MichelFest: An Introduction to Fractal Geometry

So I’m currently at The Summer School on Fractal Geometry and Complex Dimensions at Cal Poly San Luis Obispo. The event is a conference/summer school held in honour of the 60th birthday of Michel Lapidus of U.C., Riverside, a French Mathematician, my mentor, editor of The Journal of Fractal Geometry, and one of the world’s leading fractal geometers. Ergo, the event has been nicknamed “MichelFest”

In honour of Professor Lapidus, I will spend the next few posts discussing the basics of Fractal Geometry, some historical developments, and Lapidus’ many contributions to the field. We will begin with an (quasi) informal introduction, followed by the rigorous mathematics of fractal geometry. If you already know the basic premises of a fractal, you may freely skip to III. Box Dimension.



I. Introduction

So what is a “fractal” anyways? The short answer is: “you usually know one when you see one”. The long answer is, well, we’ve no formal definition for the object, but there is certainly some denumerable criterion that generally identify a fractal. Here’s a few such properties for a fractal (c.f. Fractal Geometry, Falconer), F :

  1. F is self-similar. This simply means, if you pick any piece of the fractal and zoom in by some scale, the original image pops back up. (An example of this will be given in a second.)
  2. Although F has an intricate structure, it can be easily described mathematically.
  3. F is created through some iterative process.
  4. Classical geometry just won’t cut it to describe F , say, as some locus.
  5. The local geometry of F is messy or impossible to describe. There’s some sort of weird lack of smoothness locally.
  6. F is not easily measured by classical measures, or the answer to its measure is unintuitive and clumsy.

Let’s look visually at some two and three dimensional fractals, and then discuss the mathematics behind some, and how to describe fractal geometry.

We begin with perhaps the most famous fractal, the Mandelbrot Set, which is also one of the most intricate, yet remarkably simple to describe analytically.


Many of you may have seen this image before, but the self-similarity is not immediately obvious. To understand what is meant by this self-similarity, take a look at the following link: of a Mandelbrot Zoom, in which the animation zooms in on a part of the fractal, eventually you begin to see copies of the original set, and the animation ends by returning to the original set.

What most people do not know is the simple description of the Mandlebrot Set, M , is described recursively as follows:

Let c \in \mathbb{C} , and let x be a complex variable. Define f_c(x) = x^2 + c. Now let

x_1 = f_c(c) = c^2+c

x_2= f_c(x_1) = x_1^2+c


x_{n+1} = f_c(x_n) = x_n^2+c

If \{x_n\}_{n \in \mathbb{N}} is bounded (in complex modulus), c \in M.

That’s it. One simple equation applied recursively, and one simple criterion give a complete characterization of every point in the Mandelbrot Set.

Now, way back in the day (the day being circa 1883), Georg Cantor was advancing the Continuum Hypothesis, which, in terms of cardinals, \aleph_0 and \aleph_1, it says that there is no set with cardinality between that of the natural numbers and that of the real numbers, i.e. there is no set S

{\aleph_0} < |S| < 2^{\aleph_0} = \aleph_1.

Georg Cantor, in his attempts to find such an S, developed the proverbial mathematical fractal (it is the most famous in mathematics because it is simple to describe and has very interesting properties) known as the Cantor Set.

Visually, the Cantor set has a simple appearance defined via iteration


Where the n^{th} iteration of the Cantor Set is the n^{th} layer from the top, e.g. the 1^{st} iteration is the solid black bar, and the 2^{nd} iteration is the two black pieces in the second row.

The Cantor Set has a simple mathematical description. Let C_1 := [0,1]. Remove the middle third of the interval to get C_2 = [0,\frac{1}{3}] \cup [\frac{2}{3},1], and repeat this process indefinitely. For example, C_3 = [0,\frac{1}{9}] \cup [\frac{2}{9},\frac{1}{3}] \cup [\frac{2}{3},\frac{7}{9}] \cup [\frac{8}{9},1].

Define the Cantor Set, C to be

C = \bigcap \limits_{n=1}^{\infty} C_n.

Another description of the Cantor set is in terms of base 3 expansion; namely, that the Cantor Set is precisely them points in [0,1] with base 3 expansion containing no 1’s. A moment’s thought proves this to be so. Take some x \in [0,1] and compute it’s base 3 expansion, of the form x = .a_1a_2a_3 \ldots . If a_1 = 0 , the point is in the first third of interval, if a_1 = 1, the point is in the second third of the interval, and if a_1 = 2 , the point is in the last third of the interval. Note this is precisely what “base 3 expansion” means. Now note if a_1 = 1, x \notin C , by construction. Repeat the same process for a_2 , and it is clear that if a_k = 1 for any k x \notin C .

Ergo, we have the description

C = \{x  \in [0,1] : a_k \in \{0,2\} \, \, \text{for all} \, \, k \} .

Analogous to the Cantor Set, in two dimensions we have the Sierpinski Carpet


with final limit


Other examples include the Sierpinski Gasket


or the Koch Snowflake


If one refers to the original list of proposed fractal criterion, it is rather clear that these objects are all rather strange, yet seem to share some basic characteristics. The issue with which we will begin our study of Fractal Geometry concerns two simple questions:

How does one assign a size, or measure, to a fractal object? 

Are fractals truly n dimensional objects for n \in \mathbb{N}?

The answers to the questions are by no means easy, and will require machinery from analysis, topology, measure theory, and complex analysis, amongst other areas.

II. Size is Relative to Dimension

How long is a square? How much volume does a line have? What is the area of a sphere?

You may respond: “that’s absurd! those aren’t feasible questions to ask!”. Well this is true, but also not true.

  1. How long is a square? In some sense, this is a logically ill-worded question, in other senses, it is infinitely “long”.
  2. How much volume does a line have? In this case it seems reasonable to say it has 0 volume.
  3. What is the area of a sphere? huh? Surface area? 4 \pi r^2. But do we really mean the area of the boundary? No, we mean the area of the whole sphere. Thus we should again, probably, say that it has infinite area. 

These examples illustrate a basic measure-theoretic principle: size is relative to the dimension of the measure we are taking. In order to specify the size of something, we should know two things (i) the “dimension” of the object we are measuring, (ii) the “dimension” of the measure we are using. In the example of the length of a square, we are applying a one dimensional measure to a two dimensional object. To get a better indication of size, a good rule of thumb is to apply a r-dimensional measure to an r-dimensional object for r \in \mathbb{R}.

Wait. Did you just say r \in \mathbb{R} instead of r \in \mathbb{N}? Is that a typo?

No, no it’s not 😉 . And that lack of typo is precisely the issue of fractal geometry.



III. Box Dimension

Let us return to the archetypal fractals previously mentioned, and see why taking a standard measure of “length” or “area” or “lebesgue measure”, simply will not do.

Let C denote the Cantor Set. Note C is uncountable, compact, and perfect. What is the 1-D Lebesgue measure, m of C? Well since the measure of closed sets is often difficult to compute, we use the following elementary property, given A \subset X, both Lebesgue measurable,

m(X \setminus A) = m(X) - m(A).

Observe that at iteration n , we remove 2^n open intervals of measure 3^{-n}. Ergo, the measure of the complement of the Cantor Set is

m([0,1] \setminus C) = \sum \limits_{k=1}^{\infty} (\frac{2}{3})^k = 1 ,

so we conclude m(C) = 0.

Using a similar argument, we see that the 2-D Lebesgue Measure of the Sierpinski Carpet is 0, and the 1-D measure is\infty . We can also calculate the perimeter of the Koch Snowflake to be \infty .

These calculations suggest that the measure we are taking does not have the same dimension as the objects involved, since all three of the above objects are compact subsets of \mathbb{R}^n and should feasibly have some measure.

The first attempt to remedy this issue is the box dimension. Let A : = [0,1] \times [0,1]. Let C_\epsilon (x_i) be a closed box centered at x_i (ball under the sup norm) of side length \epsilon. Let N_\epsilon(A) be the smallest number of boxes required to cover A , i.e.

N_{\epsilon}(A) := \inf \# \{M : A \subset  \bigcup \limits_{i=1}^{M} C_\epsilon (x_i) \, \text{with} \, x_i \in \mathbb{R} \} .

For the example set A, let us compute a few values of N_{\epsilon}(A) .

N_{\frac{1}{2}}(A) = 4 = (\frac{1}{2})^{-2}

N_{\frac{1}{3}}(A) = 9 = (\frac{1}{3})^{-2}

N_{\frac{1}{1000}}(A) = 1,000,000 = (\frac{1}{1000})^{-2}

The pattern may be clear by now. Regardless of how N_{\epsilon} (A) changes, it is always related to \epsilon by

N_{\epsilon} (A) = \epsilon^{-D} ,

where D seems to be the dimension of A . Ergo, at least for the set A , it seems fruitful to define the box dimension \text{dim}_{B}(A) as

\text{dim}_{B}(A) = \frac{\log(N_\epsilon)}{-\log(\epsilon)} =2.

This leads to the formal definition of the box dimension. We define the upper box dimension of a set A as

\text{dim}^{*}_{B}(A) = \limsup \limits_{\epsilon \to 0^{+}} \frac{\log(N_\epsilon)}{-\log(\epsilon)}

and the lower box dimension of a set A as

\text{dim}_{*B}(A) = \liminf \limits_{\epsilon \to 0^{+}} \frac{\log(N_\epsilon)}{-\log(\epsilon)}

if \text{dim}^{*}_{B}(A) = \text{dim}_{*B}(A) , the limit exists and we define the box dimension box dimension of a set A as

\text{dim}_{B}(A) = \text{dim}^{*}_{B}(A) = \text{dim}_{*B}(A)

or an equivalent formulation, if you’re familiar with Big O notation,

\text{dim}_{B}(A) = \inf \{ \alpha \geq 0 : N_\epsilon(A) = O(\epsilon^{-\alpha}) \, \text{as} \, \epsilon \to 0^{+} \},

essentially we look to compare the growth of the counting function in regards to \epsilon. Here are some properties of the box dimension, which I present without proof except for a few sketches , but most are straightforward to prove:

  1. If E \subset A \text{dim}_{B}(E) \leq \text{dim}_{B}(A) .
  2. If E \subset \mathbb{R}^m, \text{dim}_{B}(E) = \text{dim}_{B}(\overline{E}) .
  3. If U \subset \mathbb{R}^m is open and bounded, \text{dim}_{B}(A) = m.
  4. If E \subset [0,1], 0 \leq \text{dim}_{B}(E) \leq 1.
  5. 0 < \text{dim}_{B}(C) = log_{3}(2) < 1 .
  6. 1<  \text{dim}_{B}(\text{Sierpinski Carpet}) = log_{3}(8) <2
  7. If F is a finite set, \text{dim}_{B}(F) = 0 .

Proof ideas: (1.) N_\epsilon(E) \leq N_{\epsilon}(A) for all \epsilon. (2.) When covered with closed boxes, the boundary of an open set is already contained. Suppose not, then every box is strictly in the interior. The complement of a box in the set is open, and thus since, we have a finite number of boxes (if we had an infinite number it is clear the dimension of the set and its closure are equal), the union must not be a covering. (3.) Clearly \text{dim}_{B} \geq m since it contains a box, which has constant counting function. Since it is contained in a compact box, by monotonicity, we have \text{dim}_{B} \leq m. The details and the rest are left as exercises.

You may have already discovered a few issues with this notion of dimension using the above properties. For example, the box dimension has huge issues with dense sets that are otherwise small; we may easily calculate that \text{dim}_{B}(\mathbb{Q} \cap [0,1]) = 1 . By (5.), the Cantor Set has dimension log_{3}(2). Thus, we have an uncountable set that has smaller dimension than a countable set! That’s not a good sign. Moreover, this dimension is clearly not given by any measure, since if it were, it would be countably additive, but \mathbb{Q} is the countable union of singletons, each with dimension 0, while \text{dim}_{B}(\mathbb{Q} \cap [0,1]) = 1 .


As this post is getting a tad lengthy, I’ll end it here. The next post will begin with some ideas for better notions of the dimension of a fractal, and discuss the most important concept of Iterated Function Systems. As indicated by (5.) and (6.), it seems we should construct a measure of the Cantor Set and Sierpinski Carpet that is of their respective, non-integer, dimensions. In the next post, we will do precisely this, by introducing the Hausdorff Measure.




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