A slope just off

With the slope at 2/5, drag “Turns round the hole”: on the fifth turn the line meets its starting point exactly, the whole line turns gold, and the corner reads “closed after 5 turns”. Move the slope to 0.4003 and the first five turns look the same; the fifth misses by only 0.0015, yet all the way to turn 89 no later turn comes any closer, let alone closes. The corner compares the line's position after each complete turn and keeps the closest return so far: the q in (p, q) is the turn, p is the integer nearest to qk, and “Missed by” is how far qk is from p, counting one turn round the tube as 1; p/q is the fraction that return stands for. Every value on the slider is a terminating decimal, which is a fraction, so the line does close in the end: 0.4003 is 4003/10000, and it takes ten thousand turns. Choose “One over the golden ratio” and the slope becomes a true irrational, 0.618…: a new closest return comes on turns 1, 2, 3, 5, 8, 13, 21, 34, 55 and 89, nearer each time and never exactly back.