FractionalGeometry-Chap2

Three pieces each generated by two distinct

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Unformatted text preview: x2 )2 + (y1 − y2 )2 ? Certainly, the Euclidean distance is more familiar, but calculations with max are simpler. How do we know that points nearby in Euclidean distance also are nearby (though to be sure the distance may be different) in the max distance? Consider a point R = (x0 , y0 ). In the Euclidean distance, the points within a given distance r of R is the disc of radius r and center R; in the max distance it is the square of side length 2r and with center R. Each square with center R contains a disc with center R; each disc with center R contains a square with center R. That is, these two metrics have compatible notions of closeness. Using these notions of distance, we have Theorem 2.9.1 The function A : (P n , D) −→ (K(R2 ), h) taking a list on parameters to the attractor of the IFS with those parameters, is continuous. To get some understanding of how IFS parameters relate to the topology of the attractor, we explore two examples. In the first, we modify θ and ϕ for one transformation of the IFS for an equilateral gasket. In the second example, we consider all the topological types of attractors in a limited collec...
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