FractionalGeometry-Chap2

# Reections reverse orientation rotations preserve

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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.

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