Cantor Set and Fractals

Question

I have read that the Cantor set is considered a fractal. I am referring to the Cantor set in which the middle third of a real line is removed recursively. I see that this is recursively defined, but the other "visual" or "intuitive" fractal properties are not apparent (if they exist). For example, I like to think of a fractal as some geometrical object with scale invariance such that if you "zoom in" on the object it will look the same/similar.

Maybe because the Cantor set has a lower dimension perhaps it is not woven into my brain as a classic fractal such as the Koch Snowflake or the Sierpinski triangle. I do see that Cantor's set would look the same if you kept "zooming in" but I do not see the irregular aspects or the complexity that is usually inherent with fractals.

Perhaps some formal definitions would clear up my confusion. Is Wikipedia's definition of fractal the standard? It isn't even clear what is the definition of fractal.

1. Where fractal properties does Cantor's set have?
2. What is the argument that Cantor's set is a fractal?

Answer 1

If you restrict your sight to $[0,\frac{1}{3}]$ then the picture of Cantor's set is exatly the same as it is in the whole $[0,1]$. Again, restrict to $[0,1/9]$ and you get the same picture. I think this is the main property of a fractal: a picture which repeats itself.

If you want to see another approach, then Cantor's function may answer your question: http://en.wikipedia.org/wiki/Cantor_function

The Cantor function is closely related to the Cantor set. The Cantor set C can be defined as the set of those numbers in the interval $ [0,1] $that do not contain the digit 1 in their base-3 (triadic) expansion, except if the 1 is followed by zeros only (in which case the tail $1000\ldots$ can be replaced by $0222\ldots$ to get rid of any 1). It turns out that the Cantor set is a fractal with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume). Only the D-dimensional volume $H_D$ (in the sense of a Hausdorff-measure) takes a finite value, where $D =\log(2)/\log(3)$ is the fractal dimension of C. We may define the Cantor function alternatively as the D-dimensional volume of sections of the Cantor set $$f(x)=H_D(C \cap (0,x))$$

Answer 2

The Cantor set has a very special property. It is the unique, up to a homeomorphism, compact metric space without isolated points which is zero dimensional.

One consequence of this property is that every uncountable compact subset of the Cantor set is homeomorphic to the Cantor set.

While homeomorphism doesn't tell us much about self similarity like we like to think about in the case of the Koch snowflake, it does mean that every uncountable compact subset, if we "zoom in a little bit" would look like the Cantor set itself.

I'd think that this make the Cantor set self-similar enough to be considered a fractal.

Answer 3

You ask: "Is Wikipedia's definition of fractal the standard?" and right near the top of Wikipedia's page of fractals, we see the following definition:

A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers.

The statement that the fractal dimension may "fall between the integers" really adds nothing but, other than that, I would say that this is fairly standard; it is unquestionably the definition that was put forward by Mandelbrot around 1975 when he coined the term "fractal". He did not refer to "fractal dimension" at that time but, rather, the "Hausdorff-Besicovitch dimension" as he put it. In fairness, the usefulness of this definition has been debated with even Mandelbrot himself feeling that it might not be inclusive enough. Nonetheless, this comparison of dimension is central in fractal geometry. Gerald Edgar calls his great book, Measure, Topology, and Fractal Geometry, a meditation on the definition.

Taking this to be the definition, we can definitely say that the Cantor set satisfies it. If by "fractal dimension" you mean similarity dimension, then the Cantor set has fractal dimension $\log(2)/\log(3)$, since it's composed of two copies of itself scaled by the factor three. Also, the set is regular enough that any reasonable definition of fractal dimension agrees with that computation. (Well, any real-valued defintion.)

Topological dimension is a trickier thing, actually. It's inductive in nature. Totally disconnected sets (like single points, finite sets, or notably the Cantor set) have dimension zero. Higher dimensions are defined in terms of lower dimensions. The space we live in is three dimensions because balls in this space have a surface that is two dimensional. Because of this inductive nature, topological dimension always yields an integer.

When you write that you "do not see the irregular aspects or the complexity that is usually inherent with fractals", I think you might have a bit of a mis-understanding about fractal geometry. The Cantor set is indeed regular but, then so are all the strictly self-similar sets studied in classical fractal geometry - the Koch curve, the Sierpinski triangle, the Menger sponge, and countless others all display this regularity. Indeed, it's exactly this regularity that allows us to understand them.

To emphasize this regularity, and how it appears in not just the Cantor set, compare the following zooms of

The Cantor set

The Koch curve

Now, of course, there are "irregular" fractals - or, at least, less regular fractals. Examples include random version of self-similar sets, examples that arise from number theory, and examples arising from complex dynamics (like Julia sets). It's not their irregularity that makes these objects fractal, however. On the contrary, its the regularity that we can find that allows us to analyse these objects to the point where we can characterize them as fractal. Of course, this analysis is bit harder with these less regular examples.