<Text-field style="Heading 1" layout="Heading 1" spaceabove="8">Introduction</Text-field>

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Fractional Dimension and Space-Filling Curves (with iterated function systems)

Mark D. Meyerson U.S. Naval Acdemy U.S.A. mdm@usna.edu

Abstract. By using complex numbers to represent points in the plane, and the concept of iterated function system, we efficiently describe fractal sets of any dimension from 0 to 2 and continuous curves that pass through them. Maple's animation feature allows us to make "movies" that show the transition through different dimensions.

Segments and curves are typically thought of as 1-dimensional objects, while a square region is a 2-dimensional object and a cube is a 3-dimensional object. Note that cutting a square into small identical copies with 1/3 the linear measurement requires 9 copies to fill the square. Cutting a cube into small cubes of 1/2 the linear measurement requires 8 copies to fill the square. Cutting a segment into pieces of 1/5 the linear measurement requires 5 copies to fill the segment. Thus, we are led to the formula: N=1/s^D, where N is the number of copies needed after shrinking by a factor s a set of dimension D. This type of dimension is called "self-similar dimension" -- it can only be applied to particularly nice sets that can be cut up into shrunken versions of the original.

In the following, we will show how to create a family of so-called fractal sets that have such self-similarity. They can be created with any desired dimension. We will show how to do so to get a smoothly changing family of fractal sets with every dimension from 0 to 2. As a bonus, we will show how to pass a continuous curve through these fractal sets, so that we see, at the end of the animation, how to get a space filling curve - in this case the image of a segment that completely fills up a square.

<Text-field style="Heading 1" layout="Heading 1" spaceabove="8">Initialization and Complex Number basics</Text-field>

restart:

Maple uses capital I to represent imaginary number i. Complex numbers are of the form a + b*I (a and b are called the REAL and IMAGINARY parts respectively) and all the usual arithmetic properties hold, with:

I^2;

Complex number addition is like vector addition:

with(plots):

a:=complexplot([0,2+I],thickness=5,scaling=constrained):

b:=complexplot([0,1+2*I],thickness=5,scaling=constrained):

c:=complexplot([0,3+3*I],thickness=5,scaling=constrained):

display(a,b,c);

Complex number multiplication: multiply distance from origin, and add angles:

(2+I)*(1+2*I);

d:=complexplot([0,(2+I)*(1+2*I)],thickness=5,scaling=constrained):

display(a,b,d);

Mulitplying by i ( = I ) rotates by 90 degrees:

(2+3*I)*I;

a:=complexplot([0,(2+3*I)],thickness=5,scaling=constrained):

b:=complexplot([0,I],thickness=5,scaling=constrained):

c:=complexplot([0,(2+3*I)*I],thickness=5,scaling=constrained):

display(a,b,c);

Conjugating flips a complex number across the real (x) axis - it negates the imaginary part:

conjugate(2+3*I);

a:=complexplot([0,(2+3*I)],thickness=5,scaling=constrained):

b:=complexplot([0,conjugate(2+3*I)],thickness=5,scaling=constrained):

display(a,b);

<Text-field style="Heading 1" layout="Heading 1" spaceabove="8">Iterated Function Systems and Fractals</Text-field>

The use of iterated function systems has only become popular in the last 20 years.

First we define a function system by shrinking by 1/2 toward each corner of the unit square.

f1:=z->z/2; f2:=z->(z-I)/2+I; f3:=z->(z-1-I)/2+1+I; f4:=z->(z-1)/2+1;

We pick an arbitrary set of points - roughly like the 5 spotted side of a die:

pts:=[0,.8+.1*I,.8+.8*I,.1+.8*I,.4+.5*I];

And plot them: