The set of the previous section is an example of a Cantor set. Often, a Cantor set is described as the result of an iterative process, much in the same way we described the Koch curve in §2. We'll describe the most common Cantor set, the ``middle-thirds Cantor set'', this way now.

Begin with the interval [0, 1] as the set . Now remove the open interval (,) from to obtain

It may not be clear at first that even exists. It seems we are taking just about everything out when we remove the intervals. Indeed, the total length of intervals we remove is

+ + + +...= 1.

That is, the total length of the segments removed is equal to the segment we
started with! For this reason, a set like is sometimes called a
``Cantor dust''. But notice that the endpoints of each interval of
must be in , because they never get removed at any
level. Thus, is nonempty, since it contains at least the points
0, 1,,,,,,,,,.... And
it contains quite a few more points, as well.
To see that, we can view in a manner similar to the
description of the set given in
§5.2. However, is more naturally
described in base 3, rather than base 10. Let's briefly review what
we mean by this.

When we express a number *x* in base 10, we are expressing it as sums of
powers of ten. Thus, when we write
*e*^{5} 148.413, we are actually saying that

Returning to the Cantor set , note that when we remove the
interval
(1/3, 2/3), we are removing all the numbers whose ternary
expansion has a 1 immediately after the decimal place.^{5.17}At the next stage, we remove those points which have a 1 in the
second place after the decimal, and so on. Thus, in the limit, we
obtain the following alternate description of :

What is its box dimension? The description by the sets
give us an effective way to calculate the limit: at the
*n*^{th} stage, we have 2^{n} intervals of length 3^{-n}.
Using this, we can immediately calculate that

= = 0.63093.

You may have noticed the strong similarity in the construction of the
Koch curve of §2 and the middle-thirds
Cantor set . In fact, notice that the intersection of
with its base (the initial segment ) is exactly
the middle-thirds Cantor set.

- ... human
^{5.14} - Some Native American peoples used base 20, and the Babylonians were very fond of base 12 and base 60 (consider how we measure time).
- ... this
^{5.15} - Maple will convert integers to other bases, using a command such as convert(148, base, 3). One has to work a little bit to convert fractions-- can you think of a way to do this?
- ... binary.
^{5.16} - Hexadecimal is convenient for use with a binary system, because we can group blocks of 4 binary digits to obtain one hexadecimal digit (by convention, the letters A-F are used to represent the integers 10-15). This is quite handy in computer applications, because a single byte is represented by 8 binary digits (bits), or two hex digits. Before 8-bit bytes became standard, octal (base 8) was very common as well. Octal is much less convenient for 8-bit quantities, however, because 8 bits don't break up nicely into groups of 3 bits.
- ... place.
^{5.17} - As in decimal expansions, some points have two
representations. Thus, the point 1/3 can be written either
as .1
_{3}or as .0222.... As in the construction of in §5.2, we keep such points in our set.

2002-08-29