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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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 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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