Cantor function

         We have already discussed washing machine as a function, which is specified by “to clean,” rather than by a math equation.   We present here another such word function.  It is the Cantor function, named after the German mathematician Georg Cantor (1845-1918), which is described by the function diagram of figure 7.

Figure 7.  Cantor function

Figure 8.  Cantor rod

After cutting it into three pieces of ¹/₃meter long each, we throw away the middle rod piece and keep only the two end rod pieces for the first iteration. Now, we apply the Cantor function to each of the 2 end rod pieces of ¹/₃ meter. What we get is 4 rod pieces of length ¹/₉ (= ¹/₃ x ¹/₃) meter after the second iteration, two from each of the two rod pieces of ¹/₃meter of the first iteration. After the third iteration, there are 8 rod pieces of ¹/₂₇ (= ¹/₃ x ¹/₃ x ¹/₃) meter each, as shown in figure 8. It is evident that much of the initial rod gets thrown away as we apply the Cantor function to the remaining rod pieces. This is seen in figure 9; total length of the remaining rod pieces decreases as the Cantor iteration is applied.

Figure 9. Total length of Cantor rods

We summarize in table 4 the number of remaining rod pieces and its total length after the first three Cantor iterations. Following the trends of the first three iterations, it is not hard to guess results of the tenth iteration, at which there are 1024 rod pieces but the total length is only 0.0173 meter. After iterating the Cantor function (infinitely) many times, the rod is being cut up to (infinitely) many thin slices, yet each slice becomes so thin that the total length of them all decreases to zero. For a long time, the Canter set has remained as a weird idea in mathematics; how can infinitely many things add up to almost nothing? But we will see that the eerie pattern of Canter set turns up quite unexpectedly in fractal geometry and chaos.

    
Table 4.  Cantor rod