root of 1/2ócall it 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.

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