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Third Base

Brian Hayes

Turning to Ternary Dust

Although numbers are the same in all bases, some properties of numbers show through most clearly in certain representations. For example, you can see at a glance whether a binary number is even or odd: Just look at the last digit. Ternary also distinguishes between even and odd, but the signal is subtler: A ternary numeral represents an even number if the numeral has an even number of 1s. (The reason is easy to see when you count powers of 3, which are invariably odd.)

Figure 3. Ternary structure . . .Click to Enlarge Image

More than 20 years ago, Paul Erdo"s and Ronald L. Graham published a conjecture about the ternary representation of powers of 2. They observed that 22 and 28 can be written in ternary without any 2s (the ternary numerals are 11 and 100111 respectively). But every other positive power of 2 seems to have at least one 2 in its ternary expansion; in other words, no other power of 2 is a simple sum of powers of 3. Ilan Vardi of the Institut des hautes études scientifiques has searched up to 26973568802 without finding a counterexample, but the conjecture remains open.

The digits of ternary numerals can also help illuminate a peculiar mathematical object called the Cantor set, or Cantor's dust. To construct this set, draw a line segment and erase the middle third; then turn to each of the resulting shorter segments and remove the middle third of those also, and continue in the same way. After infinitely many middle thirds have been erased, does anything remain? One way to answer this question is to label the points of the original line as ternary numbers between 0 and 0.222.... (The repeating ternary fraction 0.222... is exactly equal to 1.0.) Given this labeling, the first middle third to be erased consists of those points with coordinates between 0.1 and 0.122..., or in other words all coordinates with a 1 in the first position after the radix point. Likewise the second round of erasures eliminates all points with a 1 in the second position after the radix point. The pattern continues, and the limiting set consists of points that have no 1s anywhere in their ternary representation. In the end, almost all the points have been wiped out, and yet an infinity of points remain. No two points are connected by a continuous line, but every point has neighbors arbitrarily close at hand. It's hard to form a mental image of such an infinitely perforated object, but the ternary description is straightforward.

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