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.
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