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Fractals, iterations and the Sierpinski Triangle an iterative approach

Central Arizona College Science Night at San Tan Campus

Fractals

What are they?Fractals – A term introduced by the French Mathematician Benoit Mandelbrot to describe objects that have ‘Fractional Dimension’.

-Dictionary of Mathematics, Penguin 1998

In other words the object’s dimensions are not a Natural number.

a better definition for a fractal from Wolfram Alpha

Fractal - A fractal is an object or quantity that displays self-similarity on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales and are generated by some infinitely repeated process.

The first definition for fractals defines them to be “objects that have ‘Fractional Dimensions’”?

What does that mean?

Let’s begin this explanation for fractional dimensions with observations of 1-, 2-, and 3- dimensional objects and the number of exact copies

that are created when those objects are doubled?

As it turns out there is a relationship between an objects dimension and the number of exact copies of the original object that are produced

when the object is doubled. We will use this relation ship to develop a better understanding of fractals and fractional dimensions.

Note: The number of copies will include the original object.

Let’s now observe what happens to 1-, 2-, and 3- dimensional objects when they are doubled.

One-dimensional object: line segment (it’s one dimension is only the length of the line segment)

Line segment doubled

The number of copies of the original object is 2

Two-dimensional object: a rectangle (the two dimensions for a rectangle are it’s length and width)

A rectangle doubled

The number of copies of the original object is 4

Three-dimensional object: a rectangular Prism(the three dimensions for a rectangular prism are it’s length, width and height)

A rectangular prism doubled

The number of copies of the original object is 8

Lets review our procedure and observed dataProcedure: double (multiply by 2) each d-Dimensional object (1-, 2-, and 3- dimensional) and record the number of copies of the original object produced (including the original) after doubling.

Double DimensionNumber of

Copies2 1 22 2 42 3 8… … …2 d n

After some rigorous mathematics we find the relationship between an objects dimensions(d) and the number of copies produced(n) by doubling

the object to be

If an object of d-dimensions(d) is doubled then the number of copies of the original object created can be found using the following relationship

We will use this relationship to determine the dimension of the following object.

Let’s now consider an equilateral triangle and use our established doubling procedure to determine the dimension of the following

object.

Equilateral Triangle:

Equilateral Triangle Doubled:

The number of copies of the original object is 3

The equilateral triangle was doubled and 3-copies of the original were produced, which means

To determine the value of the objects dimension (d) [note: the objects dimension is the exponent] we will employ some algebra and logarithms.

The dimension of this object is approximately 1.585

Observe what happens when the object (an equilateral triangle) is doubled for two iterations

Doubled first iteration:

Doubled second iteration:

Keep these in mind you will see them again

Sierpinski Triangle

Definitions:Iteration – applying a function repeatedly, using the previous output as the new input.

Example: using the following function f(x) =2x and x=1 for the first , find the value after four iterations

f(1) = 2First iteration: f(2) = 4second iteration: f(4) = 8Third iteration: f(8) = 16Forth iteration: f(16) = 32

Remember our “better” definition for fractals from Wolfram Alpha?

Fractal - A fractal is an object or quantity that displays self-similarity on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales and are generated by some infinitely repeated process.

Iterations provide us with an infinitely repeating process.

Sierpinski TriangleRule: remove the middle part of the triangle(s)

Initial object:

Sierpinski Triangle

Rule: remove the middle part of the triangle(s)

First iteration:(application of the rule)

Sierpinski TriangleRule: remove the middle part of the triangle(s)

Second iteration:

Sierpinski TriangleRule: remove the middle part of the triangle(s)

Third iteration:

Did you recognize the objects?

To generate a Sierpinski Triangle the one rule (Rule: remove the middle part of the triangle(s)) is iterated. The iteration of the rule produces an object that is similar to the object we obtained from doubling an equilateral triangle and in fact has the same fractal dimension,

Sierpinski Triangle has a fractal dimension approximately equal to 1.585.

Triangle Doubled [] Sierpinski Triangle [

Fractals in Nature

The Mandelbrot Set