Unformatted text preview: b).
r
1/2
1/2
1/2
1/4 s
1/2
1/2
1/2
1/4 θ
0
π/2
−π/2
0 ϕ
0
π/2
−π/2
0 e
0
1
0
3/4 f
0
0
1
3/4 In general, rotations and reﬂections are not equivalent. Reﬂections reverse orientation, rotations preserve orientation. For sets with reﬂectional symmetry,
some rotations and reﬂections are equivalent.
Fractal (c) is made of three copies of itself, each scaled by 1/2. The lower
left piece has the same orientation as the whole, the lower right is reﬂected, the
upper right both reﬂected and rotated. This fractal is produced by this IFS
table. 29 2.1. ITERATED FUNCTION SYSTEM FORMALISM
r
1/2
1/2
1/2 s
1/2
1/2
1/2 θ
0
0
π/2 ϕ
0
0
π/2 e
0
1/2
1/2 f
0
1/2
1/2 Note there is no reason to limit ourselves to fractals as subsets of R2 .
Example 2.1.2 An IFS in 3 dimensions.
In Fig. 2.5 we see the equilateral Sierpinski tetrahedron generated by these
transformations.
xxx
,,
T1 (x, y, z ) =
222
1
xxx
+
T2 (x, y, z ) =
,,
, 0, 0
222
2√
3
1
xxx
+
T3 (x, y, z ) =
,,
,
,0
222
4√
4
1
33
xxx
+
,,
,
,
T4 (x, y, z ) =
222...
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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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