Vol. 8 No 7 - Pi Mu Epsilon

where N represented the number of parts and r represented the scaling
factor. Consider one side of an equilateral triangle on the triadic*
curve (Figure 5.1-5.2).
Let this length -- call it k -- be the frame of
reference. Imagine splitting the length into thirds (Figure 5.3) and
constructing another equilateral triangle in the middle one-third
(Figure 5.4). Notice that there are now four sides each with length
one-third the original length i. Hence, our total length is now equal
to:
(4 segments) xl/(the number of segments).
The dimension of this structure for If = 4 and P = 1/3 is:
Since the decimal equivalent of the expression is about 1.2618 we have
a fractional dimension. Figure 5.5 shrinks the segment to the original
scale and places it back where we originally removed the segment.
Figure 5.6 demonstrates the completed curve if the preceding process is
used on the other segments a, b, and e of the original figure.
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Figure 5.2
Figure 4
From THE FRACTAL GEOMETRY OF NATURE
by Benoit B. Mandelbrot.
Copyright 1977, 1982, 1983.
Reprinted with the permission of
W. H. Freeman and Company.
Figure 5.3 I I
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