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normal walking. We established that such physiologic time series are fractal; they
satisfy scaling, and the scaling represents a kind of long-time memory in complex
physiologic webs.
2.6
Problems
2.1 The Fibonacci sequence
Fibonacci scaling has been observed in the dimensions of the Parthenon as well as in
vases and sculptures of the same period. The golden mean was made famous by the
mathematician Leonardo Fibonacci (Filius Bonaci), also known as Leonardo of Pisa,
who was born in 1175 AD. The problem he solved was as follows. Given a sequence
of integers such that an integer is the sum of the two preceding integers, what is the
limiting ratio of successive integers? The algebraic expression of the problem given the
sequence of integers labeled by a subscript is
u n + 1 =
u n +
u n 1 .
Show that u n + 1 /
φ
u n equals
as n becomes large.
2.2 An infinitely long line
Consider a curve that connects two points in space. It is not unreasonable to assume
that the length of the curve connecting the two points is finite if the Euclidean dis-
tance between the two points is finite. However, let us modify the argument for
the Cantor set in Figure 2.13 and not concentrate on the mass, but instead increase
the length of the line segment by not removing the trisected region, and replac-
ing this region with a cap that is twice the length of the interval removed. In this
way the unit line segment after one generation is 4/3 units long. If each of the four
line segments in the curve is now replaced with the trisecting operation the second-
generation curve has a length of 16/9. What are a and b in this scaling operation
and consequently what is the fractal dimension of the line segment? If the opera-
tion is repeated an infinite number of times, determine the dimension of the resulting
curve. Show that the length of the curve given by L
N 1 D diverges as
(
r
) =
Nr
=
r
0.
2.3 The generalized Weierstrass function
(a) Show that the GWF given by ( 2.38 ) satisfies the scaling relation ( 2.33 ). Solve the
scaling equation to obtain the coefficients in the solution ( 2.36 ).
(b) The GWF is so elegant pedagogically because it allows the student to explore
explicitly all the scaling concepts being described. Write a computer program to
evaluate the series ( 2.38 )for a
=
4 and b
=
8
.
(1) Graph the function separately
for the intervals
1
<
n
<
1
,
2
<
n
<
2 and
3
<
n
<
3 and discuss the result-
ing curves. (2) Compare the graph with 0
<
W
<
0
.
8 and 0
<
t
<
1 with the
graph with 0
<
W
<
0
.
2 and 0
<
t
<
0
.
12 and discuss. (3) Superpose the algebraic
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