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 , we remove the open interval corresponding to the “middle third” to get . Next, we remove the open middle third of each remaining interval, to get , a union of four closed intervals. Continuing this process, we arrive ultimately at the ternary set
The initial stages of the construction are shown in the figure below.
- The set is “large“: it is uncountable, making it large compared to the rational numbers.
- The set is “small“: it cannot have a positive length: at each stage, the length is decreased by a factor and the limit of is zero.
- is self-similar: If we scale it up by a factor of 3, we obtain two pieces, each identical to itself.
- is fractal, with a fractal dimension of
- is a perfect set: For every point in , there are other points arbitrarily close to it; there are no isolated points.
- is nowhere dense: The interior of the closure is empty.
- 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 is . Summing these, the total length rermoved is 1. This implies that the remaining length is 0. Technically, the set is of Lebesgue measure zero.
(2) Size: for any point in , we construct a binary number as follows: if is to the left of a middle third removed at stage the th digit is 0. If to the right, the th digit is 1. Clearly this gives a one-to-one correspondence between and all binary numbers in so must be uncountable.
(3) Self-similar: Scaling by 3 maps all numbers in the interval to in such a way that the original set is reproduced. The elements of in give a copy of shifted to the interval .
(4) Fractal: In coordinate geometry, if all axes are scaled by a factor , the length of a line segment is increased by , the area of a square by , the volume of a cube by , etc. In general, a -dimensional set scales as . But we have seen that scaling by doubles the set. So or .
For discussion of (5), (6) and (7) see topology texts.
Suppose . Starting with , we remove from the centre of each component an open interval that is a fraction of its length,leaving two closed intervals of length . Iterating this process, the length remaining at stage is . This tends to zero. The result is a ternary set that, expanded by a factor yields two copies of itself. So
By choosing correctly, we can obtain a set of any fractal dimension between 0 and 1.
The choice or 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 .
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 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 we remove () from each interval. The total length removed is
and the length remaining is also 1/2. Such a set, with positive measure, is called a fat fractal. The set is a particular example of a Smith-Volterra-Cantor or SVC set.
Volterra used such a set to define a function with a remarkable property: is differentialbe everywhere on 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.