In 1904 the Swedish mathematician Helge von Koch proposed a method for constructing a "snowflake" curve.44 Like the Sierpinski curve, the Koch curve is a closed limit curve of infinite length that bounds a region of finite area.45 But unlike the Hilbert and Sierpinski curves, the Koch snowflake is a fractal curve that is not plane-filling.
The method of construction that Koch proposed begins with an equilateral triangle with sides of unit length. Each side is then trisected, and each middle segment is replaced by a smaller equilateral triangle whose sides measure 1/3. The middle segments are deleted, resulting in a Star of David. This process of trisecting the resulting sides and replacing them with smaller triangles is repeated ad infinitum.46 Figure 2.3 shows three curves in this sequence.
Figure 2.3
Koch Snowflakes of Order 1, 2 and 4
Cesaro devised an alternate method for constructing the Koch snowflake in
1905. His construction begins with a larger regular hexagon and proceeds by
displacing the midpoints of each side inward. The limit snowflake curve
lies between these outer approximations and the inner approximations of the
original
Each iteration of Koch's algorithm increases the length of the curve by a factor of 4/3. Thus it is easy to see that the curve's length approaches infinity as the order of the curve increases without bound. The nth approximation, Cn, has four times as many sides as Cn-1. Therefore the number of triangles to be added at each stage quadruples, while the triangles added to Cn have 1/9 the area of those added to Cn-1. Thus the areas enclosed by successive approximations to the Koch snowflake form an infinite geometric series with common ratio 4/9. As n approaches infinity, the area of Cn approaches 8/5 that of the original triangle.48 Thus the Koch snowflake is an infinitely long curve that encloses a finite area.
The Koch snowflake clearly illustrates the concept of fractal dimension.
Each side is split into four new sides;
The Koch curve can be generalized in a number of ways. Any regular
polygon can be used initially, and many methods for sectioning the sides
exist in addition to trisection. The basic algorithm has also been applied
in dimensions other than
two.52 An enormous
variety of non-intersecting fractal curves and surfaces can be generated by
this type of recursive
44 Gardner, "'Monster' Curves," 124.
45 Joel E. Schneider, "A Generalization of the von Koch Curve," Mathematics Magazine 38, no. 3 (1965): 144.
46 Schneider, 144.
47 Benoit B. Mandelbrot, Fractals: Form, Chance, and Dimension (San Francisco: W. H. Freeman, 1977), 37.
48 Schneider, 144.
49 Batty, 32.
50 Rose, 24.
51 Mandelbrot, Form, Chance, and Dimension, 40.
52 Schneider, 144.
53 Gardner, "'Monster' Curves," 133.
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