Farey Fraction Pattern Graph
If you haven't studied Farey Fractions before, it's a good idea to start with this applet http://www.geogebratube.org/student/m12717 to get a sense of what they are. At each y value, the applet plots all the Farey fractions with y as denominator. For example, when y=12, the Farey fractions plotted are at x= 1/12, 5/12, 7/12, and 11/12, whereas when y=13, all fractions x=1/13, 2/13, ... , 11/13, 12/13 are plotted. Move the slider to change the maximum value of the denominator -- you'll see some pretty patterns. Check the box and move the new slider to see some curves. Try inputting some curves of your own (be sure to simplify so there are no y's in the denominator of what you enter, and also make sure not to write xy -- use x*y or x y).
What patterns do you notice? As you move the slider to show various curves, why do some curves have more points on them than others? What are the equations for some of the other curves you see? Try inputting them (see note above on format).