Chapter 2
Iterated function systems:
fractals as limits
Often a first exposure to fractals consists of googling “fractal geometry,” finding
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 first question is “How do these rules make that
picture?” The answer to that question is the first topic of this chapter, if an
IFS 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 Fig. 2.1.
This shape, the Sierpinski gasket, is one of the
simplest examples of a selfsimilar fractal. The middle image, a magnification
of a small portion of the left side, is indistinguishable from the left. The right
is a magnification of a portion of the middle. In our minds, this process can be
continued forever.
Figure 2.1: Successive magnifications of portions of the gasket.
The reappearance, under increasingly high magnification, of copies of the
whole shape, suggests two things.
First, some sort of limiting process is in
volved here. Second, it is not the limiting process familiar from calculus, where
under magnification a smooth curve approaches its tangent line. Something dif
ferent is happening here: the level of complexity remains about constant under
magnification. So we must determine what sort of limit produces fractals.
25
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
26
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
bracketleftbigg
x
y
bracketrightbigg
=
bracketleftbigg
r
cos(
θ
)
−
s
sin(
ϕ
)
r
sin(
θ
)
s
cos(
ϕ
)
bracketrightbiggbracketleftbigg
x
y
bracketrightbigg
+
bracketleftbigg
e
f
bracketrightbigg
(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
,y
2
),
d
parenleftBigg
T
bracketleftbigg
x
1
y
1
bracketrightbigg
,T
bracketleftbigg
x
2
y
2
bracketrightbigg
parenrightBigg
≤
t
·
d
parenleftBigg
bracketleftbigg
x
1
y
1
bracketrightbigg
,
bracketleftbigg
x
2
y
2
bracketrightbigg
parenrightBigg
,
(2.2)
and for any number
s<t
,
d
parenleftBigg
T
bracketleftbigg
x
1
y
1
bracketrightbigg
,T
bracketleftbigg
x
2
y
2
bracketrightbigg
parenrightBigg
>s
·
d
parenleftBigg
bracketleftbigg
x
1
y
1
bracketrightbigg
,
bracketleftbigg
x
2
y
2
bracketrightbigg
parenrightBigg
for at least one pair of points (
x
1
,y
1
) and (
x
2
,y
2
).
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land, Metric space, Fractal, ifs, Iterated function system

Click to edit the document details