EXPLORING FRACTALS ON THE WEB

History

• Early 1900’s:  Koch, Sierpinski, Cantor
• 1918 -1920:   Julia, Fatou
• 1970’s:   Mandelbrot, Hubbard, Douady, Barnsley, Feigenbaum, etc.

Description

• The geometry of nature
• Has self-similarity on every scale
• Generated by iteration, using a starting point (seed), and a rule. In the iteration, the output of the previous computation is used as the input for the next.  The resulting sequence is called the orbit of the seed. The limiting stage of the iteration is called the fate of the orbit, or the attractor.           

Applications

• Fluid dynamics, physics, stock market analysis, seismology, data encoding, meteorology, human physiology, medicine, geology, astronomy, music...

Fractal Dimension

• S: scaling factor
• N: number of similar copies
• D: fractal dimension
• D = log(N)/log(S)

Geometric iteration

Koch curve 

 http://www.shodor.org/interactivate/activities/koch/index.html

• Seed: A line segment one unit long,
• Rule: Replace the middle third with two segments each 1/3 unit long.

• Length of limiting Koch curve approaches infinity (each segment is 4/3 of previous segment)
• Continuous everywhere, but differentiable nowhere

• 3: scaling factor
• 4: number of similar copies
• D: fractal dimension
• D = log(4)/log(3) or about 1.26 

Sierpinski triangle

• Seed: An equilateral triangle one unit per side
• Rule: Join the midpoints of the sides, and remove the middle triangle.

• Area approaches 0. (Each triangle has ¾ the area of previous triangle.)

• 2: scaling factor
• 3: number of similar copies
• D: fractal dimension
• D = log(3)/log(2) or about 1.59 

Peano Curve (a space-filling curve)

http://www.math.umass.edu/~mconnors/fractal/generate/peano.html

• Seed:  A line segment 1 unit long
• Rule:   Replace the line segment with 9 pieces, all 1/3 unit long.
           
• 3: scaling factor
• 9: number of similar copies
• D: fractal dimension
• D = log(9)/log(3) = 2

• Example of a one-dimensional curve filling a two-dimensional space.

Cantor set  (perhaps the first fractal studied)  

http://www.shodor.org/interactivate/activities/cantor/index.html

• Seed:  A line segment one unit long
• Rule:  Divide the segment three equal segments and remove the middle one.  

• 3: scaling factor
• 2: number of similar copies
• D: fractal dimension
• D = log(2)/log(3) or about .63

Random Iteration   

http://www.shodor.org/interactivate/activities/chaosgame/index.html

• Seed:   Any point inside an equilateral triangle
• Rule:   Roll a die, and depending on what number comes up, move the seed half the distance to the appropriate vertex.

Quadratic Iteration

Julia sets (1918 - 1920):

http://math.bu.edu/DYSYS/explorer/tour2.html

http://www.shodor.org/interactivate/activities/julia/index.html

• The ‘filled’ Julia set is the set of all seeds whose orbits do not escape to infinity under iteration of x² + c.    (The Julia set is the boundary of the filled Julia set.)

• Seed:  any complex number x_o
• Rule:  x_n+1 = x_n² + c, where x_n is a complex number.

• For every complex value of c, there is a different Julia set.

Mandelbrot set (1980) 

http://math.bu.edu/DYSYS/explorer/tour1.html

http://math.bu.edu/DYSYS/applets/M-setIteration.html

• The set of complex c-values for which the orbit of zero does not escape to infinity under iteration of x² + c.

• Seed:  x_o = 0
• Rule:  x_n+1 = x_n² + c, where x_n is a complex number.

Connection between Julia sets and the Mandelbrot set

http://math.bu.edu/DYSYS/applets/JuliaIteration.html

http://math.bu.edu/DYSYS/explorer/page1.html

• Given a Julia set x² + c, if the orbit of zero (the critical orbit) escapes to infinity, then the Julia set is fractal dust (a Cantor set); if the critical orbit doesn't escape to infinity, then the Julia set is completely connected. Therefore, the Julia set for any c in the Mandelbrot set is completely connected, whereas a Julia set for any c not in the Mandelbrot set is a cloud of dust. For this reason the Mandelbrot set has been called a dictionary of all the Julia sets.