Unformatted text preview: upper left pieces are oriented diﬀerently from the lower left and upper right,
which share the orientation of the whole shape. Because of the symmetry of the
pieces, these orientations can be achieved through reﬂections or rotations. For
example, the lower right piece can be obtained by reﬂecting the lower left piece
across the y axis and then translating by e = 1. Reﬂections across the y axis
can be obtained by making r negative; reﬂections across the xaxis by making
s negative. Consequently, fractal (b) can be obtained by
r
1/2
1/2
1/2
1/4 s
1/2
1/2
1/2
1/4 θ
0
0
0
0 ϕ
0
0
0
0 e
0
1
0
3/4 f
0
0
1
3/4 Rotations, too, can produce these orientations. With the given formulation
of the transformations, rotations ﬁx the origin, and positive angles are counterclockwise. Note also the order of the components of these transformations:
scalings and reﬂections ﬁrst, rotations second, and translations last.
Using rotations, another set of transformations can produce fractal (...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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