Chapter 2
Iterated function systems:
fractals as limits
Often a Frst exposure to fractals consists of googling “fractal geometry,” Fnding
a program to generate fractals, selecting some example from a preset menu, and
clicking the RUN button. The image drifts into being, dot after dot dancing
across the screen in furious pattern. If the rules generating this fractal can be
viewed with the program, the Frst question is “How do these rules make that
picture?” The answer to that question is the Frst topic of this chapter, if an
I±S program had the highest google rank.
Before starting this relatively long analysis, a moment’s consideration of a
selfsimilar fractal suggests a general direction for the proof. Look at the fractal
shown in the left of ±ig. 2.1. This shape, the Sierpinski gasket, is one of the
simplest examples of a selfsimilar fractal. The middle image, a magniFcation
of a small portion of the left side, is indistinguishable from the left. The right
is a magniFcation of a portion of the middle. In our minds, this process can be
continued forever.
±igure 2.1: Successive magniFcations of portions of the gasket.
The reappearance, under increasingly high magniFcation, of copies of the
whole shape, suggests two things. ±irst, some sort of limiting process is in
volved here. Second, it is not the limiting process familiar from calculus, where
under magniFcation a smooth curve approaches its tangent line. Something dif
ferent is happening here: the level of complexity remains about constant under
magniFcation. So we must determine what sort of limit produces fractals.
25
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CHAPTER 2. ITERATED FUNCTION SYSTEMS
2.1
Iterated function system formalism
Based on work of Mandelbrot [102] and Hutchinson [81], and popularized by
Barnsley [5], iterated function systems are a formalism for generating fractals
and for compressing images. First we recall some background on transformations
of the plane.
In the plane, a general linear transformation plus translation can be written
as
T
b
x
y
B
=
b
r
cos(
θ
)
−
s
sin(
ϕ
)
r
sin(
θ
)
s
cos(
ϕ
)
Bb
x
y
B
+
b
e
f
B
(2.1)
That is, any real 2
×
2 matrix can be expressed as the 2
×
2 in eq (2.1). (See
Prob. 2.1.1.)
Let
d
(
,
) denote the Euclidean distance. A transformation
T
is a
d
contraction
with contraction factor
t
, 0
≤
t <
1, if for all points (
x
1
,y
1
) and (
x
2
2
),
d
p
T
b
x
1
y
1
B
,T
b
x
2
y
2
B
P
≤
t
·
d
p
b
x
1
y
1
B
,
b
x
2
y
2
B
P
,
(2.2)
and for any number
s < t
,
d
p
T
b
x
1
y
1
B
b
x
2
y
2
B
P
> s
·
d
p
b
x
1
y
1
B
,
b
x
2
y
2
B
P
for at least one pair of points (
x
1
1
) and (
x
2
2
).
For example, if
T
(
x,y
) = (
x/
2
,y/
2), then
d
p
T
b
x
1
y
1
B
b
x
2
y
2
B
P
=
r
±
x
1
2
−
x
2
2
²
2
+
±
y
1
2
−
y
2
2
²
2
=
1
2
d
p
b
x
1
y
1
B
,
b
x
2
y
2
B
P
for any pair of points. Consequently, this
T
is a contraction with contraction
factor 1
/
2. A slightly more di±cult example is given in Prob. 2.1.2.
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 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land, Metric space, Fractal, ifs, Iterated function system

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