MATH2916 A Working Seminar
University of Sydney School of Mathematics and Statistics
Exploring Fractal Dimension
Dibyendu Roy
SID:480344308
June 2, 2019 MATH2916 Exploring Fractal Dimension Dibyendu Roy
1 Motivations
We all have an intuitive understanding of roughness, and can easily tell that a jagged rock is rougher than a marble. We can similarly understand that the Koch curve is "rougher" than a straight line. To make this definition of "roughness" or "fractal-ness" more rigorous, we introduce the idea of fractal dimension.
What would such a dimension be defined as? It makes sense to attempt to define such a "dimen- sion" to fill the gaps in the standard integers that we use to define dimension conventionally. We know that an object in one dimension is a line and an example of a two dimensional object is a square. But what would an example of a 1.26 dimensional object look like? In this text we will explore the nature of fractal dimension culminating in a rigorous definition that can be applied to any object.
2 Compass dimension
2.1 The coastline paradox
Let us imagine that we are are a cartographer trying to find the length of the coastline of Britain. As the coastline is not a regular mathematically defined shape it would make sense for us to try and find a series of approximations that would converge to the true value after a large number of iterations.
Let us consider the following method for evaluating the perimeter of a shape. We take a compass (or more appropriately a divider) and open it up to a fixed distance s. We then walk the compass along the perimeter of the shape and count the number of steps n we take to get around the coastline. If we multiply the number of steps we took with the length of the compass we get and approximation of the length of the perimeter N(s) = ns. Intuitively it is obvious that as s → 0 we should have N(s) converging to the perimeter of the shape in question.
Fig. 1: Sequence of compass steps for approximating the perimeter of the circle
Before we try applying this to the coastline of Britain let us see if this method works, taking a circle with radius 1000km. A sequence of tracings of the compass is shown in Fig. 1. Note as the compass size decreases the traced shape in red converges to the circle. As in Fig. 3a we observe as s decreases N(s) converges to approximately 1000π km ≈ 3141km as expected. We also see that the corresponding paths traced out be stepping round the circle are regular polygons that converge to the shape of the circle.
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Fig. 2: Sequence of compass steps for approximating the perimeter Britain
Now let us apply it to the coastline of Britain. As we see in Fig. 2 we are tracing around the coast with smaller and smaller compass lengths. The results are shown in Fig 3b. We note that the perimeter doesn’t seem to converge to anything! It seems like the smaller we make s, N(s) will keep increasing without bound.
Steps n Size s (km) N(s) (km) Size s (km) N(s) (km) 12 258.82 3106 500.00 2600 24 130.53 3133 100 3800 48 65.40 3139 54 5770 96 32.72 3141 17 8640 (a) Circle (b) Britain
Fig. 3: Results from the compass dimension
It is clear that the circle is approaching the expected value while the coastline shows no signs of converging. We suspect that there is some kind of power law between the length coastline and the precision of the measurements. The precision of our compass steps can simply be expressed 1 as s as larger precision corresponds to smaller steps. Creating a log-log plot as shown in Fig. 4 gives us a linear relationship between the log(N(s)) and log(1/s).
Therefore we can fit a line to these data points and find their gradient, call it d. We see that the gradient of the line is 0 for the circle while for Britain it is 0.36. We can now hypothesise that we can use d to get a measure for roughness or "fractal-ness" of a shape.
To wrap up the coastline paradox we see that the coastline of Britain really has in a sense infinite length, the higher precision our measurements have the larger the coastline seems to get without bound. In reality however it is clear to see that in measuring the coastline we will run into physical limitations, not of the least considering that distance is largely undefined under the plank length.
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Britain d ≈ 0.26 Circle d ≈ 0.00 ) 3.8 Perimeter
( 3.6 10 log 3.4 −2.8−2.7−2.6−2.5−2.4−2.3−2.2−2.1 −2 −1.9−1.8−1.7−1.6−1.5−1.4−1.3−1.2−1.1
− log10(Compass length)
Fig. 4: Log-Log plot of the Perimeter vs Compass length for Britain and the Circle
2.2 The Koch curve
Now let us evaluate d for the Koch curve. Since the Koch curve is a well defined object we can use more convenient compass steps that reflect its construction. We take a Koch curve of length 27 for easy computation and trace around it with a compass. Figure 5a shows the corresponding diagrams to each step in the iteration. The results are shown in Fig. 5b.
Size s Lines n N(s) 27 1 27 9 4 36 3 16 48 1 64 64 (a) Tracing the Koch curve with a compass (b) Table of results
Fig. 5: Compass dimension of the Koch curve
The power law is even more clear here as we can see the increasing powers of 4 and and the decreasing powers of 3 as we go down the column N(s). Also note that this sequence is clearly 4 escaping to infinity as we multiply by 3 for each iteration. If we repeat the steps that we did above with creating the Log-Log plot and finding the gradient as in Fig. 6 we get a gradient of d ≈ 0.26. This aligns with our preconceptions that the Koch curve is more fractal like than a straight line with d ≈ 0
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Koch curve d ≈ 0.26
3.8
) 3.6
3.4 Perimeter ( 10 3.2 log
3 −3.2 −3 −2.8−2.6−2.4−2.2 −2 −1.8−1.6−1.4−1.2 −1 −0.8−0.6−0.4−0.2 0 0.2
− log10(Compass length)
Fig. 6: Log-Log plot of the Perimeter vs Compass length for the Koch curve
2.3 Discussion
We have discovered a parameter d as the compass dimension that seems to encode the roughness of an object. However there are a few issues, firstly it doesn’t agree with any of our current notions of dimension as it gives a circle dimension zero. Secondly we cannot apply this method to shapes that are not made up of lines, so finding the dimension of a ball would be impossible. Let us try investigating the properties of conventional shapes to define dimension in a more general way.
3 Self similarity dimension
3.1 Scaling Factors
In order to define a dimension to align with our intuition we need to find a link between dimension and the properties of common objects. We know that when we scale a line by a 1 factor of 2 we will need 2 of those lines to create the original unit line. Similarly scaling 1 down a cube by factor 2 means that we will need 4 of them to create the original cube, a graphical representation is shown in Fig. 7a. Therefore we can create a simple table showing the relationship between scaling factor s and the number of pieces needed to make the original 1 object again a. We can generalise this to a scaling factor of n to make the power relation even more clear as shown in Fig. 7b. Using this relationship we can now create the equation linking the s and a in relation to the dimension of the object Ds such that: 1 log a a = or Ds = (1) SDs log 1/S
The equation on the right therefore provides a definition of dimension with respect to the scaling factor, s, and the number of pieces we split an object into, a. Let us now see what happens when we apply this dimension to fractals.
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Object Scale s Pieces a
1/2 2 Line 1/n n 1/2 4 Square 1/n2 n2 1/2 8 Cube 1/n2 n3
(a) Graphical representation (b) Table of the scaling factors
Fig. 7: Scaling factors of familiar objects
3.2 The Koch curve (again)
Using the definition of dimension we just created let us try applying this to the Koch curve and see what results we get. As in Fig. 8 we see that the Koch curve is made of 4 smaller 1 pieces each a size 3 of the original.
Fig. 8: Self similarity of the Koch curve
Plugging in this number to the equation that we just found before we see that: Ds(Koch) = log 4 log 3 ≈ 1.26
This number looks quite familiar! In the last section we found that d(Koch) = 0.26. In fact it is possible to show that: Ds = 1 + d
3.3 Discussion
This self similarity dimension is a marked improvement on the compass dimension as it is based off the scaling factors of conventional shapes, thereby agreeing with the conventional definitions of dimension. Also we can apply this definition to shapes in any dimension such as cubes which compass dimension cannot do. Also it is really easy to compute since self similar fractals are easy to break up to find the scaling factors. However it is impossible to find the
6 of 14 MATH2916 Exploring Fractal Dimension Dibyendu Roy dimension of a shape that is not self similar such as the coastline of Britain, even through it is fractal like. In order to cover all the bases we need to extend the definition to be more general, combining the compass dimension and the self similarity dimension together.
4 Box counting dimension
4.1 Introduction
What happens when we want to find the fractal dimension of an irregular object that is not mathematically well defined of periodic? It is intuitive to come to the conclusion that we need a definition that is computationally based, and that is where box counting dimension comes in.
Box counting dimensions is used in a wide variety of applications as it represents the easiest method to evaluate the fractal dimension of an arbitrary object. It represents an extension of the compass dimension and can be easily applied to objects with arbitrary topological dimen- sion.
4.2 The definition
Say we want to find the box counting dimension of some object, k with topological dimension N . First we place k on to a grid with size s and count the number of "squares" that go through k, call it N(s). After taking samples of N(s) at various s, we then construct a Log-Log plot of log(N(s)) vs log(1/s) and find the gradient of the line of best fit Db, (much like the compass dimension) giving us the dimension of the object. Lets see what happens when we apply this process to a few examples.
4.3 Circle and the Koch curve
To gain a stronger understanding of how this definition of dimension behaves, let us look at a "normal" shape, the circle. In Fig. 9 we see the progression of the approximations of the circles with smaller grid sizes. Observe that as s decreases out image looks more and more like a circle.
Fig. 9: Box counting dimension of a circle
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Now lets see what happens to the Koch curve when we apply the same process. So placing the curve onto sucessively smaller grids is shown in Fig. 10
Fig. 10: Box counting dimension of the Koch curve
Now we can create a plot of log(N(s)) (N(s) is the number of red boxes) vs log(1/s) (s is the size of the grid). This graph is shown in Fig. 11. Note that the gradient for the circle is 2 which agrees with the conventional dimension. The Koch curve has gradient 1.26, which agrees with the self similarity dimension.
Koch curve Db ≈ 1.26 Circle Db ≈ 2.00
10
8 )) s ( N
( 6 10
log 4
2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4
− log10(s)
Fig. 11: Log-Log plot of the Precision vs Number of squares for the Koch curve and a Circle
4.4 Discussion
The box counting dimension finally gives us a definition of dimension that is easy to work with and easily generalisable to all kinds of shapes. This is what makes it he most used definition of dimension in the real world as it is relatively simple to computationally evaluate and can be done quickly and reliably. However useful the box counting dimension might be, it lacks the rigour that seems necessary when defining something as fundamental as the definition of dimension.
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5.1 Introduction
The Hausdorff measure can be viewed as an extension of the box counting dimension and allows us to calculate the dimension of arbitrary shapes in any dimension and metric space. In this section we will explore the Hausdorff dimension. This is split into two parts. First we look at the Hausdorff measure, which represents the concept of generalised "length", "area" and "volume" in any metric space, and then we will look at the definition of the dimension its self. However before we begin we must introduce some mathematical concepts in order to make these definitions make sense.
5.2 Metric space
A metric space is the ordered pair of a set M with some defined metric or distance function d : M × M → R. Let us call the distance between x ∈ M and y ∈ M by the function d(x, y). It must satisfy the following criteria:
1. Identity of indiscernibles d(x, y) = 0 ⇔ x = y
2. Symmetry d(x, y) = d(y, x)
3. Triangle inequality d(x, z) ≤ d(x, y) + d(y, z)
For example in R2 we can define: q 2 2 d((x1, y1), (x2, y2)) = (x1 − x2) + (y1 − y2)
The Hausdorff dimension makes use of this distance funcion and an easy way to develop an intuitive understanding is to try and visualise what is happening in R2.
5.3 The infimum
The infimum of a set is defined as the largest lower bound of that set. So it is the largest number that is smaller than or equal to all the other numbers in the set. As an example for the set {1, 2, 3, 4}: inf{1, 2, 3, 4} = 1 Clearly 1 is less than or equal to all the numbers in the set. A more complicated example is shown below: 1 1 1 1 inf : n ∈ = inf , , ,... = 0 n N 1 2 3
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We see that 0 is the greatest lower bound even though 0 is not in the set its self (its a limiting point of the set).
5.4 Balls and the open cover
A ball at point x and radius r in some metric space M is denoted by:
B(x, r) = {m ∈ M : d(m, x) < r} (2)
Effectively this is just the generalised circle in any metric space. So in R it is a line, in R2 it is a circle and in R3 it is a sphere etc.
The open cover of a set S is the the union of a series of open sets Ui such that:
∞ [ S ⊂ Ui (3) i=1 So the open cover of a the set of points in a rectangle (gray) by a set of balls (red) would is represented in Fig. 12
Fig. 12: Open covering of the grey rectangle using balls
5.5 The Hausdorff Measure
We will represent the Hausdorff measure of the set S in dimension d as Cd(S). Before we go d any further let us define Cδ as:
( ∞ ) d X d [ Cδ (S) := inf ri : S ⊂ B(xi, ri) and (4) i i=1 Let us break this statement down to better understand what is going on. Note that the S∞ expression S ⊂ i=1 B(xi, ri) is just saying that the sequence of balls B(xi, ri) form an open cover of S. We also have the added restriction that ri < δ i.e. all the balls have in the open cover have radius less than δ. So expressing equation 4 in English we can say:
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• The infimum of the set... inf{...}
X d • Formed by the sum of the radii of balls to the power d where ... ri i
∞ [ • The balls form an open cover of the set S such that... ⊂ B(xi, ri) i=1
• All the balls have radius less than δ. ri < δ
Finally we can define the measure Cd(S) its self as:
( ∞ ) d d X d [ C (S) = lim Cδ (S) = lim inf ri : S ⊂ B(xi, ri) and ri < δ (5) δ→0 δ→0 i i=1 All we are doing now is taking the limit as the radii of the balls go to zero. It is quite clear that there are many similarities between this definition of measure and other concepts such as the Lebesgue measure and Reinmann sums as they both involve covering objects using infinitely small pieces. Figure 13 shows the open cover of the coastline of Britain using smaller and smaller circles
Fig. 13: Open covering of the coastline of Britain with successively smaller balls
Now that we know what the Hausdorff measure is we can move on to the definition of the Hausdorff dimension.
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5.6 Hausdorff dimension
The most fascinating fact about the definition of the Hausdorff measure is that we can prove: ∞ for d < D (S) Cd(A) = H 0 for d > DH (S)
This means that Cd(A) will either be 0 or ∞ unless d is some specific value. So we can define the Hausdorff dimension (DH (S)) to be:
n d o DH (S) = inf d|C (S) = 0
All this equation says that we are defining the dimension d to be the number such that Cd(A) is just on the border between 0 and ∞. As we have learnt the definition let us try and evaluate the Hausdorff dimension for the Koch curve.
5.7 The Koch curve (again)2
To better understand how the Hausdorff dimension works I will present a (slightly less then rigorous) method to find the dimension of the Koch curve. We will consider a Koch curve with a tip to tip length of 1. Figure 14 shows a sequence of open coverings of the Koch curve. We start off with a single circle of diameter 1.
Fig. 14: Open coverings of the Koch curve
1 In the next step we cover 4 smaller copies of the Koch curve with circles of diameter 3 . Fig. 15 shows the relationship between the radii and the number of balls required for the coverings (note that this reminiscent to the self similar properties of the the Koch curve). it is possible to show that this sequence of coverings indeed the most efficient but the proof is outside the scope of this paper.
Now we can find the Hausdorff measure of the Koch curve using the definition in equation 5:
Therefore we have now found the Hausdorff measure of the Koch curve with respect to dimen- sion d as:
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Iteration Radius r Number of balls N 1 1 1 2 1/3 4 3 1/9 16 ...... k 1/3k 4k
Fig. 15: Table of the open coverings of the Koch curve
Cd(Koch) The Hausdorff measure ( ∞ ) X d [ Expanding using the definition of = lim inf ri : S ⊂ B(xi, ri) and ri < δ δ→0 i i=1 the Hausdorff measure ( 1 d 1 ) Using the set of open covers that = lim inf 4k × : < δ δ→0 3k 3k we found before ( 4 k 1 ) = lim inf : < δ Simplifying δ→0 3d 3k ( 4 k ) Re-stating the condition for the = lim inf : k > − log3(δ) δ→0 3d radius to solve for k Cases
4 • If 3d ≥ 1 the infimum of the set will be at the minimum 4 d− log3(δ)e 4 k = lim or lim value of k δ→0 3d k→∞ 3d 4 • If 3d ≤ 1 then we take the largest value of k
4 K Make the substitution shown and = lim (K = d− log3(δ)e or k) K→∞ 3d the corresponding change in limit
4 K Cd(Koch) = lim (6) K→∞ 3d
As this is the limit of a simple power series observe that:
4 K ∞ for 4 > 1 Cd(Koch) = lim = 3d (7) K→∞ 3d 4 0 for 3d < 1
d Therefore in order to find the Hausdorff dimension (DH ) we must find the point where C (Koch) 6= 0, ∞. It is clear to see that this occurs when: 4 = 1 3DH (Koch)
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Re-arranging give us: log(4) D (Koch) = (8) H log(3)
Which agrees with all the previous definitions of dimension!
5.8 Discussion
The Hausdorff definition of dimension is a rigorous definition of dimension that we can apply to any metric space and it represents gives us the answer to what fractal dimension can be defined as. However the actual calculations using this definition are tedious at best and generally impssible. As such it serves to be more of a theoretical tool than a method for computation. It cab be though to be much like the relationship between the Reinmann sum to integration, it is a theoretical tool but people generally don’t solve integrals using Reinmann sums. In addition there do exist cases where the Hausdorff dimension disagrees with the box counting dimension. An example of this is the set Q ∩ [0, 1] it has Hausdorff dimension of 0 but box counting dimension 1. Regardless it is an important tool in understanding the nature of dimension and how we can extend it to arbitrary objects in arbitrary metric spaces.
6 Conclusion
We have seen four different definitions of fractal dimension. Compass dimension gives us a way to look at the roughness of lines and solves the coastline paradox. The self similarity dimension uses the scaling of objects to define dimension. Box counting dimension provides us with a consistent and computationally simple method of finding the dimension of any object. Finally the Hausdorff dimension extends box counting and provides a rigorous understanding of the nature of dimension. Now we have a much deeper understanding of the concept of dimension. In fact fractal dimension is deeply related to the natural world (such as the coastline of Britain) and gives us insight into the world of strange shapes.
7 References and software
• Peitgen, H., Jurgens, H. and Saupe, D. (n.d.). Chaos and fractals. Springer, p.Chapter 4.
• Mathematica 12
• Geogebra 5
• mailto:[email protected], B. (2019). Make-a-map : a geological map of Britain and Ire- land | Geology of Britain | British Geological Survey (BGS). [online] Bgs.ac.uk. Available at: http://www.bgs.ac.uk/discoveringGeology/geologyOfBritain/makeamap/map.html [Ac- cessed 2 Jun. 2019].
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