The circle [i]A[/i] rolls around the fixed circle B on the outside. The ratio between the radii of A and B is given by the whole number e, either [math]e=\frac{B}{A}[/math] or [math]e=\frac{A}{B}[/math].[br][i][b]A[/b][/i] completes a [b]small cycle[/b] when it rotates a full turn about its center, and completes a [b]large cycle[/b] when it returns to its starting configuration (a configuration is the set of positions of all points of [i]A[/i]).[br][br]Question: How many small cycles in a large cycle? [br]Hint: Check the "Show Markers" checkbox to see the result [math]n=\cdots[/math].[br][br]Extensions: When the ratio [i]e[/i] is extended to non-integer, the problem becomes more complicated. Let [i]N[/i] be the number of small cycles in a large cycle, called "the size of large cycle". Let's derive [i]N[/i] from [i]e[/i], fist in case [i]e[/i] being rational, then irrational.