Periodic trajectory

Let us imagine that a volleyball player can hit a ball to prescribed heights without missing a beat. Figure 1 shows the trajectory (path) of a ball reaching the height of 3, say, meters after each hit, and thus representing the decimal digits of ¹/₃ = 0.333… This is period 1, for the ball returns to the same height after each hit.

Figure 1. Period 1

Now, figure 2 depicts the alternating trajectory of a ball reaching the heights "6" and "3" of the decimal expression ⁷/₁₁ = 0.636363… It is period 2.

Figure 2. Period 2

For the decimal expression of ³/₇ = , figure 3 shows the trajectory of repeating digits (4, 2, 8, 5, 7, 1), and hence period 6.

Figure 3. Period 6

Although the volley ball analogy is helpful to visualize trajectory of a certain period, one wonders if it is possible for any one to hit a ball so precisely over and over. This is because what lurks in our mind is the question of stability to be discussed presently.

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