• I draw reduced fractions between \(-1\) and \(1\), including the endpoints. The q slider sets the largest denominator. At 1, we get circles for \(-1\), \(0\), and \(1\). At 0, I hide the circles, as a zero denominator would be undefined.
• Each circle gets a copy on the other side of its number line with the same radius.
• The second slider draws \(N\) lines through the screen’s center, spaced \(\frac{180^\circ}{N}\) apart.

Circles on the same side of one line still never overlap, but copies from different lines can (and do!) overlap. The pattern scales with the screen and uses faint strokes so it can sit behind the text.