Ternary Variations

Georg Cantor (1845-1918) was led, through his study of trigonometric series, to distinguish between denumerably infinite sets like the rationals and uncountable sets like the reals. He introduced a set that is an abstract form of what we now call Cantor’s Ternary Set. In fact, the ternary set had been studied some ten years earlier by the Dublin-born mathematician Henry Smith and, independently, by the Italian Vito Volterra. General sets of this form are now called Smith-Volterra-Cantor sets (SVC sets).
Construction of the Ternary Set

To construct the ternary set we proceed iteratively. Starting with the unit interval {C_0 = [0,1]}, we remove the open interval {I_1 = (\frac{1}{3},\frac{2}{3})} corresponding to the “middle third” to get {C_1 = [0,\frac{1}{3}]\cup[\frac{2}{3},1]}. Next, we remove the open middle third of each remaining interval, {I_2 = (\frac{1}{9},\frac{2}{9})\cup(\frac{7}{9},\frac{8}{9})} to get {C_2}, a union of four closed intervals. Continuing this process, we arrive ultimately at the ternary set

\displaystyle C = \bigcap_{n=0}^{\infty} C_n = [0,1] - \bigcup_{n=1}^{\infty} I_n \,.

The initial stages of the construction are shown in the figure below.

The first seven stages in constructing the Cantor set.

The first seven stages in constructing the Cantor ternary set.

Properties of {C}

  1. The set {C} is “large“: it is uncountable, making it large compared to the rational numbers.
  2. The set {C} is “small“: it cannot have a positive length: at each stage, the length is decreased by a factor {\frac{2}{3}} and the limit of {(2/3)^n} is zero.
  3. {C} is self-similar: If we scale it up by a factor of 3, we obtain two pieces, each identical to {C} itself.
  4. {C} is fractal, with a fractal dimension of {(\log 2 /\log 3 )}
  5. {C} is a perfect set: For every point in {C}, there are other points arbitrarily close to it; there are no isolated points.
  6. {C} is nowhere dense: The interior of the closure is empty.
  7. {C} is totally disconnected.

We will not prove these properties, which are demonstrated in many standard texts on point set topology. But a few remarks are apposite.

(1) Length: The length removed at stage {n} is {2^{n-1}/3^n}. Summing these, the total length rermoved is 1. This implies that the remaining length is 0. Technically, the set {C} is of Lebesgue measure zero.

(2) Size: for any point {x} in {C}, we construct a binary number as follows: if {x} is to the left of a middle third removed at stage {n} the {n}th digit is 0. If to the right, the {n}th digit is 1. Clearly this gives a one-to-one correspondence between {C} and all binary numbers in {[0,1]} so {C} must be uncountable.

(3) Self-similar: Scaling by 3 maps all numbers in the interval {[0,\frac{1}{3}]} to {[0,1]} in such a way that the original set {C} is reproduced. The elements of {C} in {[\frac{2}{3},1]} give a copy of {C} shifted to the interval {[2,3]}.

(4) Fractal: In coordinate geometry, if all axes are scaled by a factor {S}, the length of a line segment is increased by {S^1}, the area of a square by {S^2}, the volume of a cube by {S^3}, etc. In general, a {D}-dimensional set scales as {S^D}. But we have seen that scaling {C} by {S = 3} doubles the set. So {3^D = 2} or {D = (\log 2 /\log 3 ) \approx 0.631}.

For discussion of (5), (6) and (7) see topology texts.

Generalization

Suppose {0 < r < 1}. Starting with {[0,1]}, we remove from the centre of each component an open interval that is a fraction {r} of its length,leaving two closed intervals of length {s=(1-r)/2}. Iterating this process, the length remaining at stage {n} is {(1-r)^n}. This tends to zero. The result is a ternary set that, expanded by a factor {1/s = 2/(1-r)} yields two copies of itself. So

\displaystyle \left( \frac{1}{s} \right)^{\!\!\!^D} = 2 \qquad\mbox{or}\qquad D = \left( \frac{\log 2}{\log(1/s)} \right) \,.

By choosing {s} correctly, we can obtain a set of any fractal dimension between 0 and 1.

The choice {r = 4/5} or {s = 1/10} is interesting: the first stage removes all numbers except those whose decimal expansion begins with 0 or 9. (A technicality: numbers with terminating decimals are assumed to be represented by infinite expansions, e.g. 0.1 = 0.0999…). At the second stage, only numbers beginning with .00, .09, .90 or .99 remain. Ultimately, we obtain all numbers with decimal expansions containing only zeros and nines. The fractal dimension of this set is {(\log 2 /\log 10 ) =\log_{10}2=0.3010}.

Fat Fractal. The SVC set.

So far, all the ternary sets were of measure zero. It is possible, and quite easy, to modify the construction procedure so that the resulting set {S} is of positive measure. For example, at stage 1 we remove 1/4 from the middle. At stage 2 we remove 1/16 (rather than 1/8). At stage {n} we remove ({1/2^{2n}}) from each interval. The total length removed is

\displaystyle \frac{1}{4} + \frac{2}{16} + \frac{4}{64} + \dots = \frac{1}{4}\left( 1 + \frac{1}{2} + \frac{1}{4} + \dots\right) = \frac{1}{2}

and the length remaining is also 1/2. Such a set, with positive measure, is called a fat fractal. The set {S} is a particular example of a Smith-Volterra-Cantor or SVC set.

The first six stages in constructing the Smith-Volterra-Cantor set.

The first six stages in constructing the Smith-Volterra-Cantor set.

Volterra used such a set to define a function {V(x)} with a remarkable property: {V(x)} is differentialbe everywhere on {[0,1]} but its derivative, although bounded, is not (Riemann) integrable. This defies the Fundamental Theorem of Calculus. As a consequence of results like this, a completely new method of integration, Lebesgue integration, was developed.

But that is another story, to which we shall return.


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