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Unformatted text preview: y the random algorithm, but not by the
deterministic algorithm, unless the initial shape is a point. Figure 2.38: Two real snowﬂakes
Take the origin to be the center of the snowﬂake, and consider the left image
of Fig. 2.39. From our experience with the fern, we might think we need seven
rules: one for the center and one for the end of each of the six arms. Simpler
would be one rule for the center, one for an arm, and then rotation by 60◦
to build the other ﬁve arms. However, rotation is not a contraction and so
our proof of the convergence of the determinisitc algorithm no longer is valid.
Worse still, if one of the transformations is a rigid rotation, the deterministic
algorithm will not converge. The initial shape will appear in its original size
with each iteration. If the initial shape is a ﬁlled-in circle, then every generation
will contain a circle, rotated π/3 from the position of the circle in the previous 73 2.6. IFS FORGERIES OF NATURAL FRACTALS generation. See the...
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