A spirograph is generated by one circle rolling inside another. The pen traces a hypotrochoid — a curve defined by three numbers: the radius of the fixed outer circle R, the radius of the rolling inner circle r, and the distance of the pen from the center of the rolling circle d.
The equation.
x(t) = (R−r)cos(t) + d·cos((R−r)t/r)
y(t) = (R−r)sin(t) − d·sin((R−r)t/r)
One equation. Every path this exhibit can draw comes from adjusting three numbers inside it. The circle, the star, the rosette, the never-closing dense fill — all the same family.
Whether your path closes depends on whether R/r is rational.
If R/r reduces to a fraction p/q in lowest terms, the curve closes after exactly q full revolutions and traces p loops. If R/r is irrational, the curve never closes — it fills the annulus asymptotically, approaching every point without ever returning to any of them. You cannot determine which kind of path you are on by looking at your current position. You have to complete the traversal.
Closure sounds the same regardless of how long it took.
A path that closes in 3 revolutions and one that closes in 47 both arrive at the same sonic resolution — the same chord, the same landing. Only the time between departure and arrival differs. The chord does not know how long you were gone.
The irrational paths do not fail.
They are not incomplete. They are a different kind of complete — one that doesn't fold back, that keeps finding new territory, that never mistakes where it has been for where it is going. Both kinds are generated by the same equation. Both are valid traversals. The difference is not better or worse. It is rational or open.