jurnal matematika

given by θ = 0, 2π, 4π, 6π, . . . for x > 0 and θ = π, 3π, 5π, 7π, . . . for x < 0 as can be seen in figure 1.
Thus, the angles of the intersections of the spiral with the horizontal axis are as follows:
θk = kπ (2)
where k = 0, . . . , N and in the limit we let N → ∞. The corresponding horizontal locations of these crossing points are
therefore:
xk = (−1)kbe−aθk = (−1)kbe−akπ (3)
and thus in terms of their absolute values these are:
|xk| = be−aθk (4)
Therefore, the crossing scales lk which are the spacings between successive crossing points can be expressed as:
lk = |xk − xk+2| = (−1)k(xk − xk+2) = b(e−akπ − e−a(k+2)π) (5)
where the reason the indices are k and k + 2 is because every pair of successive crossing points occurs on either the right
or the left side of the spiral as in figure 1. We show in figure 2 a schematic of the crossing scales lk and the corresponding
pairs of successive crossing points along the intersection through the center of the spiral.
The probability density function (pdf) of the crossing scales can be written therefore as a sum of delta functions because
all of the crossing scales in equation 5 have distinct values. Thus, the pdf of crossing scales can be expressed as:
p(l) =
1
N
N−1
k=0
δ(l − lk) (6)
for finite N, with the understanding that the analysis of the crossing scales for the entire logarithmic spiral correspond to
the limit N →∞.
We can now use the following general relation derived by the author (Catrakis 2000) which relates directly the pdf p(l) of
crossing scales to the fractal dimension D(λ) as a function of the coverage scale λ, i.e. the partition scale corresponding
to successively partitioning the entire interval containing all the point crossings, as follows:
D(λ) = 1 −
λ
∞
λ p(l)dl
λ
0
∞
λ p(l)dldλ
(7)
Substituting from equation 6 for the pdf p(l) of crossing scales of the logarithmic spiral, we obtain the following result for
the fractal dimension D(λ) which we express for clarity this way:
D(λ) = 1 − λA(λ)
B(λ)
(8)
where the functions A(λ) and B(λ) are given respectively by:
A(λ) =
∞
λ
p(l)dl =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
0 , λ>l0
kN
, lk−1 > λ > lk , k = 1, 2, . . . , N − 1
1 , λ<lN−1
(9)
and,
4