Investigating the Coastline Paradox
1: Introduction
As a runner and a frequent traveller, I often research whether the islands I am travelling to are
small enough that I could run around them in a reasonable amount of time. I was born in Bali, in
the country of Indonesia, an archipelagic country filled with thousands of little islands. Whilst we
do not have an impressive landmass, my country has a lot of coastlines. Whilst area is easily
measured in acres, trying to systematically measure a coastlines length appears to be more
difficult than one may first assume. From my research I discovered that the value for various
coastline’s lengths can vary significantly from site to site For Indonesia, the CIA World Factbook
measures the length as 54,720km[13] whilst the World Resources Institute states it as
95,181km[14], a massive difference. At first I wanted to investigate how geographers approach
measuring the length of a coastline length. Does someone just run around the island with a
GPS? Can it be done through Google maps? However, after further investigation, I quickly
realized how this problem is more complicated than I first imagined.
2: The Coastline Paradox
2.1 What is the Coastline Paradox?
Anyone that has been to a beach will know that there is no clear line between the ocean and the
land, as it is instead constantly changing due to the nature of the ocean. In search for a way to
measure the length of a coastline, problems arise. If you attempt to measure the distance by
walking around the island; where do you start and which path do you take? Coastlines naturally
change overtime due to do the weather and erosion. An estimate of a coastline is generally
found by approximating complex curves as straight lines. However, this approximation actually
leads to another issue, which is that the size of the coastline increases as you decrease the size
of the ruler. As the ruler becomes really small, the length of the measured coastline approaches
infinity, as more complexity is added.
This paradox is known as the ​Coastline Paradox and was first recorded by Lewis Fry
Richardson and expanded on by Benoit Mandelbrot in his paper ​How Long Is the Coast of
Britain? ​[2] in 1967. In it, Mandelbrot explained the problem with trying to map coastlines:
Seacoast shapes are examples of highly involved curves such that each of their
portions can in a statistical sense can be considered a reduced-scale image of the
whole. This property will be referred to as "statistical self-similarity.” To speak of a
length for such figures is usually meaningless... As even finer features are taken
into account, the total measured length increases, and there is usually no
clear-cut gap or crossover, between the realm of geography and details with
which geography need not be concerned.” ​[2]
2.2 Aim of IA
In this IA, I will, firstly, investigate the concept of the fractal dimension and, secondly,
attempt to answer the question: ​What are the fractal dimensions of the Indonesian islands;
Sulawesi and West Nusa Tenggara? These are both islands which appear to have
complicated coastlines. I was also tempted to pick Sulawesi because the island just
experienced an earthquake and tsunami. Events like these highlight how useless the
measurement of coastline “length” is, as the shape of an island edge is changing all the time.
3: Investigating the Concept of the Fractal Dimensio​n
3.1 The Complexity of the Bali’s Coastlin​e
As a preliminary study, I decided to investigate this phenomenon by looking at Bali’s coastline
using Google Maps. As shown if Figure 2, Bali’s coastline does appear to embody the statistical
self symmetry that Mandlebrot described.
Fig 1. The Island of Bali at different magnifications to show the increasing complexity of the
coastline
From these images, there appears to be no point at which the island’s coastline “smoothens”
out, as each zoom reveals more complexity. This presents a problem for someone who is trying
to approximate the length of a coastline using a ruler, as it becomes impossible to pick a point at
which the complexity can be simplified as a straight line. The conclusion is that coastlines
cannot be analysed effectively using ideas from Euclidean geometry, where if you zoom in
enough curved lines are essentially straight lines whose lengths can be easily measured (Fig 2).
This was indeed Mandelbrots conclusion. Eddie Woo wrote that Mandelbrot found this
“Euclidean vision of the world deeply unsettling” because “reality is not made up of smooth
unbroken lines. It is filled with jagged edges and bumpy surfaces, lines that have been broken
and split into millions of different pieces and have little resemblance to Euclid’s divine
shapes”[4].
Fig 2 Circle the demonstrates the main idea in Euclidean geometry where curves are
approximated to show straight lines [4]
Mandelbrot proposed a new way of describing a coastline, in which he introduced the idea of
the fractal dimension. The fractal dimension assumes an alternative definition for the concept of
a dimension. Mandelbrot proposes that such shapes with “infinite” parameters can be
interpreted using the fractional dimensions in which a shape can have a non-whole number
dimension. (e.g 1.585 dimensions). For a coastline, the number would fall between 1 and 2
where the closer the number is to 2, the more jagged line. Smooth coastlines such as the one of
South Africa are said to have really smooth coastlines and thus have a fractal dimension of 1.02
[4].
3.2 What is a fractal dimension?
Mandelbrot saw the definition of dimension as; an n
dimensional shape is one that when scaled by a factor
1
n
x , the shape creates x copies of itself. For example,
when a square plane (n=2) is scaled by a factor
2
1
3
, it
creates 9 copies of itself, which is 3 . That power of 2 is
its dimension. Fig. 4 summarises this for the three
dimensions we normally associate with when thinking in
terms of regular Euclidean geometry.
Fig. 3 Summary of how fractal
dimensions are derived [1]
This can be represented through the equation:
D
( r1 ) = N
r = scale f actor
D = shapes dimension
N = number of copies of itself created
Using log laws, this can be rearranged to find the D:
D
( r1 ) = N
D
log ( 1r ) = log N
D log
1
r
D=
D=
= log N
log N
log 1r
log N
log 1r
This equation suggests an alternative way of looking at dimensions and challenges the common
intuition of what a dimension is.
3.3 Calculating the Dimensions of Sierpinksi’s Triangle and Koch Snowflake
In order to deepen my understanding of fractal dimensions, I will attempt to investigate the
fractal dimensions of various man-made fractals, the most common examples being the Koch
Snowflake and the Sierpinski Triangle. In order to find the fractal dimensions, the shapes will be
scaled down by a given scale factor, and then counted for the number of copies of itself that
were created.
Fractal Dimension of the Sierpinski Triangle
Fig 4 Sierpinski's Triangle​ [6]
D=
D=
D=
log N
log 1r
log N
log r
log 3
log 11
( )
2
D=
log 3
log 2
D ≈ 1.585
Fractal Dimension of the Koch Snowflake
Fig 5 Koch Snowflake scaled down by a third [7]
D=
D=
log N
log 1r
log 4
log 11
2
D=
log 4
log2
D ≈ 1.262
3.4 The Connections of Fractals with Coastlines
The Koch Snowflake has a “finite” area with an “infinite” perimeter. These properties are similar
to the coastline which has a finite area and a perimeter that approaches infinity. Rather than
measuring the length, the fractal dimension is able to measure the complexity of the edge.
Thus, rather than attempting to estimate the length of the island's coastlines, I will attempt to
calculate the fractal dimension of two different Indonesian islands, using two different
techniques; the Hausdorff Method [9] and the Box Plotting technique [10]. Before that, I will
attempt to find the fractal dimension of Britain, to check that my methods work. This value was
calculated in Mandelbrots paper (≈ 1.25) [2]. If I am able to achieve a fractal dimension similar to
his, it will indicate that my method is an effective way of calculating fractal dimensions, as well
as give me a sense of my degree of error in carrying out the method. I will trial two different
methods of calculating the fractal dimension, and evaluate from there which method is the most
suitable for calculating the fractal dimension of 2 Indonesian Islands.
4.1: The Hausdorff Method
This method for measuring the fractal dimensions of shapes with complicated edges was
introduced by Felix Hausdorff [9] and was used by Mandelbrot when he was trying to determine
the fractal dimension of the coast of Great Britain. It involves using a ruler with a length G and
approximating several straight lines between points on a coastline. The final length of coastline
will be the length of the ruler (G), multiplied by the number of rulers it takes to cover the entire
coastline (n).
In his paper, Mandelbrot described how “L depends greatly upon G”. As you make G smaller,
you are able to pick up on more finer details, and thus L increases. How much the length of the
approximated coastline increases and the length of the ruler decreases is related by the
equation:
L = M G1−D [2]
L = length of coastline
M = proportionality constant
G = length of ruler
D = f ractal dimension
Using log laws this equation can be turned into the form y = mx + c :
L = M G1−D
log L = log M G1−D
log L = log M + log G1−D
log L = log M + (1 − D)log G (1)
This can be plotted as:
y = log L
m = (1 − D) (2)
Thus
D = 1 − m (2)
x = log G
c = log M
4.2: The Hausdorff Method Applied to Britain’s Coastline
Using photoshop, I placed rulers along Britain's coastline, each time using different lengths (G),
and developed these four figures. Then I counted the number of rulers placed around the
coastline to find L, the length of the coastline. L is thus equal to the length of the ruler multiplied
by n, the number of rulers placed around the coastline.
Table 1
Length of Ruler
(G) (k
200
100
50
25
log (G)
2.301
2.000
1.699
1.298
Number of rulers
(n)
12
32
72
172
Length coastline
(L)
(=G*n) (km)
2400
3200
3600
4300
Log (L)
3.380
3.505
3.556
3.633
Solving for Fractal Dimensio​n
D = m − 1 (2)
From the graph, m =− 0, 2694
D = 1 − (− 0.2694)
D = 1.2694
D ≈ 1.27
4.3 Evaluating the Accuracy of
My Application of the Hausdorff
Meth​od
The value I obtained had a difference of 0.02 compared to Mandelbrots value of 1.25[2]. This is
only a 1.6% error. This difference is due to the nature of the method. The placement of rulers
along the coastline is approximated by eye. A person other than me would place the rulers in a
different location. Based off how rulers are placed, the number of rulers that can fit along the
coastline can change, resulting in a different fractal dimension. Thus, in my opinion, is an error
of 1.6% is insignificant given the method I utilised. In addition, the high R2 (0.96852) also
suggests that my placement of the rulers was precise and that the method was followed
accurately. However I will still trial the Box Method to see if it can bring me even closer to
Mandelbrot’s value.
5.1 The Box Counting Method
The second method I will use to get an estimate of the fractal dimension is the Box Counting
method, also known as the ​Minkowski–Bouligand method[10]. This method is similar to the last,
as it considers how much detail is added when the scale factor is changed. The procedure
involves superimposing grids of different calibre (size) over an image and then recording the
number of blocks the coastline passes through for each calibre. With a finer grid, more detail is
supposed to be picked up, and the method can be modeled based on the equation:
N = csD [10]
N = number of boxes
c = proportionality constant
s = scale f actor
D = f ractal dimension
Using logs this equation can be rearranged in to the form y = mx + c .
log N = log csD
log N = log c + logsD
log N = log c + D logs
log N = D logs + log c
y = log N
m=D
x = logs
c = log c
The fractal dimension D will thus be the gradient of this graph.
5.2 Attempting Box Plot Metho​d
The most effective method would be to perform the method using programing to create a really
fine grid whose boxes can be counted accurately by the computer. However I do not possess
the skills to do so, but still attempted the method using photoshop. I began by placing a grid
over the island of Britain and then highlighted all the boxes in green which touched the coast
line. For each successive plot, I shrunk each box down by a scale down factor of 2. Then I
counted the number of boxes by eye. In order to keep track of the amount of boxes, I would
double highlight a box every ten counts.
Table 2
Scaled
down 0
Factor (S)
2
4
8
log (S)
1.863
2.326
2.704
73
212
506
1.863
2.326
2.704
Number
Boxes (N)
Log (N)
1.477
of 30
1.477
As m = D , the fractal dimension is calculated as 1.376.
5.3 Evaluation of Box Counting Metho​d
Whilst 1.38 is within a 10% error of Mandelbrots value 1.25[2], it is not nearly as accurate as the
Hausdorff method. There are indeed reasons that can explain this error. The use of the
boxplotting method becomes more accurate as a finer grid is used. For me, the problem with
using a finer grid is that it increases the number of boxes that need to be accounted for, and
increases the possibility in accidentally miscounting the number of boxes. This problem can be
solved through computer programming. My selection of grid size was probably too small to
accurately approximate the fractal dimension, and I have no experience in programming to have
the skills that would allow me to generate data using a finer graph. The process of counting the
boxes is more tedious and inaccurate than the Hausdorff method, and thus it is the Hausdorff
method that I will utilize in order to find the fractal dimension of two Indonesian islands.
6.Determining the fractal dimensions of 2 Indonesian Islands
Using the Hausdorff method, the fractal dimension was calculated for two Indonesian islands.
6.1 Sulawesi
Table 3
Length of
Ruler (G) (k
150
100
50
25
log (G)
1.477
1.863
2.326
2.704
Number of
rulers (n)
26
40
85.5
181
Length
coastline (L)
(=G*n) (km)
3900
4000
4425
4525
Log (L)
3.591
3.602
3.646
3.656
Solving for Fractal Dimension
D = m − 1 (2)
From the graph, m =− 0.09
D = 1 − (− 0.09)
D = 1.09
6.2 West Nusa Tenggar​a
Table 4
Length of Ruler
(G) (km)
50
25
12.5
6.25
log (G)
1.699
1.398
1.097
0.796
Number of rulers
(n)
15
33
82
170
Length coastline
(L)
(=G*n) (km)
750
825
1025
1062.5
Log (L)
2.875
2.916
3.010
3.026
Solving for Fractal Dimensio​n
D = m − 1 (2)
From the graph, m =− 0.18121
D = 1 − (− 0.18121)
D = 1.18121
D ≈ 1.18
​7: Discussion and Conclusion
To conclude, I found that Sulawesi and West Nusa Tenggara have a fractal dimension of 1.09
and 1.18 respectively. My computation of the coastline of Britain was close to 1.6% accuracy. I
believe that I applied the method in the same way for Suluwesi and West Nusa Tenggara,
therefore, I will use this error in calculating the error of the islands fractal dimensions.
Table 5
Fractal Dimension
% Error
Absolute Error
Max
Value
Min
Value
Sulawesi
1.09
1.6
0.0174
1.1074
1.0726
West Nusa
Tenggara
1.18
1.6
0.01888
1.1612
1.1988
Qualitatively, looking at West Nusa Tenggara, it does appear to have a more complex coastline
which folds in on itself more than Sulawesi which thus explains the larger fractal dimension.
Both have fractal dimensions lower than that of Great Britain, implying that they are smoother
and less jagged than Britain's coastline.
The Hausdorff method is done by eye, and is therefore not infallible. This is the reason for the
error. An interesting way of expanding this IA would be to ask other people to calculate the
fractal dimension using the Hausdorff method, and compare these results. This also could be
expanded by using the Boxploting method, but done by utilizing a computer program rather
than by hand. Additionally, I only found the fractal dimension of 2 coastlines, and Indonesia has
14,000 potential islands that could be analysed.
I first met the concept of fractals in grade 6, when we learned about the Sierpinski triangles. I
found them mathematically elegant and pleasing to draw. To learn how the concept of fractals is
be applied mathematically to real life was fascinating for me. Also the ability to bring in logs and
regression equations really taught me how different areas of math can be woven together to
solve complicated problems.
Through my investigation, I also discovered various ways in which the fractal dimensions is
applied in different areas of science. The fractal dimension is an effective way of describing the
roughness/complexity of real world objects. Grant Sanderson, a software engineer from
Stanford said that a fractal dimension seems to be “the core differentiator between objects that
arise naturally, and those that are just man made”[7]. Prof T. Bhaskara Reddy’s paper (2017)
analysed how fractal dimensions can be used in plant morphology to “characterize the
structure/architecture of medicinal leaves”[11]. Wolfgang Bauer’s paper (2010) described how
the fractal dimension can be used in cancer cell analysis [12]. This academic exercise was a
great introduction for me into calculating and interpreting the fractal dimension, a skill that I
could possibly utilize later on in my academic pursuit.
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