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More Generalizations (Page 8)

Now, let’s generalize in a different direction. An inscribed square need not touch the enclosing square at the midpoint of a side. We can, in fact, choose any point on a side to place a vertex of the inscribed square. There are various ways we could parameterize this notion, including the rotation of the inner square relative to the outer one. I chose to do so in terms of the fraction of the base of the outer square to the left of a vertex of the inner square. If we refer to this measure as the fraction, the fraction for the designs we have seen so far has been 0.5. Figures with fractions deviating a bit from 0.5 seem slightly “off.” Here we have changed colors, and, using 10 segments and an offset of 10, have set the fraction to 0.65: When the fraction nears 0 or 1, we get a spiral effect. In this case, the fraction is only 0.1: Notice that we need more squares here—the above figure uses 50—as each inscribed square is only slightly smaller than the enclosing square.    