Beauty in Mathematics? A Study in Fractals

A fractal is an intricate geometric figure that contains infinitely many smaller copies of itself.  In fact, when magnified over and over again, a fractal image always seems to look the same.  These images have long been appreciated for their striking beauty and mathematical complexity.

Two famous fractal images, the Sierpinski Triangle and the Fractal Plus, are shown below.

We will investigate two methods for fractal generation:
deterministic methods and random (or chaotic) methods.

Deterministic methods follow a predetermined rule.  In this method, an initial image (or seed) is chosen and then the rule is carried out over and over again. The resulting sequence of images is called the orbit.

For the Sierpinski Triangle:        
Seed:  Equilateral Triangle
Rule:   Reduce image by 50%, make 3 copies and arrange in the pattern shown below.

For the Fractal Plus:
Seed:  Square
Rule: Reduce image by 33.333%, make copies and arrange in the pattern shown below.    

Random methods are also referred to as chaos games.  In these games, a movement scheme and vertices are defined and then “played” according to some random process.   Use the information below and http://math.bu.edu/DYSYS/applets/fractalina.html to generate the fractals.  Be sure to think about how the given movement scheme and vertices were determined. 

For the Sierpinski Triangle:
Movement Scheme:  1/2
Vertices:  (-120,-120), (120,-120), (0, 134)
Random Process:  Roll a die.

For the Fractal Plus:
Movement Scheme: 1/3
Vertices: (0,0), (150,0),(0,-150), (0,150),(-150,0)
Random Process:  pick 1 card from A,2,3,4, 5.

Assignment: For the fractals shown below.

I. Determine the seeds and rules for the fractals below.  Use PowerPoint to generate orbits. 

II. Determine the vertices, movement scheme, and random process for the fractals below.  Use Fractalina at http://math.bu.edu/DYSYS/applets/fractalina.html to generate the fractals

Application of Fractals 

In the mid 1970’s Benoit Mandelbrot made an important mathematical discovery.  He realized that fractal geometry rather than Euclidean geometry better describes many of nature’s objects.  Ordinary geometric constructions (circles, squares, etc.) were not very useful in analyzing the intricate patterns found in ferns, snowflakes, clouds, coastlines and tumors.

The two fractals below, Barnsley ’s Fern and Koch snowflake, have been generated by mathematical methods.