Iterated Function
Systems
Larry Riddle
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Koch Snowflake


Niels Koch
von Koch




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Construction
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 Start with an equilateral triangle T. Scale T by a factor of 1/3 and place 3 copies along each of the three sides of T as illustrated in the diagram below to form a new image S(1). Next scale T by a factor of 1/9 = (1/3)^2 and place 12=4*3 copies along the sides of T(1) as illustrated to form the image S(2). For the next iteration, take 48=4*12 copies of T scaled by a factor of 1/27=(1/3)^3 and place them around the sides of S(2) to form the image S(3). Continue this construction. The Koch Snowflake is the limiting image of the construction.

The sequence of sets S(1), S(2), S(3), .... that are constructed this way form a Cauchy sequence in the Hausdorff metric, and the limit is the Koch snowflake. Here is a proof.

Leslie Reid, Missouri State University, provided the following IFS code for the Koch snowflake. Use 12 maps. The first 6 consist of scaling by a factor of 1/3 and respective translations by (0,4), (-sqrt(3),1), (sqrt(3),1), (-2sqrt(3), -2), (0,-2), and (2sqrt(3),-2). The other 6 maps consist of scaling by a factor of 1/3, rotation by 180 degrees, and translation by the opposite of the vectors listed above. This produces the following image.

snowflake

The boundary of the snowflake consists of three copies of the Koch curve placed around the three sides of the initial equilateral triangle. It can be constructed by the following L-system:

Angle 60
Axiom F++F++F
F --> F-F++F-F

The Koch snowflake is also known as the Koch island.

The length of the boundary of S(n) at the nth iteration of the construction is 3*(4/3)^n*s, where s denotes the length of each side of the original equilateral triangle. Therefore the Koch snowflake has a perimeter of infinite length.

The area of S(n) is . Letting n go to infinity shows that the area of the Koch snowflake is . Therefore the snowflake has a finite area bounded by a perimeter of infinite length! Here are the details.

 
  1. Barcellos, Anthony. "The Fractal Geometry of Mandelbrot," College Mathematics Journal.
  2. Camp, Dane. "A Fractal Excursion," Mathematics Teacher, April 1991, 265-275.
  3. Edgar, Gerald A. Measure, Topology, and Fractal Geometry, Springer-Verlag, 1990.
  4. Lauwerier, Hans. Fractals: Endlessly Repeated Geometrical Figures, translated by Sophia Gill-Hoffstadt, Princeton University Press, 1991.
  5. Mandelbrot, Benoit. The Fractal Geometry of Nature, W.H. Freeman and Co. 1983.
  6. Peitgen, Heinz-Otto, Hartmut Jurgens and Dietmar Saupe. Fractals for the Classroom, Part One: Introduction to Fractals and Chaos, Springer-Verlag New York, Inc. 1990.
  7. Prusinkiewicz, Przemyslaw and James Hanan. Lindenmayer Systems, Fractals, and Plants, Lecture Notes in Biomathematics #79, Springer-Verlag 1989.
  8. Prusinkiewicz, Przemyslaw and Aristid Lindenmayer. The Algorithmic Beauty of Plants, Springer-Verlag, 1990.
  9. Walter, William Grey. The Curve of the Snowflake, W. W. Norton & Co. Inc., NY, 1956 (science fiction novel).
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