A cycle expresses the idea behind the wheel - something that repeats over and over again. Pick a single point on a turning wheel. It will move and trace out a curve. To see this go to: sine, and click on "spin". If you want to learn more about the curve (the sine wave or sinusoid) go to Sine.

A question about a cycle in math has bothered people for quite a while. It involves taking a whole number (an integer) and cutting it in half if it is even, tripling it and adding one if it is odd. The next paragraphs say this and more.

Lagarias, J., American Mathematical Monthly 92 (1985), 3-23 provides a comprehensive review of "the 3x+1 problem and all its aliases". In his words (n an integer):

The conjecture is that the sequence n, f(n), f(f(n)), ... is ultimately periodic for all n and such that there is only one final cycle 1 -> 4 -> 2 -> 1.

In sum, take an integer. If it is odd, triple it and add one. If it is even, divide by two. Repeat this using the result as the new start.

Is there any integer other than one for which the process cycles indefinitely?

Some things that are known are in More on Collatz.