FractionalGeometry-Chap2

29 some topology of ifs attractors 93 figure 259 the

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Unformatted text preview: hree functions from a specific class. Example 2.9.2 A three function topological classification problem On pgs 232-7 of [125] we find 224 attractors A of IFS {T1 , T2 , T3 } where Ti (x, y ) = (ri ·cos(θi )·x−si ·sin(ϕi )·y, ri ·sin(θi )·x+si ·cos(ϕi )·y )+(ei , fi ) (2.18) 90 CHAPTER 2. ITERATED FUNCTION SYSTEMS with ri = ±1/2, si = ±1/2, θi = ϕi = 0, π/2, π, 3π/2, and ei and fi take the values 0, 1/2, and 1, chosen so that T1 (A) ⊂ [0, 1/2] × [0, 1/2], T2(A) ⊂ [1/2, 1] × [0, 1/2], T3(A) ⊂ [0, 1/2] × [1/2, 1] Each of these transformations with si = −1/2 is equivalent to one with ri = −1/2: (+, −, 0) ↔ (−, +, π ), (+, −, π ) ↔ (−, +, 0), (+, −, π/2) ↔ (−, +, 3π/2), (+, −, 3π/2) ↔ (−, +, π/2) (2.19) where, for example, (−, +, π/2) represents ri = −1/2, s = 1/2, and θ = ϕ = π/2. This gives 8 distinct tramsformations of this type, so 83 = 512 different IFS. Yet Peitgen, J¨ rgens, and Saupe show only 224. Did they forget the other u 268? Of course not. First, none of those examples are symmetric across the y = x line, so reflecting each across that diagonal gives an additona...
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