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Unformatted text preview: left side of Fig. 2.40. If the imitial shape is a point, the
deterministic algorithm still will converge. See the right side of Fig. 2.40. Figure 2.39: Two fractal snowﬂakes
Elton [48] gave a proof that the random algorithm converges if some of the
Ti are not contractions, so long as on average the Ti are contractions. Figure 2.40: Applying the deterministic IFS algorithm to sonwﬂake rules. Left:
starting with a circle. Right: starting with a point.
The left snowﬂake of Fig. 2.39 is made with three transformations. The ﬁrst
transformation scales by 0.4: it ﬁlls the middle of the snowﬂake with a small
copy of the whole ﬂake. The second transformation makes an elongated version
of the ﬂake and translates it to the right. This is the right branch of the ﬂake.
The last transformation rotates by π/3; iterated applications of this T3 builds
the other ﬁve arms of the snowﬂake. Be careful about trying to separate out
too fully the roles of the Ti . For example, without T3 , T1 and T2 generate a
horizontal line segment....
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 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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