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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