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Fractal Geometry and Characteristics
Giorgio Scrocca - 1345887
March 8, 2016
1 Introduction and Motivation
The study of fractals has made a bold statement to the capabilities of Mathematics. Naturally
occurring phenomena can now be potentially modelled with mathematics. Complicated plant and
cell structures are found to not random, and order can be found in disorder. Although there are
some sceptics against its theory and real uses, potential uses have already begun to be observed and
studied. These include modelling the rhythm of a human’s heartbeat; population growth of trees
and forests; and even in digital art to create things like mountains and planets for animated movies.
This is discussed in Fractals - Hunting the Hidden Dimension (2012) [1]. Fractal geometry has been
discovered for a long time, but only thoroughly studied since Benoit Mandelbrot’s era when he had
access to a modern computer, and created the infamous “Mandelbrot Set”, which resembles the
body of a“beetle”.
What is a Fractal?
Benoit Mandelbrot coined the term fractal for such complex shapes and he constructed this definition
of what a fractal is, “a rough or fragmented geometric shape that can be split into parts, each of
which is (at least approximately) a reduced-size copy of the whole” [2]. Fractals contain most of, or
all of the following properties [3]:
1. The set has fine structure, it has details on arbitrary scales.
2. The set is too irregular to be described with classical euclidean geometry, both locally and
globally.
3. The set has some form of self-similarity, this could be approximate or statistical self-similarity.
4. The Hausdorff dimension of the set is strictly greater than its Topological dimension.
5. The set has a very simple definition, i.e. it can be defined recursively.
Statement four was in fact a previous definition of what a fractal is, that Mandelbrot also coined.
Although this statement holds for most fractals, it does not hold for all; such as the Hilbert Curve.
These terms and the dimensionality of a fractal will be expanded on in section 4.
2 Simple Example
To introduce the reader further into fractals we begin with a simple example. Figure 1 is the Cantor
Set discovered by Henry John Stephen Smith in 1874 and introduced by Georg Cantor in 1883. The
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basic intuition of the Cantor Set is to start with a straight line and split it into three equal parts.
Delete the middle part then place the two remaining parts one ’step’ below the original line. This
process is continued infinitely with each proceeding line as shown below. A more formal construction
is to begin with a line of length one on the unit interval [0, 1]. Then delete the middle part (1/3, 2/3)
and repeat this process with the two segments of length 1/3. Repeating this process infinitely leaves
no connected single line set, but what is the length of the Cantor Set?. Lets construct using this
process below, whilst evaluating the area we remove at each step as it leads us to an intriguing
result.
Figure 1: The Cantor Set - taken from http://stackoverflow.com/questions/6631333/how-to-
generate-a-plot-of-planar-cantor-set-in-mathematica
Iteration Size of segments Amount of segments removed
n=1 1/3 1
n=2 1/9 2
n=3 1/27 4
n=i (1/3)n
2n−1
By evaluating how much length is removed at each step, we can generalise the amount for
the iteration n = i and then calculate the summation of this. Using this information we can deduct
our answer from the total length of the original line (1unit) and see how long the cantor set really
is. We now calculate the sum of all of our segments multiplied by the area of each of these segments
below:
∞
n=1
2n−1
∗
1
3n
=
∞
n=1
2n
∗ 2−1
∗
1
3n
=
∞
n=1
1
2
∗
2
3
n
=
1/3
1 − 2/3
= 1
Notice on our last step we can evaluate the value of the summation by using the first term
in the summation over 1 minus the ratio. The reader can now observe that the total length removed
is 1, which is the size of the original line. So the cantor set has length zero. The reader may find this
concept tricky to understand at first. If the size of the set is zero, how can we characterise such a
fractal? We can instead characterise such sets by other means. Taken from Schroeder [4], he states
that Felix Hausdorff (1868-1942) constructed this formula for the dimension of such fractals:
DH := lim
r→0
log N
log 1/r
where N is the size of the fractal; r is the number of segments measured and DH is the dimension
of the fractal. So at each nth stage of the deletion process we are left with N = 2n
segments, each
of length r = (1
3 )n
so
DH =
log 2
log 3
= 0.63...
Here the proof of the Hausdorff dimension is omitted and dimensionality will be explained further
in section 4. Notice the amount of segments calculated when working out the dimension is 2n
rather
than 2n−1
as we are now counting segments we have at each step, rather than how many are lost.
As the reader can observe again in figure 1, “zooming” in on any part of the produced
shape results in a replication of the original shape, so this set can be classified as a self-similar
fractal which will be discussed further in section 3.
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2.1 How long is the Koch curve?
Another example of a self-similar fractal, but now in a 2-dimensional plane is the Koch curve, shown
below. This was first proposed by the Swedish Mathematician Helge von Koch in 1904.
Figure 2: The Koch Curve - taken from http://what-when-how.com/biomedical-image-analysis/self-
similarity-and-the-fractal-dimension-biomedical-image-analysis/
Following a basic principle, it produces a very complex fractal. A construction taken from
Schroader is to “take a segment of a straight line, and raise an equilateral triangle over its middle
third” [6] as shown in figure 2. Note that the length of this line is now 4/3 of the original, as our
so-called triangle has side length 1/3. This process is then repeated for each individual new line
segment and subsequently repeated infinitely. To begin examining the length of the Koch curve, we
will begin our construction of our Koch curve with a straight line of length one. Thus at the first
step of our construction, the length of our line will then become 1/3 ∗ 4 = 4/3 units long (as there
are now 4 segments of line of length 1/3). Then on our second step we have 42
= 16 line segments
of length 1/9, so our length is now 16 ∗ 1/9 = 1.778 to three decimal places. On our third step we
have 162
∗ 1/27 = (4/3)3
= 2.370....etc. Until our kth step we have (4/3)k
= 4k
∗ (1/3)k
. Notice
lim
k→∞
4
3
k
= ∞.
Essentially the length of the Koch curve is unmeasurable so we say it tends to infinity. This infor-
mation is useless in attempting to examine this curve further or distinguish it with other fractals.
Further to this, if one were to draw this curve on a page, the line would cover negligible area as it
is 1-dimensional. We hence also say it has zero area. In summary, it is too large to be described as
1-dimensional and too little to be described as 2-dimensional, and in fact, with proof omitted, the
Koch curve actually has a fractal dimension of around 1.262.
3 Self-Similarity
Both the Cantor Set and Koch Curve possess the characteristic of self-similarity. To provide the
reader with further intuition on this we first begin with the definition of a shape which is similar.
A shape is said to be similar to another shape if that shape can be represented as a duplicate of
the other by either scaling, rotating, sliding or ‘flipping’ the shape. It can also be called congruent
if this can be done without scaling. Falconer states that “a self similar set is one that is made up
of several smaller, similar copies of itself”, [5]. As the reader can see, this definition fits the above
example of the Cantor set and Koch Curve. However, these are two examples of exact self-similarity
when this is not always the case, as discussed below.
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3.1 Self-affine Fractals
Figure 3: A Barnsely Fern (Self-affine Fractal) - taken from https :
//en.wikipedia.org/wiki/Barnsleyf ern
An affine transformation is one such that it elongates the shape, by scaling at different
ratios or directions. For example, transforming a square into a rectangle, or circle into an ellipse.
Hence a self-affine fractal is one that is made up of smaller affine copies of itself. Thus by “zooming”
into certain sections of the fractal the reader will not see exact copies of the original shape, but
elongated ones, as shown in figure 3. Further to this area of fractals are more dissimilar fractals,
but still similar, are called statistically self-similar fractals.
3.2 Statistically self-similar fractals
If one were to construct the above example of the Koch curve, but instead of raising an equilateral
triangle through the line segment, one would flip a coin as the whether “raise” or “lower” the triangle.
And this element of chance would be followed with every step of construction, as shown in figure 4.
This is a random Koch curve.
Figure 4: The Random Koch Curve - taken from http://lsaravia.github.io/project/multifractals-in-
r/Curso1.html
As one can see it is not self-similar in the strict sense, but statistically self-similar. This
concept is more applicable with naturally occurring fractals, such as common trees or the fern shown
in figure 3. This provides further motivation for these studies in the sense that perhaps the growth
of tree structures could be predicted in advance. But these observations are in a very early stage of
study.
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4 Fractal Dimension
We have seen that the Cantor set essentially has length zero, and that the perimeter of the Koch
curve is infinite. As such, fractals can not be defined with traditional mathematics such as their
size. For example if two fractals both have infinite length, but one fractal is four times the size of
the other, it is useless to say that one fractal is measured at four multiplied by infinity and the other
one multiplied by infinity. So how can we distinguish or even compare such sets? One way fractals
can be distinguished by their dimension, which will be explained in more detail below.
4.1 Topological Dimension
We first begin by discussing properties of shapes assumed already known to the reader. The class
of dimension which is the most familiar is the Topological dimension. Imagine a straight line, which
obviously has dimension 1, and length n. If one were to increase the size of this line by scale k, the
new size of the line will be:
N = nk.
Now if this was a 2-dimensional shape, say a square, if we increased the size of this square by scale
k again, the new size would now be:
N = nk2
.
And likewise with a 3-dimensional shape, say a cube, increasing the size by scale k would be:
N = nk3
.
As we can see this follows the pattern:
N = nkd
.
Here d denotes the dimension of the shape. Notice this can be re written as
d =
log N
n
log(k)
Using this procedure, we can now evaluate the dimension of the so-called Sierpinski Triangle.
4.2 Sierpinski Triangle and it’s Dimension
Shown below is a well known fractal shape called the “Sierpinski triangle”. Made from infinitely
drawing four equally sized, smaller triangles inside the original triangle, removing the central triangle
and then repeating this process with each triangle drawn.
Figure 5: Sierpinski Triangle - taken from https://www.zeuscat.com/andrew/chaos/sierpinski.html
Starting from the original and largest triangle; when 3 smaller triangles are created, they
are at scale of 1/2 of the first triangle. This can be seen as the length of one of the sides of the
triangle is half the length of the larger triangle. Using,
N = nkd
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we can create a formula for the size of the larger triangle:
N = 3 ∗ (N ∗ (1/2)d
)
Where N is the size of the larger triangle and d is the dimension of the triangle. This basically
states that the size of the larger triangle is 3 times one of the smaller triangles (as it has an empty
space in the middle) and the size of one of the smaller triangles is scaled 1/2 of the larger to it’s
corresponding dimension. This implies:
1 = 3 ∗ (1/2d
)
3 = 2d
d =
log N
n
log(k)
= (log(3))/(log(2)) 1.584
Stated by Mellor, B, “This number d is called the similarity dimension of the figure.” [7]. Unfor-
tunately this simple method only works with fractals which pose exact self-similarity, but it is an
elementary way for the reader to be introduced into the dimensionality of fractals. Another way of
evaluating a fractals dimension is via the box counting method, explained in the following section.
4.3 Box-Counting Dimension
Many ways of evaluating the dimension of a fractal involve measuring the shape at arbitrary scales
and seeing how this measurement behaves at finer details. A simple and more general method of
evaluating the dimension of a fractal is the box-counting method.
Figure 6: Box Count Diagrams for Illustrative Purposes - taken from Falconer, K “Fractals A Very
Short Introduction” (2013)
Shown in figure 6 is this method applied to a line segment of length 1, along with a table.
The process involves splitting our shape into equal size segments or boxes and counting how many
intersect our chosen set.
Line segment of length 1:
Side length of boxes 1/2 1/4 1/8 1/16 1/32 r
Number of boxes 2 = 21
4 = 41
8 = 81
16 = 161
32 = 321
1/r = (1/r)1
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As the reader can see, the length of the line segment does not change when measured at
finer details as all we are doing is counting the amount of ‘boxes’ that intersect our set. Similarly
done for a square with side lengths of 1.
Square of side 1:
Side length of boxes 1/2 1/4 1/8 1/16 1/32 r
Number of boxes 4 = 22
16 = 42
64 = 82
256 = 162
1024 = 322
1/r = (1/r)2
The important piece of information to take away from this is that as smaller box sizes are
chosen the number of boxes that intersect our set tends to (1/r)1
for our line segment; and (1/r)2
for our square segments. Thus our dimensions are 1 and 2 respectively. Now we will apply this
method to the Sierpinski triangle shown in figure 6.
Sierpinski Triangle:
Side length of boxes 1/2 1/4 1/8 1/16 r
Number of boxes 3 = 21.585
9 = 41.585
27 = 81.585
81 = 161.585
≈ (1/r)1.585
N(r) =
1
r
d
log N(r) = log
1
r
d
log N(r) = d ∗ log
1
r
This leads us on to the formal definition of the box counting dimension:
DB := lim
r→0
log N(r)
log 1/r
,
[8] where N(r) is the number of boxes used in our count and 1/r is the side length of one of the
boxes.
Practically this method can pose some problems. If our boxes are too large, they may not
be enough intersections of our fractal, and we are unable to evaluate. Or using too little boxes would
pose problems on fractals like the barnsely fern or other naturally occurring fractals. If we examine
the barnsely fern deep enough, instead of replications of the outer leafs, we notice small stubby teeth
which will skew results. The range of scales suitable for evaluation is called the ”range of fractality”
[9]. One way round this would be to take measurements at multiple box sizes, plotting on a graph
log 1
r on the x axis and logN(r) on the y axis to formulate a gradient to this line. This will result
in an average for the intersects and form an approximate dimension. This becomes applicable when
measuring the coastline of Britain for example.
Further to the two methods described of evaluating a fractals dimension, there are many
more complex models. We now only know how to evaluate self-similar fractals with our first method,
or fractals on a grid with our box counting method, but what about non self-similar 3-dimensional
fractals? Unfortunately further methods are omitted due to complexity.
5 Famous Sets of Fractals
More complex and artistic fractals are the infamous Julia and Mandelbrot sets. The Julia set, which
comes from the formula devised by Gaston Julia; and the Mandelbrot set, formula devised from
Benoit Mandelbrot; are often pictured throughout the study of fractals for their complex nature
and artistic beauty. The sets are so complex, their properties are still not fully understood, yet
they are both defined so simply. They follow the same characteristics we have discussed including
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self-similarity and non-integer dimension. It is easy to think of them as iterated functions across the
complex plane, but they will be discussed in more detail now.
5.1 The Julia Set
We begin by examining the iteration function given as:
z → z2
+ c,
where z is an arbitrary number on the complex plane, and c is a pre defined complex number. The
way to think of this function is to first define our c value, and then evaluate every possible z value
and observe how this number converges or diverges. The simplest form of this is if we take c = 0.
Therefore, this essentially continually squares an arbitrary complex number:
z → z2
.
We now evaluate this iteration function by inserting every number on the complex plane. As the
reader will already know, the magnitude of the complex number is squared and the argument is
doubled, when a complex number is multiplied by itself. So for instance if we take 3 + i we get:
3 + i → 8 + 6i → 28 + 96i → −8432 + 5376i → ...
As the reader will observe, this number diverges. However, if we take say 0.3 + 0.1i, and substitute
this into our iteration function we get:
0.3 + 0.1i → 0.08 + 0.06i → (2.8 ∗ 10−3
) + (9.6 ∗ 10−3
)i → ...
As the reader can see, this quickly converges to zero. Obviously, if we square a complex number
greater than one, our chosen number diverges, and if it is less than one, this converges to zero. To
represent our Julia set we now plot every arbitrary number on the complex plane and shade it in
based on how it converges. Blue being a z value which diverges and black being a z value which
converges. Using this representation method we are left with a circle of radius 1 and centre at the
origin.
Figure 7: Julia Set: c = 0 and c = 0.1 + 0.2i respectively - Created on Fractal eXtreme
Now say we take c = 0.1 + 0.2i, our c value slightly shifts our iteration function and we are
left with the boundary shown in figure 7. This leaves us to form a formal definition of a Julia set.
Taken from Falconer “The set of initial complex numbers, thought of as points in the plane, from
which the itinerary does not wander off to infinity is called the filled-in Julia set and the boundary of
the filled-in Julia set is called the Julia set of the function” [10]. But the two examples just discussed
are relatively simple examples. Here are more examples of Julia sets and their corresponding c value.
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Figure 8: Multiple Ex-
amples of Julia Sets -
taken from Falconer, K
“Fractals A Very Short
Introduction” (2013)
One can observe now how complex the Julia sets can become and in
fact hold properties earlier discussed. They are self-similar on all scales
and hold infinite beauty. As the reader has now observed, this set is very
sensitive to its initial condition. TALK ABOUT FIXED POINTS? Points
are shaded dependent on how fast that point iterates past a certain chosen
point past the origin, often chosen at 5 . The greater the c value, the
greater the change in the Julia set, until it ceases to become connected.
A set is connected if a path can join any two points in the set. As the
reader can see from figure 8, all Julia sets except for (g) are connected
sets. But for which values of c provide us with a “disconnected” Julia
set? This leads us on the infamous Mandelbrot set, which answers such
questions and is argue-ably even more artistic and complex.
5.2 The Mandelbrot Set
As an alternative to the method just described, which produces the Julia
set, we could instead examine the function z → z2
+c with instead a fixed
z and examine this with a varying c value. We will examine arbitrary
c values for the fixed z value 0 + 0i. If our chosen itinerary does not
diverge we will colour the point c = a + bi black, and if it does diverge
will we leave this coloured white. One can create a rough image of the
Mandelbrot by deeming a value to be coloured black if it does not diverge
within the first 100 itineraries. Alas, the Mandelbrot set is now shown
in figure 9. The reader is also urged to explore this set at their own
will. This can be accessed through cinema-tics on You-tube by searching ‘Mandelbrot Zoom’, or
on various other means of software. Figure 10 also further explores this complex set with different
colours created on “fractal eXtreme”. By zooming in, one can see in certain areas replication of the
original ‘beetle-like’ shape, and at very high magnifications one can also see replications of certain
Julia sets. There are an array of windmill and sea-horse looking shapes throughout, yet it is defined
so recursively. The variety of shapes is unending.
Figure 9: Mandelbrot Set - taken from http://i.stack.imgur.com/kZK91.png
Figure 10: Mandelbrot Set Magnified Four Times - created on Fractal eXtreme
This leads us on to what is sometimes called the ‘Fundamental Theorem of the Mandelbrot
Set’, taken from Falconer (2013, p.77). This helps us further understand the Julia set.
‘The point c is in the Mandelbrot set precisely when the Julia set of the function z → z2
+c
is connected.’[11]
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This was proved independently by Gaston Julia and Pierre Fatou in 1919. As mentioned earlier, a
set is connected if a path can be made between any two points in that set. If an arbitrary c value
lies within the Mandelbrot set then the Julia set of this c value is connected. And if the c value is
outside the Mandelbrot set, then its Julia set is totally disconnected. In fact a Julia set is either
connected or totally disconnected, we never have several disconnected counterparts. This is why the
Mandelbrot set helps us further understand the Julia set.
Further exploration of the Mandelbrot set
One thing to point out is that the Mandelbrot set is not regarded as a fractal, as it’s area is positive.
Unlike the Cantor Set or Koch Curve which have indistinguishable lengths and areas respectively,
the set has countable area and can not be regarded as a fractal. Even though fine structure is found
on arbitrary scales, most of the Mandelbrot set appears to just be ‘black area’. It is the boundary
of the Mandelbrot set which is regarded as a fractal. It was in 1998 when a Japanese researcher
Mitsuhiro Shishikura showed that although the boundary has area 0, it has a dimension of 2, which
is very expressive of it’s complexity. The proof of this has been omitted due to it’s length.
6 Real World Fractals
Up to now, most of the fractals discussed are only valid in a theoretical world of mathematics, but
of course real life fractals can be much different. It is only in theory can one construct a fractal
infinitely, and view it at arbitrary scales. None the less, techniques and characteristics discussed
still help us further understand naturally occurring fractal phenomenons. Not all scientific models
are perfectly accurate but still help us understand the world. Newtonian mechanics for instance, a
lot of other variables are involved in mechanics which are not accounted for. Despite this, we can
still make accurate models of real life things.
Bodily structures and the Lungs
Many fractal branch networks are found in our body. These are included in our breathing, blood
circulation and nervous system. In particular fractal networks are found in our lungs. Our ‘windpipe’
splits into two when it meets the lungs, which is split into further and further branches within our
lungs. In fact adult lungs are only around 12 inches long and 5 inches wide, however, due to the
fractal nature of our breathing network, our windpipes have a surface area of around 100 yards.
This fractal nature is what achieves such a high area in small volume and makes our lungs and other
bodily networks a lot more efficient.
Rhythm of the Heartbeat
As one may know, our heart does not beat like a ‘metronome’. This can be seen more clearly if
one were to record the human heart rate and present the results via a line graph. There are lots of
variations in a humans heartbeat and lots of tiny ‘bumps’. Observing the nature of the heart beat
shows characteristics of self-similarity and also fractal dimension. In fact a healthy human heart
rate represented on a line graph has a dimension of around 1.5. Fractal analysis is used, alongside
other methods to diagnose problems with the heart or blood circulation. Extensive research is also
being done to use fractal analysis of the heart to diagnose certain cancers or tumours a lot more
early on.
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Efficient Electronic Systems
In order for antennas to receive waves of information they must be at a length corresponding to
that wave, typically at 1
2 or 1
4 of that wave length. This proposes the problem of spaces, fitting
large antennas on top of houses or fitting micro sized ones inside of portable devices. Making these
antennas fractal shaped significantly reduces their size, similar to structures within our body. In
short, they can help make technology, including popular phones, a lot smaller.
7 Conclusion
To summarise, although vigorous studying of fractals only began in Benoit Mandelbrot’s era, it is a
vast and ever complicating subject in which the reader in recommended to study further as this is
just an introduction to the subject. It is a subject which is not short of motivation and is definitely
not short of artistic beauty. One may think, if so many natural things can be modelled by fractals,
what limits does this have on anything being modelled by fractals? Or perhaps our planet is merely
an atom which is part of a small rock on an even larger planet in a universe, and the atoms we are
made up of are much more smaller planets with life forms so microscopic we will never see. Although
this is now just hear-say, it is interesting to ponder on.
References
[1] Nova, “Fractals - Hunting the Hidden Dimension”, Youtube, (2012),
<https://www.youtube.com/watch?v=s65DSz78jW4>.
[2] Mandelbrot, B, “Fractals: Form, Chance and Dimension”, W.H.Freeman Co Ltd, (1975).
[3] Falconer, K, “Fractal Geometry: Mathematical Foundations and Applications”, John Wiley
—& Sons, (1990).
[4] Schroeder, M, “Fractals, Chaos, Power Laws”, Dover Publications, INC, (1991), p10.
[5] Falconer, K, “A Very Short Introduction to Fractals”, Oxford University Press, (2013), p18.
[6] Schroeder, M, “Fractals, Chaos, Power Laws”, Dover Publications, INC, (1991), p7.
[7] Black, M, “Fractals: Self-Similarity and Fractal Dimension”, Loy-
ola Marymount University, (2013), Maths 198, p8 < http :
//myweb.lmu.edu/bmellor/courses/Symmetry/FractalDimension − Spring2013.pdf >.
[8] Grassberger, P, “Phys. Lett. A, 97”, Department of Physics, University of Wuppertal, Ger-
many, (1983), p224.
[9] Falconer, K, “A Very Short Introduction to Fractals”, Oxford University Press, (2013), p49.
[10] Falconer, K, “A Very Short Introduction to Fractals”, Oxford University Press, (2013), p70.
[11] Falconer, K, “A Very Short Introduction to Fractals”, Oxford University Press, (2013), p77.
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