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In view of the stability consideration, we propose here an alternate analogy for the periodic decimal digits. Consider a super ball that bounces for a long time without ever losing its height, after it is dropped on a hard floor. It is, in fact, so perfectly elastic that not much energy is being lost while bouncing off the floor. Suppose a super ball is dropped onto the center hollow of a concave floor. It will then bounce along the vertical path at the center hollow, and thereby tracing a steady trajectory of period 1, as illustrated in the right frame of figure 6. Any path deviating from the vertical will eventually get shoved into the vertical path by the reason of stability.
Figure 6. Stable fixed point of period 1
We next consider two super balls bouncing on a doubly hollowed floor in figure 7. Then, a continuous snapshot of two super balls reaching different heights but with the same individual period would give rise to a stable composite trajectory of period 2. In principle, we can then visualize a stable trajectory of period n, which is made up of n super balls bouncing simultaneously on a multiply concaved floor.
Figure 7. Stable fixed point of period 2
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